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Mathematical Programming, Series B manuscript No.(will be inserted by the editor)

Mixed-Integer Optimal Control under Minimum Dwell Time Constraints

Clemens Zeile1 · Nicolò Robuschi2 · Sebastian Sager1

Received: date / Accepted: date

Abstract Tailored Mixed-Integer Optimal Control policies for real-world applications usually have to avoidvery short successive changes of the active integer control. Minimum dwell time (MDT) constraints expressthis requirement and can be included into the combinatorial integral approximation (CIA) decomposition,which solves mixed-integer optimal control problems (MIOCPs) by solving one continuous nonlinear pro-gram (NLP) and one mixed-integer linear program (MILP). Within this work, we analyze the integrality gapof MIOCPs under MDT constraints by providing tight upper bounds on the MILP subproblem. We suggestdifferent rounding schemes for constructing MDT feasible control solutions, e.g., we propose a modificationof sum-up rounding (SUR). A numerical study supplements the theoretical results and compares objectivevalues of integer solutions with a priori lower bounds.

Keywords Mixed-integer linear programming · optimal control · discrete approximations · switcheddynamic systems · approximation methods and heuristics · minimum dwell time constraints

Mathematics Subject Classification (2010) 49J15 · 49M25 · 49M27 · 90C11 · 93C30

1 Introduction

Optimal Control Problems with integer control choices have been investigated in many research articlesduring the last few decades, since they naturally arise in a range of applications. In order to apply the receivedcontrol policy in practice, additional switching constraints are usually needed, such as minimum dwell timerequirements that describe the necessity of activating a integer control for at least a given minimal durationif at all and analogously for deactivation periods. Recent case studies of MIOCPs with MDT considerationscan be found, e.g., for pesticide scheduling in agriculture [2], electric transmission lines [15], solar thermalclimate systems [8] and hybrid electric vehicles [28]. MDT constraints attracted also a lot of attention aspart of MILPs, see [27] for a study of unit-commitment problems and [23] for a corresponding polytopeinvestigation.

One common approach to tackle MIOCPs under MDT constraints is to consider two separate levels ofoptimization. At the upper level, a mode insertion gradient is usually evaluated in order to fix a sequence ofactive system modes with promising cost function value. At the lower level, the algorithm aims at minimiz-ing the cost functional with respect to the switching times and continuous control input, if available. Suchapproaches can be found in [37,11,4,1,13] and are usually referred to as transition-time optimization or vari-able time transformation method. We remark such switched systems have also been intensively investigated

Clemens [email protected]

Nicolò [email protected]

Sebastian [email protected]

1 Department of Mathematics, Otto-von-Guericke Universität Magdeburg, 39106 Magdeburg, Germany2 Department of Mechanical Engineering, Politecnico di Milano, Via Giuseppe La Masa 1, 20156 Milano, Italy

2 Clemens Zeile, Nicolò Robuschi, Sebastian Sager

with respect to stability under average dwell time, see [18]. Another approach to include MDT constraintsinto MIOCPs is to apply dynamic programming, see [9], which is, however, computationally expensive.

Similar to the two-stage optimization method, the CIA decomposition [29,33], also known as embeddingtransformation technique [5], is based on the idea to solve the problem in several optimization steps. The firstidea is to use the partial outer convexification method that allows us to reformulate a problem with integercontrols into an equivalent problem with binary controls. Second, discretizing the MIOCP in the spirit offirst-discretize-then-optimize methods with, e.g., Multiple Shooting [6] or Direct Collocation [34] results inan mixed-integer nonlinear program (MINLP). The latter becomes an NLP, if the integrality requirementon the binary controls is dropped. After solving this NLP, the resulting relaxed binary control values areapproximated in the CIA step - which is an MILP - with binary control values. This work is based on theMILP decomposition due to its advantages:

1. MINLPs fall generally into the class of NP-hard problems so that using approaches which bypass thedirect solving of such problems is computationally favorable.

2. Convergence results have been proven for MIOCPs without MDT constraints in the sense that, under mildasssumptions, the provided solution becomes abritrarily close to the optimal solution with discretizationgrid length going to zero [29,32,24]. Moreover, the solution of the NLP represents a useful a priori lowerbound on the objective.

3. An MILP enables the option to include intuitively a large variety of combinatorial constraints. Numericalcase studies showed that the resulting feasible solution is close to the a priori lower bound in case theapplied combinatorial constraints are not too restrictive [7,8].

For a generalization of the CIA decomposition to more optimization steps and MILP variants see [38] aswell as to the PDE constraint case see [16,25,17]. Recently, an extension of the algorithm with the penaltyalternating direction method has been made [14], which can be regarded as a feasibility pump [12] variantfor MIOCPs. The main idea of the CIA problem is to minimize the integrated control deviation between therelaxed binary control α and binary control ω , where the latter is variable and the former is assumed to begiven from the NLP step. This deviation is commonly referred to as integrality gap and is defined as

infω∈Ω

supt∈T

∥∥∥∥∫ t

t0α(τ)−ω(τ)dτ

∥∥∥∥∞

≤C(nω) · ∆ , (1.1)

where C(nω) is a constant depending on the number of controls nω , T the time horizon and ∆ the maxi-mum discretization grid length. Instead of minimizing the integrality gap by means of solving an MILP, fastrounding algorithms can be applied such as SUR [29] and next forced rounding (NFR) [19], whose providedbinary solution still converges to the relaxed binary control solution with vanishing grid size. We summarizetheir integrality gap bounds in Table 1.

Next-Forced Rounding (NFR) Sum-Up Rounding (SUR) SUR with vanishing constraints

C(nω ) = 1, see [19],nω

∑i=2

1i , see [22], bnω/2c, see [26].

Table 1 So far known integrality gap bounds for binary control approximation algorithms. By vanishing constraints we mean mixedstate-control constraints.

1.1 Contribution

We conduct a theoretical analysis of the integrality gap in the presence of MDT constraints. In particular, weprove the upper bound

infω∈Ω

supt∈T

∥∥∥∥∫ t


∥∥∥∥∞

≤ 2nω −32nω −2

(CUD + ∆),

where CUD denotes the maximum of the minimum up (MU) and minimum down (MD) duration of all con-trols. We show that this bound is tight for MU time constraints. As a consequence of this result, the tightestpossible bound for the CIA problem is C(nω) =

2nω−32nω−2 . The proof is constructive, as we introduce a gen-

eralization of the NFR scheme to the MDT setting. We present a rounding modification so that MD times

Mixed-Integer Optimal Control under Minimum Dwell Time Constraints 3

can be included explicitely allowing us to deduce an improved integrality gap bound for this case. In thesame spirit, we modify SUR so that MDT requirements are satisfied by the provided binary solution. We testour algorithms on the tree tank problem from the http://mintOC.de library [31] and evaluate how largethe spread between provided integer solution and a priori lower bound becomes with increasingly restrictiveMDT constraints.

1.2 Outline

We give a problem definition of the MIOCP in Section 2 and describe the proposed CIA decompositionalgorithm with (CIA) as subproblem in Section 3. A generalization of NFR to the MDT setting is presentedin Section 4. It provides the tools to derive upper bounds on the integrality gap in Section 5. The commonlyused SUR scheme can be extended to include also MDT constraints, which we present in Section 6 togetherwith a discussion on its integrality gap. Section 7 provides a numerical example and we finish this articlewith conclusions in Section 8.

2 Mixed-Integer Optimal Control Problem

Since our work is based on the CIA decomposition approach, for which first-discretize-then-optimize and(partial) outer convexification approaches are fundamental, we formulate our problem at hand directly inthe discretized and (partial) outer convexified setting, but refer to [29,20] and, more recently, to [22,21] forextensive descriptions of partial outer convexification, the continuous MIOCP setting and its relation to thediscretized problem.

2.1 Definition of Binary and Relaxed Controls

Let [n] := 1, . . . ,n, [n]0 := 0∪ [n], for n∈N. We consider problems on a given time horizon T := [t0, t f ]⊂R and we introduce a time discretization by the ordered set GN := t0 < .. . < tN = t f denoting a grid withN intervals and lengths ∆ j := t j− t j−1, ∆ := max j ∆ j as well as ∆ := min j ∆ j for j ∈ [N]. Furthermore, weuse Gauss’ bracket notation, i.e., bxc := maxk ∈ Z | k ≤ x, x ∈ R, and analogously for dxe.

Definition 1 (Special order set one constraint (SOS1) and Matrix sets W,A) We express the requirementthat the columnwise entries of a matrix (mi, j) ∈ [0,1]nω×N sum up to one by

∑i∈[nω ]

mi, j = 1, for j ∈ [N], (SOS1)

and call it in the sequel special order set one (SOS1) constraint. Based on this constraint, we define the sets

W :=

w ∈ 0,1nω×N | w fulfills (SOS1)., A :=Conv(W ),

where we mean by Conv(W ) the convex hull of W.

Throughout this paper we assume a problem involving nω ≥ 2 binary controls that are piecewise constantand we introduce these in the following definition.

Definition 2 (Binary ω and relaxed controls α) Let the vector of binary control functions ω on the simplexand its corresponding vector of relaxed control functions α be defined by their function space domains

Ω := ω : T →0,1nω |ω(t) = w·, j, for t ∈ [t j−1, t j), j ∈ [N], t j ∈ GN ,w ∈W,A := α : T → [0,1]nω |α(t) = a·, j, for t ∈ [t j−1, t j), j ∈ [N], t j ∈ GN ,a ∈ A,

where w·, j denotes the jth column of w.

Note that we do not specify the values ω(tN),α(tN) in the above definition and may have altered the definitionto ω(t) = w·,N , for t ∈ [tN−1, tN ] and similarly for α , but we omitted this part, since the control problem tobe discussed is independent of the grid point values, as we shall see.

http://mintOC.de


2.2 Optimal Control Problem Class

We take interest in the discretized binary control problem (DBCP) with MDT constraints and its naturallyrelaxed problem, the discretized relaxed control problem (DRCP), which arises from replacing ω by α .

Definition 3 (Problems (DBCP) and (DRCP)) Let a MU time CU ≥ 0 and a MD time CD ≥ 0 be given.Consider the set of possible switching times S := GN\t f and for ts ∈S we define C

t fU := minCU , t f − ts

and similarly Ct fD := minCD, t f − ts. Let further ωi(t−s ) := lim

t→t−sωi(t) denote the one-sided limit to ts from

below. We refer to the following general problem class as (DBCP)

minx,ω∈Ω

Φ(x(t f ))

s. t. x = f0(x(t)) + ∑i∈[nω ]

ωi(t)fi(x(t)), for t ∈T \S , (2.1)

x(t0) = x0, (2.2)

ωi(ts + τ) ≥ ωi(ts)−ωi(t−s ), for i ∈ [nω ], ts ∈S , and τ ∈ [0,Ct fU ), (2.3)

1−ωi(ts + τ)≥ ωi(t−s )−ωi(ts), for i ∈ [nω ], ts ∈S , and τ ∈ [0,Ct fD ), (2.4)

where we minimize a Mayer term functional Φ ∈ C1(Rnx ,R) over the binary controls ω and differentialstates x ∈W 1,∞(T ,Rnx) with fixed initial values x0 ∈ Rnx . Constraint (2.1) expresses the dynamical systemas switched system in partial outer convexified form, i.e., as a sum of a drift term f0 and control specificfunctions fi both out of C0(Rnx ,Rnx). We define a control i ∈ [nω ] to be active at time point t ∈ T , if andonly if ωi(t) = 1 is true and the other way around for inactive controls. If a binary control is active after aswitch ts, it has to stay active for a time period of at least CU as required by (2.3), whereas (2.4) enforces theanalogous case for deactivating a control. Finally, we define (DRCP) as the canonical relaxation of problem(DBCP) where we optimize over α ∈A instead of ω ∈Ω .

We remark that our study assumes no mode specific MU times Ci,U or MD times Ci,D, but may includethem by setting CU = maxi∈[nω ]Ci,U and accordingly CD = maxi∈[nω ]Ci,D, even though this simplificationmay result in suboptimal solutions.

Without loss of generality, we omit in our problem defintion further constraints and continuous controlsu ∈ L∞(T ,Rnu). See [30] for a discussion of extensions and [25] for PDE constraints.

3 Combinatorial Integral Approximation Decomposition

We recapitulate the CIA decomposition algorithm to solve MIOCPs in Fig. 1. This approach is justified byan approximation result in the situation without MDT constraints establishing that the objective integer gapof (DBCP) compared to (DRCP) depends linearly on the integrated control deviation under certain regularityassumptions [29,24].

(MIOCP) (DBCP) (DRCP) (CIA)convexification

time discretization

relaxation

α ∈A

solving→ a

rounding a→ w

Fig. 1 Schematic representation of the CIA decomposition. MIOCPs can be equivalently reformulated into their partially outer convexi-fied counterpart problem which is thereafter transformed into (DBCP) via a temporal discretization. Next, allowing a convex combinationa in (SOS1) yields the relaxed problem (DRCP). After solving this problem we obtain the values a, which are then approximated withbinary values w in the rounding problem (CIA). Finally, we use w as fixed variables for solving (DBCP) as a continuous variableproblem.

Thus, we are interested in an admissible ω with integrated control deviation bounded by the grid lengthin the sense of

infω∈Ω

supt∈T

∥∥∥∥∫ t


∥∥∥∥∞

≤C(nω) · ∆ , (3.1)


as already mentioned in the introduction, so that the solution of (DRCP) can be arbitrarily well approximatedby an integer solution through refining the discretization grid [32]. The SUR scheme constructs solutionsω with this property; in fact, Kirches et al. [22] showed C ∈ O(log(nω)) as conjectured in [32]. Hence,the integrality gap provided by this rounding scheme is going to be arbritrarily small with arbritrarily smallgrid length, which is why SUR is often applied in the CIA decomposition as rounding step after solving(DRCP). Rather than using SUR, it has been proposed [33] to formulate the problem (3.1) as an MILP for animproved approximation and to be able to consider also different norms (such as the Manhattan norm) andcombinatorial constraints. Therefore, we recall its definition and state its MDT variants.

Definition 4 (Problems (CIA), (CIA-U), (CIA-D), and (CIA-UD)) Let a∈A be given after solving (DRCP).Then, we define the problem (CIA) to be

minθ ,w∈W

θ (3.2)

s. t. θ ≥± ∑l∈[ j]

(ai,l−wi,l)∆l , for i ∈ [nω ], j ∈ [N]. (3.3)

Let Jk(C1) := k∪ j | t j−1 ∈ GN ∩ [tk−1, tk−1 +C1) represent the MDT C1 ≥ 0 relevant intervals frominterval k ∈ [N] on. We formulate the MDT constraints as follows.

wi,l ≥ wi,k+1−wi,k, for i ∈ [nω ], k ∈ [N−1], l ∈Jk+1(CU ), (3.4)1−wi,l ≥ wi,k−wi,k+1, for i ∈ [nω ], k ∈ [N−1], l ∈Jk+1(CD). (3.5)

The (CIA) problem with added MU time constraint (3.4) is in the remainder referred to as (CIA-U). Similarly,let us (CIA) with added MD time constraint (3.5) call (CIA-D) and (CIA) with both (3.4) and (3.5) (CIA-UD).

Clearly, (CIA) is a reformulation of minimizing (3.1). In Section 4 we are going to elaborate upper boundsfor (3.1) in the presence of MDT constraints by investigating its (CIA) variants. Since the lower bound on itsobjective is trivially zero (and can be reached), bounds always refer to upper bounds in this article.

With respect to the constraints (3.4) and (3.5) we stress that there are other, in many cases computationallymore efficient, formulations of MDT constraints, e.g. in the spirit of extended formulations [23]. But, as wepropose to solve (CIA-UD) by means of tailored rounding heuristics and a branch and bound algorithm, wewould benefit neither numerically nor theoretically from alternative MILP formulations and we thereforeskip the presentation of these.

4 Dwell Time Next Forced Rounding

NFR is an algorithm that can approximately solve (CIA) in O(nω N2) [19] and provides solutions of (CIA)with a maximum objective of ∆ . It is based on the idea to activate on each interval the control which isadmissible and next forced. An control is admissible in this context means that, when activated on the currentinterval, its control deviation between accumulated relaxed and binary values would not be less than thenegative maximum grid length ∆ . A control is next forced, on the other hand, if it is the one whose controldeviation becomes the earliest greater than the maximum grid length with relaxed values a of future intervalstaken into account. We will formally define these concepts in this section.

We introduce dwell time next forced rounding (DNFR) as a generalization, aiming for a scheme thatconstructs MDT constraint feasible solutions and bounds for the (CIA) objective and its MDT variants. Ourrounding extension is threefold:

1. Depending on a given MDT C1 ≥ 0, we group the intervals into subsequent dwell time blocks and weapply the rounding scheme on these blocks so that exactly one control is active for each block. As weshall see, the MU time condition is fulfilled by this procedure. We choose a generic MDT C1, which wespecify later by setting C1 =CU , C1 =CD or similar.

2. This altered scheme can also be applied in the presence of a MD time constraint, but, since this require-ment is less restrictive than fulfilling a MU time, we propose a modification that provides a better bound.We introduce the binary constant χD so that a down time rounding mode can be turned on by settingχD = 1. We specify the concept of a down time forbidden control, i.e., a control that can not be active ona certain dwell time block due to its down time.

3. We introduce a generic rounding threshold factor C2 > 0 instead of using always C2 = 1 as in the NFRscheme.


We are going to elaborate several definitions needed for the DNFR in this section and begin with adefinition of sequences of intervals that are grouped into blocks in the presence of MDT constraints.

Definition 5 (Dwell time block interval sets) Let a MDT C1 ≥ 0 be given. We define iteratively the dwelltime invoked interval sets Jb and their last indices lb for b = 1, . . . ,nb and with l0 := 0:

Jb := lb−1 +1∪ j | t j−1 ∈ GN ∩ [tlb−1 , tlb−1 +C1),lb := max j | j ∈Jb,

where nb := minb | lb = N represents the number of interval blocks.

In the following we will sometimes write loosely block instead of dwell time block, for shortening our lan-guage. Next, we establish the lengths of dwell time blocks.

Definition 6 (Dwell time block length) Let a family of dwell time block interval sets Jbb∈[nb] be given.We denote by Lb the length of dwell time block b ∈ [nb] and name the maximum, respectively minimum,length of all dwell time blocks L , respectively L , i.e.,

Lb := tlb − tlb−1 , b ∈ [nb],

L := maxb∈[nb]

Lb, L := minb∈[nb]

Lb.

By the definition of dwell time blocks, we see that Lb depends both on the time discretization GN and C1.If C1 ≤ ∆ , then, the blocks are in fact the grid intervals, i.e., L j = ∆ j, j ∈ [N] and nb = N. As soon as C1 > ∆

holds, there is at least one block b with length of two subsequent intervals Lb = ∆ j +∆ j+1, j ∈ [N− 1].Overall, one recognizes that L increases monotonically with increasing C1.

The DNFR scheme relies crucially on the block dependent accumulated control deviation, which is whywe introduce it as auxiliary variable in the next definition.

Definition 7 (Accumulated control deviation θi, j,Θi,b,γi, j,Γi,b) Let a ∈ A and w ∈W. For controls i ∈ [nω ]we define the interval j ∈ [N] varying accumulated control deviation as

θi, j :=j

∑l=1

(ai,l−wi,l)∆l , γi, j :=j

∑l=1

ai,l∆l−j−1

∑l=1

wi,l∆l .

and further define θi,0 := 0 for all i ∈ [nω ]. We introduce for blocks b ∈ [nb] the analogous variables

Θi,b := θi,lb , Γi,b :=Θi,b−1 + ∑j∈Jb

ai, j∆ j.

In the sequel, we sometimes write forward control deviation for control i in order to distinguish γi, j,Γi,b fromthe (accumulated) control deviation θi, j,Θi,b.

Remark 1 If a small MDT C1 ≤ ∆ is given, θi, j equals trivially Θi, j and the same holds for γi, j and Γi, j.Nevertheless, with a MDT of C1 > ∆ one needs interval and block related variables to be able to clearlydistinguish between both values.

Remark 2 Let w ∈W and we denote by θ(w) its (CIA) objective. With the above definition, we concludeθ(w) = max

i∈[nω ], j∈[N]|θi, j|. Generally, one can notice that the maximum of the |θi, j| values must be assumed at

an interval before a switch happens (i.e., w·, j 6= w·, j+1) or on the last interval, since |θi, j| is monotonicallyincreasing with constant w·, j and increasing j. Hence, with constant w·, j on the dwell time blocks, it alsoholds θ = max

i∈[nω ],b∈[nb]|Θi,b|.

Next, we define different types of control variable activations which depend on prior variables choicesand the relaxed control values a.


t0 t1 t2 t3 t4 t5 t6 t7 t8 t9

0

0.25

0.5

0.75

1

ωi

αi

· · ·

∆1 ∆2 ∆3 ∆4 ∆5 ∆6 ∆7 ∆8 ∆9

L1 L2 L3

Control values

0 1 2 3 4 5 6 7 8 9

0

0.5∆

1∆

θi

γiΓi,1

Γi,2

Γi,3

Θi,1 Θi,2Θi,3

N

Accumulated deviation

J1 J2 J3

Fig. 2 Left: example binary and relaxed control values on nine intervals ∆ j , respectively three blocks Lb. Right: corresponding ac-cumulated control deviation. The forward control deviation w.r.t. intervals γi is greater or equal to θi with equality if the control i isinactive. Also, Γi,b is greater or equal to Θi,b, where the latter equals via definition the last θi before block b begins. Notice that Γi,2 sumsup the weighted α values for the intervals 3-6, which results in a large value.

Definition 8 (Admissible, forced, and future forced activation) Let the rounding threshold factor C2 > 0be given. The choice wi, j = 1 for i ∈ [nω ], j ∈Jb, b = 1, . . . ,nb is admissible, if it holds

Γi,b ≥−C2L +Lb.

Denote by W ba the set of admissible controls for block b. Similarly, the choice wi, j = 1 for i ∈ [nω ], j ∈

Jb, b = 1, . . . ,nb is forced, if it holds

Γi,b >C2L .

We define a control i ∈W ba on block b to be l-future forced, if it holds

Θi,b−1 +l

∑k=b

∑j∈Jk

ai, j∆ j >C2L .

with the special case l = b meaning i is actually forced on b. If the above inequality holds for l = nb, we callthe control i ∈W b

a on block b to be future forced, and group these controls into the set W bf .

Definition 9 (Minimum down time forbidden control) We introduce the constant χD ∈ 0,1. If the CIAproblem involves a MD time constraint with CD > ∆ , we set χD = 1 and otherwise χD = 0. We define iDb , b =3, . . . ,nω , as the index of the control that has been activated on block b−2 and deactivated on block b−1 -if such a control exists:

∃i ∈ [nω ] : wi, j = 1, j ∈Jb−2 ∧ wi, j = 0, j ∈Jb−1 ⇒ iDb := i.

Then, let I Db denote the χD dependent set of the minimum down time forbidden control

I Db :=

iDb , if χD = 1, and b≥ 3,/0, otherwise.

Note that I Db is either the empty set or contains exactly one control index. It may seem unintuitive to declare

only one control per block as minimum down time forbidden, because a sufficiently large chosen MD timecan comprise more than two intervals and therefore more than one control could be minimum down forbiddenon certain blocks. We will explain later in Section 5.4 why such a fine granular definition is not necessaryand would lead to weak bounds.

Finally, we use these control properties to declare the DNFR scheme in Alg. 1. We iterate over all dwelltime invoked blocks (line 2) and check on each block, if there is a forced control and activate it in this case(line 3-4). Otherwise, we test if there is an earliest future forced control and if so, we set it to be active (line5-8). Else, the algorithm selects the control with the maximum forward control deviation (line 9-13). In casethe MD mode is turned on by setting χD = 1, we exclude the set I D

b from our control selection task (line 3,5, 11).


tb j tb j+1 tb j+2 tb j+3 tb j+4

0

0.25

0.5

0.75

1

ωi

αi

· · · · · ·

Lb j Lb j+1 Lb j+2 Lb j+3

Control values

b j b j+1 b j+2 b j+3

−C2L+Lb

C2L

ΓiΘi

nb

Accumulated deviation

Fig. 3 Exemplary visualization of defined quantities. Left: binary and relaxed control values on four blocks. Right: corresponding blockaccumulated control deviation. Control i is admissible on block b j , not admissible on block b j+1, down time forbidden and b j+3-futureforced on block b j+2 as well as forced on block b j+3.

Algorithm 1: Dwell time next forced rounding algorithm for (CIA-UD)Input : Relaxed control values a ∈ A, time increments ∆ j , j ∈ [N], parameter C1,C2,χD.Output: Feasible solution wDNFR of (CIA-UD) with approximation quality depending on C1,C2,χD.

1 Initialize w = 0.2 for all dwell time blocks b = 1, . . .nb do3 if there is a control i ∈ [nω ]\I D

b with forced activation then4 Set wi, j = 1, j ∈Jb.5 else if it exists a future forced control, i.e., W b

f \I Db 6= /0 then

6 Identify the control with the earliest future forced activation (break ties arbitrarily):7 i = argmin

b(i) ∈ [nb] | i ∈W b

a \I Db , i is b(i)− future forced

.

8 Set wi, j = 1, j ∈Jb.9 else

10 Find the admissible control with maximum control deviation (break ties arbitrarily):11 i = argmaxΓi,b | i ∈W b

a \I Db .

12 Set wi, j = 1, j ∈Jb.13 end14 end15 return: wDNFR = w;

5 Tight Bounds on the Integrality Gap for (CIA) with Dwell Time Constraints

We now consider the question of how large the objective function value θ of (CIA) and its MDT variants canbe. In other words we examine

θmax := max

a∈Aminw∈W

maxi∈[nω ], j∈[N]

|θi, j| s.t. MDT constraints (3.4), (3.5).

It turns out that the DNFR scheme is particularly suitable for deriving these integrality gap bounds. Wefirst supply lemmata in Section 5.1 before proving approximation results for (CIA) by means of DNFRconstructed solutions. These results come along as two theorems in Section 5.2 with specified parameterchoices for C2 and χD. Aftwards, we deduce specific bounds for the (CIA) dwell time problems and evaluatehow tight they are in Section 5.3 and Section 5.4. We begin this section with the trivial upper bound

θmax ≤ ∑

j∈[N]

∆ j = t f − t0

and another weak result in the following remark.

Remark 3 Neglecting for a moment MDT constraints, it is known from Minimax theory [36] that

maxa∈A

minw∈W


|θi, j| ≤ minw∈W

maxa∈A


|θi, j|

holds. In the right hand side, we maximize over a for given w and check which one of the latter leads toan overall minimum objective. In this way a manipulates the control deviation to be as large as possible.


That means with given w it is possible to set aimin, j = 1, j ∈ [N], where imin is the control with smallesttotal accumulation ∑ j∈[N] wi, j∆ j, so that the (CIA) objective becomes θ = ∑ j∈[N](1−wimin, j)∆ j. With thesearguments one may derive

θmax ≤

(N−

⌊Nnω

⌋)∆ .

We omit the exact proof, since this bound is generally weak as we will see later in this section.

5.1 Lemmata for DNFR Approximation Results

We present a series of lemmata, which will be needed later in the proofs to Thm. 1 and Thm. 2.

Lemma 1 (Family of dwell time block sets) The family of dwell time block interval sets Jbb∈[nb] asdefined in Def. 5 is a partition of the set of all interval indices [N].

Proof This follows directly from the definition of dwell time block interval sets. ut

Lemma 2 (Block accumulated control deviation properties) It holds for b ∈ [nb]

∑i∈[nω ]

Γi,b = Lb, ∑i∈[nω ]

Θi,b = 0.

Proof Let us derive an auxiliary result for j ∈ [N]:

∑i∈[nω ]

θi, j = ∑i∈[nω ]

∑l∈[ j]

(ai,l−wi,l)∆l(SOS1)= ∑

l∈[ j]∆l− ∑

l∈[ j]∆l = 0. (5.1)

We use this and rearrange the sums in order to proof the first assertion:

∑i∈[nω ]

Γi,b = ∑i∈[nω ]

(di,lb−1 + ∑

j∈Jb

ai, j∆ j

)= 0+ ∑

i∈[nω ]∑

j∈Jb

ai, j∆ j = ∑j∈Jb

∑i∈[nω ]

ai, j∆ j(SOS1)= ∑

j∈Jb

∆ j

= Lb.

The auxiliary result is also useful for the second statement:

∑i∈[nω ]

Θi,b = ∑i∈[nω ]

θi,lb−1

(5.1)= 0.

ut

Lemma 3 (Accumulated difference of Γ and Θ over active controls) Let b1,b2 ∈ [nb] and we define Sb1,b2as the set of active controls between b1 and b2:

Sb1,b2 := i ∈ [nω ] | ∃b : b1 < b < b2 with wi, j = 1, ∀ j ∈Jb.Then, it holds

∑i∈Sb1 ,b2

(Γi,b2 −Θi,b1

)≤L .

Proof Using Def. 7 of Γ ,Θ and rearranging sums yields

∑i∈Sb1 ,b2

(Γi,b2 −Θi,b1

)= ∑

i∈Sb1 ,b2

(b2

∑b=b1+1

∑j∈Jb

ai, j∆ j−b2−1

∑b=b1+1

∑j∈Jb

wi, j∆ j

)

=b2

∑b=b1+1

∑j∈Jb

∆ j ∑i∈Sb1 ,b2

ai, j︸︷︷︸≤1

−b2−1

∑b=b1+1

∑j∈Jb

∆ j ∑i∈Sb1 ,b2

wi, j︸︷︷︸=1

≤b2

∑b=b1+1

∑j∈Jb

∆ j−b2−1

∑b=b1+1

∑j∈Jb

∆ j

= Lb2 ≤L .

ut


Note that Sb1,b2 is trivially the empty set, if b2 ≤ b1 + 1, but the result remains true in this case. We aregoing to employ the concept of Sb1,b2 for a contradiction in the proofs of Thm. 1 and Thm. 2.

Lemma 4 (Control with negative Γ value has not been future forced) Let (C1,C2,χD) be given andassume for the forward control deviation of a control i∈ [nω ] and a block b2 ≥ 2 after executing Alg. 1 holds:

Γi,b1 ≤C2L −L , and Γi,b1 < 0.

Then there is an earlier activation of i on some block b1 < b2 and this activation has not been b2-futureforced on b1.

Proof Note that Γi,b is monotonically increasing in b for deactivated controls i. We conclude from this andΓi,b1 < 0 that there is an earlier activation of i on some block b2 < b1. We take a closer look on the forwardcontrol deviation on block b1:

C2L −L ≥ Γi,b1 =b1

∑k=1

∑j∈Jk

ai, j∆ j−b2

∑k=1

∑j∈Jk

wi, j∆ j =b1

∑k=1

∑j∈Jk

ai, j∆ j−b2−1

∑k=1

∑j∈Jk

wi, j∆ j−Lb2 ,

and rearranging terms implies

b1

∑k=1

∑j∈Jk

ai, j∆ j−b2−1

∑k=1

∑j∈Jk

wi, j∆ j ≤C2L −L +Lb2 ≤C2L .

The last inequality shows us that i has been not b1-future forced on b2. ut

5.2 Integrality Gap Results through Dwell Time Next Forced Rounding

This section establishes a connection between DNFR and the integer approximation error of (CIA). For thiswe first examine in Thm. 1 how large the control deviation can become during a MD time phase. Basedon this result we derive in Thm. 2 that DNFR constructs (CIA) feasible solutions with objective boundsdepending on the rounding threshold C2 and whether MD time forbidden controls are allowed, i.e., χD = 1.The proofs of both theorems use the idea of assuming a control deviation greater than the one to be provenand then deriving a contradiction. Due to this similarity, we skip here the proof of Thm. 1 and present it inAppendix A, but will work out the proof of Thm. 2 in detail, since it expresses the main result.

Theorem 1 (Control accumulation of a minimum down time forbidden control) Let (C2,χD) =( 3

2 ,1)

andC1 ≥ 0 be given and assume there is a minimum down time forbidden control iD ∈I D

b on block b≥ 3 afterAlg. 1 was executed. Then, for the forward control deviation

ΓiD,b ≤ 1.5 ·L .

Proof See Appendix A.

With the last theorem we already have a statement about the control deviation for mininum down timeforbidden controls. The next result goes a step further and establishes a connection between Alg. 1 and (CIA).

Theorem 2 (Rounding gap of solution constructed by DNFR) Let the following parameter settings begiven:

I. (C2,χD) =(

2nω−32nω−2 ,0

)II. (C2,χD) =

( 32 ,1),

and C1 ≥ 0. Then, wDNFR provided by Alg. 1 is a feasible solution of (CIA) for both cases with approximationquality

θ ≤ C2L .


Proof The assertion can be shown for the parameter choices I. and II. in a very similar way, which is why weprove both cases in parallel. Since the algorithm activates for each block b ∈ [nω ] either a forced, or futureforced, or admissible control and the family of blocks is a partition [N] by Lemma 1, exactly one control isactivated on each interval j ∈ [N] and therefore the (SOS1) constraint statisfied. Hence, DNFR guaranteesfeasibility of wDNFR. If down time forbidden controls are neglected, i.e., χD = 0, wDNFR yields an objectivevalue with at most the claimed upper bound by the definition of admissible and forced activation. The sameholds for the choice χD = 1, since the control deviation does not become greater than the claimed upperbound during a MD time phase by Thm. 1 . Therefore, we need only to prove that DNFR always provides asolution. For this, we show that for each interval there is 1.) at least one admissible control and 2.) at mostone forced control.

1.) We prove by contradiction that there exists at least one admissible control: Let us assume there is noadmissible activation for block b ∈ [nb] and distinguish between the cases:I. With C2 =

2nω−32nω−2 we assume

Γi,b!<−2nω −3

2nω −2L +Lb, i ∈ [nω ].

It follows from summing up all controls and from Lemma 2:

Lb = ∑i∈nω

Γi,b < nω

(−2nω −3

2nω −2L +Lb

)=−nω

2nω −32nω −2

L +nωLb,

and subtracting nωLb from the right hand side provides

−nωL < (1−nω)Lb <−nω

2nω −32nω −2

L <−nωL .

II. If there is no MD time forbidden control on b, we can proceed as in I. Otherwise, we may have onecontrol iD which is down time forbidden. We assume all other controls are inadmissible, i.e.,

Γi,b!<−1.5 ·L +Lb, i ∈ [nω ], i 6= iD.

Hence, again by Lemma 2

Lb = ∑i∈nω

Γi,b = ∑i 6=iD

Γi,b +ΓiD,b < (nω −1)(−1.5 ·L +Lb)+ΓiD,b,

and therefore

1.5(nω −1)L − (nω −2)Lb ≤ 1.5L < ΓiD,b.

The last inequality is a contradiction to Theorem 1.We conclude that there must be an admissible activation for all blocks and thereby for all intervals.

2.) If there were more than one forced control at a time step, the algorithm would be ambiguous at line 3-4. Moreover, DNFR would provide in this case a solution that does not satisfy the upper bound on theobjective. Therefore, we prove that this case is impossible and we do it again by contradiction. Let usassume there is a block b∈ [nb] on which for the first time at least two controls i1, i2 are being forced, i.e.,

I. Γi1,b,Γi2,b >2nω −32nω −2

L , II. Γi1,b,Γi2,b > 1.5 ·L . (5.2)

In the proof of Thm. 1 we have already caused a contradiction with only one forward control deviationΓi,b greater than the rounding threshold. So, case II. is settled with this theorem and we focus on case I.for which we proceed very similarly as in the proof of Thm. 1. Let us first apply Lemma 2:

L ≥Lb = ∑i∈[nω ]

Γi,b = ∑i∈[nω ],i6=i1,i2

Γi,b + ∑i=i1,i2

Γi,b︸︷︷︸>2· 2nω−3

2nω−2L

.


Hence, it holds

∑i∈[nω ],i 6=i1,i2

Γi,b < L −22nω −32nω −2

L =−2nω −42nω −2

L ,

which implies that there is at least one control i3 such that

Γi3,b <−1

nω −2· 2nω −4

2nω −2L =− 2

2nω −2L .

Then, by Lemma 4, there is an earlier activation of i3 on some block b3 < b and this activation has notbeen b-future forced on b3. Let i3 denote the control of this property with the last activation before b.This definition implies that all controls which are active between b3 and b would become forced until b.We use again the notation

Fb3,b := i ∈ [nω ] | ∃k(i) : b3 < k(i)≤ b on which i is forced or b-future forced..In particular, it holds i1, i2 ∈ Fb3,b. For i ∈ Fb3,b\i1, i2, we apply the definition of Fb3,b and Γ :

Γi,b =Θi,b−1 + ∑j∈Jb

ai, j∆ j =b

∑k=1

∑j∈Jk

ai, j∆ j−k(i)

∑k=1

∑j∈Jk

wi, j∆ j.

Since control i was last activated on block k(i) and b- future forced on k(i), we have

Γi,b >2nω −32nω −2

L −Lk(i), i ∈ Fj3, j\i1, i2. (5.3)

For block b3 we know that i3 has been chosen, even though not being b-future forced. That implies i3 wasselected on b3 since all controls out of Fb3,b have not been admissible at this block. Hence, it results fori ∈ Fb3,b

Γi,b3 <−2nω −32nω −2

L +Lb3 , ⇒ Θi,b3 <−2nω −32nω −2

L +Lb3 . (5.4)

Now, we consider the sum of inequalities (5.3), (5.2) and sum up (5.4) over Fb3,b, which yields

∑i∈Fb3 ,b

Γi,b > 2 · 2nω −32nω −2

L +(|Fb3,b|−2) ·(

2nω −32nω −2

L −Lb2

), (5.5)

∑i∈Fb3 ,b3

Θi,b3 < |Fb3,b|(−2nω −3

2nω −2L +Lb3

), (5.6)

where b2 := argminLk | b3 < k ≤ b. When we subtract (5.6) from (5.5), we obtain

∑i∈Fb3 ,b

(Γi,b−Θi,b3

)> 2 · 2nω −3

2nω −2L +(|Fb3,b|−2) ·

(2nω −32nω −2

L −Lb2

)

−|Fb3,b|(−2nω −3

2nω −2L +Lb3

)= 2 · |Fb3,b| ·

2nω −32nω −2

L − (|Fb3,b|−2) ·Lb2 −|Fb3,b|Lb3

≥L

(2|Fb3,b|

2nω −32nω −2

−2|Fb3,b|+2)

= L

(2−|Fb3,b|nω −1

)≥L . (5.7)

In the next-to-last inequality we used Lb2 ,Lb3 ≤ L , while the last inequality holds due to |Fb3,b| ≤nω − 1. As in the proof of Theorem 1, we consult now Lemma 3 with Fb3,b = Sb1,b in order to raise acontradiction to inequality (5.7). Overall, we have shown that there is at most one forced activation perblock and thereby per interval. This completes the proof. ut

Remark 4 The proceeding in the proof of Thm. 2 shows us, on closer inspection, that DNFR provides asolution with control deviation bounded by C2L for the absence of MD time constraints, i.e., χD = 0, andany chosen rounding threshold C2 ≥ 2nω−3

2nω−2 and any block length parameter C1 ≥ 0. It implies the previouslyknown result by NFR [19], θ ≤ ∆ , evolves as special case of DNFR with C1 = 0, and C2 = 1.


5.3 Implications for the Objectives of (CIA-U) and (CIA)

Thm. 2 states only generic approximation results for (CIA) with a MDT parameter C1. We are going to assessthe consequences for (CIA-U) by specifying C1 and proving tightness of the resulting upper bound. Clearly,(CIA) can be seen as special case of (CIA-U) where CU = 0, so that results for (CIA-U) are inherited by(CIA).

Proposition 1 (Upper bound for (CIA-U)) Let a MU time CU ≥ 0 be given. Then, for (CIA-U) holds:

θ ≤ 2nω −32nω −2

(CU + ∆

).

Proof We consider the DNFR scheme with (C1,C2,χD) =(

CU ,2nω−32nω−2 ,0

). Then, wDNFR is a feasible solution

by Thm. 2 and by the property of DNFR to activate dwell time blocks of intervals with length at least C1 =CU .From the definition of block lengths we conclude L < CU + ∆ so that the assertion follows directly fromThm. 2. ut

In order to analyze the received bound with respect to tightness we introduce a grid where the MDT C1overlaps the grid points by a small ε > 0. We will determine the length of the resulting blocks in the followinglemma.

Definition 1 (Minimal C1-overlapping grid) Let us consider a non-degenerated MDT length, i.e., C1 > 0,and we choose a small ε with C1 ε > 0. Let further a time horizon [t0, t f ] be given with length at least4 ·C1, i.e.,

t f ≥ t0 +4 ·C1.

We define a minimal C1-overlapping grid GN recursively as follows

t1 := t0 +C1− ε,

t2 := t1 +C1,

t j :=

t j−1 +C1− ε, if j odd,t j−1 +C1, if j even,

for j = 3, . . . ,N−1,

where we set N−1 := max j | t j ≤ t f , so that GN consists of N intervals.

t0 t1 t2 t3 t4 · · · tN−1 tN

C1 − ε C1 C1 − ε C1 ≤C1

L1 L2

Fig. 4 Visualization of a minimal C1-overlapping grid.

Lemma 5 (Length of blocks of a minimal C1-overlapping grid) The dwell time invoked blocks Jb, b ∈[nb] of a minimal C1-overlapping grid as introduced in Def. 1 have the length

Lb = 2 ·C1− ε, b ∈ [nb−1],Lnb = t f − (t0 +(nb−1) · (2 ·C1− ε)) .

Moreover, it holds∆ =C1, L = 2 ·C1− ε.

Proof We keep in mind Def. 5 from which we deduce J1 = 1,2, because t0 +C1 (minimally) overlaps t1.The next dwell time block begins at t2 = t0+2 ·C1−ε and consists again of two intervals. This argumentationcan be extended to the first nb−1 blocks and by the definition of block lengths we conclude Lb = 2 ·C1−ε .The length of the last block Lnb is directly received by the definition of N− 1 to be the last index of thegrid point recursion before t f . Finally, the definition of a minimal C1-overlapping grid and the received blocklengths implies

∆ =C1, L = 2 ·C1− ε.

ut


With this preliminary work, we evalute whether the deduced MU time bound is sharp.

Proposition 2 (Tightness of the bound for (CIA-U)) The objective bound for (CIA-U) mentioned in Prop. 1with CU ≥ 0 be given is the tightest possible bound.

Proof Let us first consider CU > 0 and we construct an example with the desired objective value by meansof a minimal C1-overlapping grid, where we set C1 = CU . We assume a time horizon length of at least2 ·C1 · (nω − 1) so that the grid induced by Lemma 5 consists of at least nb ≥ nω − 1 blocks. Let furthera = (ai,b)i∈[nω ],b∈[nb] be given as

a =

0.5 0 · · · 01

2nω −21

nω −1· · · 1

nω −1...

.... . .

...1

2nω −21

nω −1· · · 1

nω −1

,

where the ith row indicate the values of control i ∈ [nω ] on each block b ∈ [nb] meaning that we set the ai, j-values on all intervals j ∈Jb of block b to this indicated value. Next, we discuss how the optimal solutionof (CIA−U) on the first nω − 1 blocks may be chosen. Let us calculate the control deviation if we wouldactivate a control i = 2 . . .nω on the first block:

Θi,1 =

∣∣∣∣∣ ∑j∈J1

12nω −2

∆ j−L1

∣∣∣∣∣= 2nω −32nω −2

L1 =2nω −32nω −2

(2 ·CU − ε)

=2nω −32nω −2

(CU + ∆ − ε).

In the second and third equality we used Lemma 5. On the other hand, the relaxed control values a for thecontrols i = 2 . . .nω sum up until block nω −1 to be

∑b∈[nω−1]

∑j∈Jb

ai, j∆ j =1

2nω −2L1 + ∑

b=2,...,nω−1

1nω −1

Lb

=1

2nω −2(CU + ∆ − ε)+ ∑

b=2,...,nω−1

1nω −1

(CU + ∆ − ε)

=2nω −32nω −2

(CU + ∆ − ε).

Thus, there are nω −1 controls with this control accumulation on nω −1 blocks, but activating any of thesecontrols on the first block yields already the same control deviation. That is why the objective of (CIA−U)with this a is at least 2nω−3

2nω−2 (CU + ∆ − ε), where ε is arbitrarily small. If we combine this result with Prop. 1,we get that 2nω−3

2nω−2 (CU + ∆) is the tightest possible bound. Last, we argue for the degenerated case, CU = 0,that we can create an example with length of all blocks set to ∆ and obtain the same tight bound. ut

We close this subsection with a statement about the maximum control deviation due to the (CIA) problemwithout MDT constraints.

Corollary 1 (Tight bound on the rounding gap for (CIA)) Let GN be a discretization grid with increments∆ j, j ∈ [N], N ≥ nω −1, given together with relaxed values a ∈ A. The objective of (CIA) is bounded by

θ ≤ 2nω −32nω −2

∆ ,

which is the tightest possible bound.

Proof This follows directly from Prop. 1 and Prop. 2. ut


5.4 Implications for the Objectives of (CIA-D) and (CIA-UD)

The bound received for (CIA-U) can be transferred in a straightforward manner to (CIA-D) by using C1 =CDas MDT in the DNFR scheme. However, we notice the increased degrees of freedom when dealing with MDtimes rather than MU times: only the MD time forbidden control is fixed for a certain time duration incomparison with the MU time constraint situation where all controls are fixed due to the fixed active control.With this observation, we introduced in the DNFR scheme the min down mode χD = 1 and are going todeduce in the sequel an alternative upper bound compared to the one obtained by DNFR with χD = 0. As weare going to detect, this alternative bound is independent of nω but not always an improvement, so that wedeclare the minimum of both bounds as the upper bound in the following proposition.

Proposition 3 (Bounds on the objective of (CIA-D) and (CIA-UD)) Let MU and MD times CU ,CD ≥ 0 begiven. Then

1. (CIA-D) is bounded by

θ ≤min

34

CD +32

∆ ,2nω −32nω −2

(CD + ∆

).

2. (CIA-UD) is bounded by

θ ≤

2nω−32nω−2

(CU + ∆

), if CU ≥CD,

min

32CU + 3

2 ∆ , 2nω−32nω−2

(CD + ∆

), if CD >CU >CD/2,

min

34CD + 3

2 ∆ , 2nω−32nω−2

(CD + ∆

), if CD/2≥CU .

Proof Generally, if CD >CU or if MU constraints are absent, we may apply the DNFR scheme with (C1,C2,χD)=(CD,

2nω−32nω−2 ,0

)which constructs feasible solutions for (CIA-D) respectively (CIA-UD) with objective bound

2nω−32nω−2 (CD + ∆). We are left with the other case χD = 1:

1. If we set C1 =12CD, it holds L < 1

2CD + ∆ . The DNFR scheme constructs with this MDT and the choiceχD = 1 a feasible solution for (CIA-D). Then, we deduce the bound

34

CD +32

∆ =32

(12

CD + ∆

)from Thm. 2 with the configuration χD = 1 and C2 =

32 .

2. (a) If CU ≥CD is given, we can set C1 =CU and all block lengths are at least as large as those of the MDtime CD. Therefore, the solution provided by DNFR with χD = 0 and C2 =

2nω−32nω−2 fulfills both the MU

and MD time constraint.(b) We set χD = 1 and C1 =CU , C2 =

32 , when CD >CU >CD/2 is given. By this choice, the solution of

DNFR fulfills a MD time of 2 ·CU with

2 ·L > 2 ·CU > 2 ·CD/2 =CD.

Furthermore, by setting C1 =CU it is clear that wDNFR does not violate the MU time.(c) CD/2 ≥ CU : DNFR with down time configuration χD = 1 and C1 = CD/2 ≥ CU , C2 = 3

2 can beexecuted without violating the MU time constraint.

ut

Since tightness results for the MD case are not as straightforward received as for the problem (CIA-U),we move the discussion on the quality of the bounds provided in Prop. 3 to the Appendix B.

6 Sum-Up Rounding in the Dwell Time Context

SUR is computationally less expensive (O(nω ·N)) than the DNFR scheme executed on intervals. But, on theother hand, the last section showed us DNFR constructs solutions for (CIA) with a maximum integrality gapthat is less than the one of solutions provided by SUR for nω ≥ 3 (equality for the case nω = 2). Since theSUR scheme is very often used for finding approximative solutions of (CIA), but does not necessarily fulfillMDT constraints, we discuss in this section a canonical extension of the algorithm to this setting.


6.1 Dwell Time Sum-Up Rounding (DSUR)

At first, we introduce the concept of a currently activated control and dwell time blocks that depend on theinitial interval and the MDT duration C1.

Definition 10 (Initial interval dwell time block index sets) Let a MDT C1 ≥ 0 be given. We define for allintervals k ∈ [N] the initial interval dependent dwell time index sets to be

J SURk (C1) := k∪ j | t j−1 ∈ GN ∩ [tk−1, tk−1 +C1).

Definition 11 (Currently activated control) We call a control index i currently activated at interval j =2, . . . ,N, if

wi, j−1 = 1

holds. Otherwise, or if j = 1, we call the binary control i currently deactivated.

In contrast to the DNFR scheme we are now interested in considering intervals individually and workindependently of the constant χD. That is why grouping of minimum down time forbidden controls for eachinterval into sets I SUR

j is necessary.

Definition 12 (SUR minimum down time forbidden control set) Let a MDT CD ≥ 0 be given. We definethe set of MD time forbidden controls I SUR

j as follows.

I SURj := i ∈ [nω ] | ∃k < j : t j−1 ≤ tk−1 +CD, tk−1 ∈ GN ∧ wi,k = 1.

for all intervals j = [N]. We say i ∈ [nω ] is MD time admissible on j ∈ [N], if i /∈I SURj holds.

Note that the above definition assumes implicitly fixed control variables for previous intervals [N] 3 k < j. Itholds trivially I SUR

1 = /0, because there are no MD time forbidden controls on the first interval. Moreover,the set I SUR

j may contain several controls, but at most nω −1.Next, we give a definition of the dwell time sum-up rounding (DSUR) scheme in Alg. 2. It iterates over

all intervals j ∈ [N] and selects initially the interval representing the beginning of the time horizon, wherenot yet a currently activated control exists. The control dependent MDT Ci,1 is updated in line 3 for eachiteration inside the while loop so that Ci,1 equals the maximum of MU time CU and MD time CD for acurrently activated control, and otherwise it is set to the MU time CU . Hence, the algorithm sets Ci,1 = CUfor all controls in the first while iteration. Next, in line 4, one searches for the MD time admissible controli? with maximum forward control deviation on the upcoming intervals covering the dwell time Ci?,1. If it isthe currently activated one, we fix this control to be active also on the current selected interval j and increasethe interval index (line 5-6). Else, the control is activated on the whole dwell time block represented by itsinterval indices J SUR

j (Ci?,1) and the interval index is updated accordingly (line 8-9). Lastly, DSUR updatesthe set of MD time forbidden controls for the next iteration in line 11. The algorithm stops as soon as thecontrol choice has been made for the last interval N.

Clearly, wDSUR is a feasible solution for (CIA), since exactly one control is active per interval. It is alsofeasible for (CIA-U), because whenever a currently deactivated control is activated, it dwells on active for atleast the duration CU (line 8-9). The solution satisfies also MD time constraints by the definition of I SUR

j

and altogether wDSUR is a feasible solution for (CIA-UD).We transferred the main idea from the original SUR scheme to the problem setting with MDT constraints

by selecting in each iteration the control with maximum forward control deviation. In the presence of MUtime requirements, we need to calculate this forward accumulation for the set of next intervals with lengthat least CU . For a given MD time longer than the MU time, we suggested in the algorithm to comparethe forward accumulation with length at least CD of the currently activated control with the ones of othercontrols with length at least CU . The idea behind this approach is to hinder a situation in which a control getsdeactivated, but is going to accumulate a large control deviation during its MD time forbidden period.

Remark 5 (Run time of DSUR) Alg. 2 is in O(nω ·N2), since it sums up at each interval and for all controlsthe relaxed control values a on the next dwell time induced intervals.


Algorithm 2: DSUR algorithm for (CIA−UD)

Input : Relaxed control values a ∈ A, time increments ∆ j , j ∈ [N], MU time CU , MD time CD.Output: Feasible solution wDSUR of (CIA−UD).

1 Initialize w = 0, j = 1 and I SURj = /0.

2 while j ≤ N do3 Set Cia ,1 = maxCU ,CD for the currently activated control ia, and set Ci,1 =CU for all other controls i 6= ia.4 Find the control with maximum deviation (break ties arbitrarily):

i? = argmaxθi, j−1 + ∑

l∈J SURj (Ci,1)

ai,l∆l | i ∈ [nω ]\I SURj

5 if i? = ia then6 Set wi? , j = 1 and update j = j+1.7 else8 Set wi? ,l = 1, l ∈J SUR

j (Ci? ,1);9 Update j = maxl | l ∈J SUR

j (Ci?,1)+1.10 end11 Update the set of MD time forbidden controls I SUR

j .12 end13 return: wDSUR = w;

6.2 Rounding Gap Bounds for Dwell Time Sum-Up Rounding

Kirches et al. [22] have proven the tightest possible bound on the integrality gap for the original SUR. Thisresult is beneficial for our considerations, as we can derive implications for DSUR in the absence of MDconditions.

Theorem 3 (Tight bound for SUR integrality gap, cf. Thm. 7.1, [22] ) Let wSUR be constructed from a∈ Aby means of SUR for an equidistant discretization of [t0, t f ] and let denote by θ(wSUR) its (CIA) objectivevalue. Then, the rounding gap is bounded by

θ(wSUR)≤ ∆

nω

∑i=2

1i,

which is the tightest possible upper bound.

Corollary 2 (Tight bound for DSUR integrality gap without MD times) Let CD = 0 and a MU timeCU > 0 be given. Let the time horizon [t0, t f ] with a minimal CU -overlapping grid be discretized and let wDSUR

be constructed from a ∈ A by means of DSUR. Then, the rounding gap θ(wDSUR) of its (CIA) objective valueis bounded by

θ(wDSUR)≤ (CU + ∆) ·nω

∑i=2

1i,

which is the tightest possible upper bound.

Proof As in the proof of Thm. 3 in [22] a dynamic programming argument can be applied, here with equidis-tant block length of (CU + ∆ − ε) as derived in Lemma 5. With a time horizon length of nω(CU + ∆ − ε) wemay analogously to the proof of Thm. 3 construct an example indicating that the bound is tight as follows.

ai, j =

0 if 2i+1≤ j ≤ N1/(nω +1− j/2) if j is even1/(nω +1− ( j+1)/2) otherwise

, 1≤ j ≤ N = 2nω

The DSUR scheme constructs for this example a solution which switches directly after each block with length(CU + ∆ −ε). Moreover, the controls i ∈ [nω−1] are active on block i so that the last control nω accumulatesthe asserted rounding gap until block nω . ut

Remark 6 (DSUR as generalization of SUR) The last corollary implicitly states that DSUR can be seen as ageneralization of the original SUR algorithm, since it reduces to the latter for a negligible MDT CU ,CD ≤ ∆ .

Thm. 3 allows no direct conclusion for the case with absent MU times and an active MD time CD > ∆ . Atleast, it is possible to provide worst-case examples for a in order to get a clue of how large the upper boundcan be for the DSUR rounding gap without MU times. This expresses the following proposition.


Proposition 4 (Rounding gap for DSUR without MU times) Consider nω ∈ 2,3. Let wDSUR be con-structed from a ∈ A by means of DSUR for an equidistant discretization of [t0, t f ]. Consider further an inac-tive MU time constraint, i.e., CU ≤ ∆ and let the MD time CD > ∆ . Then, the rounding gap θ(wDSUR) of its(CIA-D) objective value is bounded by the maximum possible rounding gap denoted by θ DSUR

max and the latteris at least

CD/2+(nω −2) ∆ ≤ θDSURmax .

Proof For the case nω = 2 we know from (2) in Prop. 5 in connection with Rem. 10 that CD/2 is a tight upperbound for (CIA-D) and it is therefore obvious that Alg. 2 may construct a solution with this objective value.We focus therefore on nω ≥ 3 and are going to construct an a-value whose corresponding wDSUR choiceinvolves a (CIA-D) objective value of CD/2+(nω −2) ∆ .

Let us first introduce the auxiliary constant N 3 MD := CD/∆ . In order to generate an example with aslarge control deviation θi, j as possible, we suggest that it can become large if control i is MD time forbiddenon interval j and earlier intervals. Thus, we want to activate a control shortly after other control activations,so that the latter may not yet be reactivated, but already have a large deviation value θi, j. For instance, withMD = 4 and nω = 3 and a given as

(ai, j)i∈[nω ], j∈[N] =

1 0 0 1 0 1 0 0 1 1 10 1 0 0 1 0 1 1 0 0 00 0 1 0 0 0 0 0 0 0 0

,

the values on the first five intervals are chosen according to the above idea. After the first two controls alreadyhave positive control deviation values θi, j in their MD phase, a is set so that the first control is selected bythe DSUR scheme due to large forward deviation values on the next MD intervals. This explains the valueson intervals 6-11. To sum up and illustrate this example, DSUR constructs the following binary solution:

(wDSURi, j )i∈[nω ], j∈[N] =

1 0 0 0 0 1 1 1 1 1 10 1 0 0 0 0 0 0 0 0 00 0 1 1 1 0 0 0 0 0 0

.

Notice that DSUR has to choose between two controls with equal forward control deviation on intervalsj = 2,3,7,8 and we deliberately select the worst-case option. Finally, when we take a closer look at thesecond row, we recognize

θ2,8 = 3∆ =CD/2+(nω −2) ∆ .

ut

We construct in Appendix C example values for more than three controls with the same rounding gap asabove.

Remark 7 (Rounding gap for DSUR with MU and MD constraints) Generally, when the problem settinginvolves both MU and MD time constraints, i.e., CD,CU > ∆ , the worst-case rounding gap constructed by theDSUR scheme is at least the maximum of the bounds provided in Cor. 2 and in Prop. 4.

7 Computational Experiments

We consider a three tank flow system problem with three controlling modes in order to evaluate the integralitygap in practice and to test the proposed rounding methods. It models the dynamics of an upper, middle anda lower level tank, connected to each other with pipes. The goal is to minimize the deviation of certainfluid levels k2, k4 in the middle, respectively lower, level tank. This problem type was discussed in a varietyof publications for the optimal control of constrained switched systems, e.g [10], and is taken from the


benchmark http://mintOC.de library [31]. The problem reads

minx,ω

∫ T

0k1(x2(s)− k2)

2 + k3(x3(s)− k4)2 ds

s.t. x1(t) =−√

x1(t)+ c1ω1(t)+ c2ω2(t)−ω3(t)√

c3x1(t), for a.e. t ∈ [0,T ],

x2(t) =√

x1(t)−√

x2(t), for a.e. t ∈ [0,T ],

x3(t) =√

x2(t)−√

x3(t)+ω3(t)√

c3x1(t), for a.e. t ∈ [0,T ],x(0) = x0,

1 =3

∑i=1

ωi(t), ω(t) ∈ 0,13, for a.e. t ∈ [0,T ].

(P)

The additional parameters are

k := (2,3,1,3)T , c := (1,2,0.8)T , x0 := (2,2,2)T , T := 12.

Furthermore, we add the MU and MD time constraints (3.4)-(3.5) to the three tank problem with varying CUand CD parameters.

We solve this test problem with the CIA decomposition, where we used CasADi v3.4.5 [3] to parse theNLP with efficient derivative calculation to the solver Ipopt 3.12.3 [35]. For finding the optimal solution ofthe resulting (CIA) problem and its MDT variants we used the branch and bound solver of the open sourcesoftware package pycombina1 [8].

Fig. 5 depicts the state and control trajectories with relaxed (DRCP) and binary (DBCP) control valuesand a required MU time of CU = 0.3. We remark that the binary solution’s objective value under MU timeconstraints is about 1.3% higher than its a priori lower bound obtained by the relaxed solution and about1.2% higher than the binary solution’s objective value without MU time constraints.

We illustrate in Fig. 6 the effect of an increasing MU time on the objective values of (CIA-U) and (DBCP).As expected, the finer the discretization grid and the smaller the required MDT time, the better the objecivevalues of both problems. A small MU time results in a weak restriction for (DBCP) so that its objective valueis close to the one of (DRCP), which is Φ = 8.776. But, from CU = 0.7 on, a refinement of the grid can-not compensate the MU time restriction anymore and the (DBCP) objective value is about 25% higher than(DRCP) in this case. Interestingly, this objective value increases hardly in CU ≥ 0.7 apart from one outlierinstance for N = 20 and CU = 1.2, which can be explained by the coarse grid. On the other hand, (CIA-U)objective’s value increases roughly linearly in CU on fine grids until it reaches θ ≈ 0.87, which seems tobe the maximum value for P in this setting. Thus, while low values of (CIA-U)’s objective correspond topromising objective values of (DBCP), the relationship (CIA-U) and (DBCP) appears to be quite uncorre-lated from CU ≥ 0.7 on. We received similar results for (P) with MD time constraints (not shown). We alsotested whether including the relaxed MDT constraints into the NLP or not has a significant impact on thesolution - this was not the case.

We analyze the performance of DNFR and DSUR for both MU and MD time constraints and with respectto θ in Fig. 7. The obtained solutions are compared with the global minima for (CIA-UD) from pycombina.We observe that DNFR seems to perform better for MU time constraints, while DSUR performs better forthe instances with MD time requirements. We plotted also the theoretical upper bound (UB) from Prop. 3,which is here 3

4C1, C1 = CU ,CD. As already observed for Fig. 6, the minima of (CIA-U) and (CIA-D) dohardly increase, if at all, for large MDTs and therefore diverge compared with the theoretical upper bound.We explain this behaviour by the problem specific given relaxed values, which induce here an objective valueof θ ≈ 0.87 for (CIA-U) and (CIA-D) even if no switches are used in the binary solution. We implementedDNFR and DSUR in Python 3.6 as additional solvers for pycombina. The execution of the heuristics tookfor all instances not more than 0.02 seconds on a workstation with 4 Intel i5-4210U CPUs (1.7 GHz) and7.7 GB RAM. We conclude that the heuristics can either be used to quickly generate robust solutions or toinitialize pycombina’s branch and bound algorithm with a good upper bound.

1 see https://github.com/adbuerger/pycombina

http://mintOC.de

https://github.com/adbuerger/pycombina


0 2 4 6 8 10 120

1

2

3

ω1ω2ω3

t

State and control trajectories for (DRCP)

x1x2x3

0 2 4 6 8 10 120

1

2

3

ω1ω2ω3

t

State and control trajectories for (DBCP)

x1x2x3

Fig. 5 Differential state and control trajectories for the test problem (P) with relaxed binary controls, i.e. for problem (DRCP), on theleft and approximated binary controls, i.e. for problem (DBCP), on the right with MU time CU = 0.3 and a temporal discretization withN = 1280 intervals. The objective value for (DRCP) amounts to Φ = 8.776, while the one of (DBCP) is Φ = 8.888.

101

102

103 0

1

20

0.5

N

CU

θ

(CIA-U) objective value

0.2

0.4

0.6

0.8

101

102

103 0 0.5 1 1.5 2 2.5

10

12

14

N

CU

Φ

(DBCP) objective value

9

10

11

12

13

14

Fig. 6 Objective values of (CIA-U) and (DBCP) based on the test problem (P) and different control discretizations N and MU timedurations CU .

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

CU

θ

(CIA-U) objective and heuristics

(CIA-U)DSURDNFR

UB by Prop. 1

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

CD

θ

(CIA-D) objective and heuristics

(CIA-D)DSURDNFR

UB by Prop. 3

Fig. 7 CIA objective function values θ for test problem (P) with time discretization N = 1280 and varying MU time CU (left), respec-tively varying MD time CD (right). The optimal solution is obtained via pycombina’s branch and bound algorithm and compared withthe solutions constructed by DNFR and DSUR.

8 Conclusions

In this article, we have derived tight bounds for the integrality gap of the CIA decomposition applied toMIOCPs under MDT constraints. The presented proofs are constructive and take advantage of the introducedDNFR scheme. Numerical experiments show that the CIA decomposition performs notably well in termsof objective quality for problems with small MDT requirements, since the deviation from its a priori lower


bound is negligible. For more restrictive dwell time constraints, the algorithm may provide solutions thatdiffer a lot from this bound. Hence, the constructed solution quality might be low compared to the optimalsolution. Nevertheless, when considering the runtime of MINLP solvers the CIA decomposition solution iscomputationally inexpensive and can be easily assessed with the a priori lower bound for its quality. Wehave extended the SUR scheme so that it constructs dwell time feasible solutions and tested the proposedalgorithm on a benchmark problem. The resulting (CIA) problem solutions of this and the DNFR scheme areclose to the optimal ones, which is why we propose that the DNFR or DSUR scheme may be beneficial inthe context of huge (CIA) problems or in the setting of model predictive control, where a branch and boundalgorithm struggles to find the optimal solution efficiently.

A Proof of Theorem 1

Proof We proceed via induction.Base case: We consider the first b on which a MD time forbidden control iD ∈ [nω ] appears and let us assume it holds

ΓiD ,b!> 1.5L . (A.1)

It follows from Lemma 2

1.5 ·L ≥Lb = ∑i6=iD

Γi,b +ΓiD ,b,

so that there must be a control i1 6= iD with negative forward control deviation on b:

∃ i1 6= iD : Γi1 ,b < 0.

We apply Lemma 4 to the last inequality: i1 has not been b-future forced on its last activation and we denote the time block of thisactivation with b1. In other words, we know that there is at least one block block b1 and one control which was not b-future forced onb1 and still was activated on b1. Now, we denote by i1 the control of this property with the last activation before b. By this definition, weobserve that all controls being activated after b1 would become forced until b. We notate

Fb1 ,b := i ∈ [nω ] | ∃k(i) : b1 < k(i)≤ b on which i is forced or b-future forced.

In particular, it holds iD ∈ Fb1 ,b. For i ∈ Fb1,b\iD we conclude

Γi,b =Θi,b−1 + ∑j∈Jb

ai, j∆ j =b

∑k=1

∑j∈Jk

ai, j∆ j−k(i)−1

∑k=1

∑j∈Jk

wi, j∆ j!> 1.5 ·L ,

and thereforeΓi,b > 1.5 ·L −Lk(i), i ∈ Fb1 ,b\iD. (A.2)

The last inequality holds, since control i was last activated at block k(i). For block b1 we know that i1 has been chosen, despite notbeing b-future forced. We use this observation and our assumption of b > b1 being the first block with a MD time forbidden control toconclude that all controls out of Fb1 ,b have been inadmissible on b1. Hence, it results for i ∈ Fb1 ,b

Γi,b1 <−1.5 ·L +Lb1 , ⇒ Θi,b1 <−1.5 ·L +Lb1 . (A.3)

We sum up the inequalities (A.1) and (A.2) over Fb1 ,b and similarly for (A.3), which yields

∑i∈Fb1 ,b

Γi,b > 1.5 ·L +(|Fb1 ,b|−1) ·(1.5 ·L −Lb2

), (A.4)

∑i∈Fb1 ,b

Θi,b1 < |Fb1 ,b|(−1.5 ·L +Lb1

), (A.5)

where we set b2 := argminLk | b1 < k ≤ b and notate with |Fb1 ,b| the cardinality of Fb1 ,b. Subtracting (A.5) from (A.4) results in

∑i∈Fb1 ,b

(Γi,b−Θi,b1

)> 1.5 ·L +(|Fb1 ,b|−1) ·

(1.5 ·L −Lb2

)−|Fb1,b| · (−1.5 ·L +Lb1 )

= 1.5 ·L +(2|Fb1,b|−1) ·1.5 ·L − (|Fb1,b|−1) ·Lb2 −|Fb1 ,b| ·Lb1

≥ 1.5 ·L +(2|Fb1,b|−1) ·0.5 ·L

> L . (A.6)

We used Lb1 ,Lb2 ≤L in the second inequality. We finish our calculations by considering the property of Fb1 ,b comprising all controlactivations between time blocks b1 +1 and b−1, therefore we can apply Lemma 3 with Fb1 ,b = Sb1 ,b and obtain

∑i∈Fb1 ,b

(Γi,b−Θi,b1

)≤L , (A.7)


which is a contradiction to inequality (A.6).

Step case: Let the assertion hold until any block b−1 ∈ [nb] and we prove that then the statement holds for b. Again, we consideriD ∈ [nω ] and assume it holds

ΓiD ,b!> 1.5L . (A.8)

With a similar argumentation as in the base case we deduce that there is control i1 which has not been b-future forced on block b1 < band reuse the definition of Fb1 ,b. Thus, inequality (A.2) still holds. Now, we distinguish between two cases why the controls out of Fb1,bhave not been activated on b1. If all controls i ∈ Fb1 ,b have been inadmissible on b1, we can argue as in the base case. Hence, we focuson the other case: there is an i2 ∈ Fb1 ,b, which was MD time forbidden on b1 and all other controls i ∈ Fb1 ,b\i2 were inadmissible. Bythe induction hypothesis and the previously derived inequality (A.3) we have

Θi2,b1 ≤ 1.5 ·L , Θi,b1 <−1.5 ·L +Lb1 , i ∈ Fb1 ,b\i2.

Summing up these inequalities over Fb1 ,b results therefore in

∑i∈Fb1 ,b

Θi,b1 < 1.5 ·L + |Fb1,b−1|(−1.5 ·L +Lb1

). (A.9)

Next, we argue that Fb1,b contains at least two controls: the case b1 = b− 1 is not possible, since iD is via assumption forced and MDtime forbidden on b, hence admissible on b−1. Therefore, b1 ≤ b−2 and there is a control i 6= iD, i ∈ Fb1 ,b, which is activated on b−1.Altogether we have |Fb1 ,b| ≥ 2. With this observation we subtract inequality (A.9) from (A.4):

∑i∈Fb1 ,b

(Γi,b−Θi,b1

)> 1.5 ·L +(|Fb1 ,b|−1) ·

(1.5 ·L −Lb2

)−(1.5 ·L +(|Fb1 ,b|−1) · (−1.5 ·L +Lb1 )

)= 3(|Fb1 ,b|−1)L − (|Fb1 ,b|−1) ·Lb2 −|Fb1 ,b−1| ·Lb1

≥ (|Fb1 ,b|−1)L

≥L .

Notice that |Fb1 ,b| ≥ 2 is used in the last inequality. Finally, we build again on Lemma 3, where it is justified to set Fb1 ,b = Sb1 ,b. Thus,the above inequality is a contradiction to the inequality from the lemma and we have shown that the assertion holds for all b ∈ [nb] onwhich a MD time forbidden control exists. ut

B Discussion on the Tightness of the Provided Minimum Down Time Bounds

Proposition 5 (Tightness of the bound for (CIA-D)) Let us assume the MD time constraint is active, i.e., CD > ∆ is given. Then, thefollowing is true:

1. The bound for (CIA-D) stated in Prop. 3 can not be improved by the DNFR scheme with χD = 1 for nω ≥ 3.2. The bound for (CIA-D) is tight up to at most the constant 1

4CD + ∆ .

Proof The assumption of an active MD time constraint ensures that the bound cannot be improved by the bound for MU times fromProp. 2. We use again the concept of a minimal C1-overlapping grid, here with C1 =CD/2.

1. We want to prove that the DNFR scheme with χD = 1 and C2 < 1.5 may provide solutions with a (CIA−D) objective greater thanC2 ·L . Let us consider first C2 ≤ 1.5− ε1, with 0 < ε1 ≤ 0.5. We present example values for a with a time horizon length of atleast 12 ·C1, so that at least six blocks with length L exist by Lemma 5. Let 0 < ε2 < ε1 be small and let the relaxed control values(ai,b)i∈[nω ],b∈[nb ]

be given as

a =

1 0.5− ε1 + ε2 1− ε2 2ε1−2ε2 0.5 0.5 · · · 0.50 0.5+ ε1− ε2 0 1−2ε1 +2ε2 0.5 0.5 · · · 0.50 0 ε2 0 0 0 · · · 00 0 0 0 0 0 · · · 0...

......

......

.... . .

...0 0 0 0 0 0 · · · 0

.

With this example values we discuss the thereby constructed DNFR solution as well its objective quality.- First block: i1 is 2-future forced and activated.- Second block: Both i1 and i2 are 4-future forced. The DNFR algorithm breaks ties arbitrarily, so that activating i2 is legitimate.- Third block: i1 is MD time forbidden, while i2 is not admissible. DNFR activates therefore i3.- Fourth block: i1 is activated, since it is forced.- Fifth block: We have in the meantime Θi1 ,4 = (0.5+ε1−2ε2)L and Θi2 ,4 = (0.5−ε1 +ε2)L . Since ε2 satisfies ε2 < ε1, both

controls are 6-future forced on the fifth block. Let DNFR activate i2.- Sixth block: i2 is still MD time forbidden and can not be active, which implies

Θi1,6 = (0.5+ ε1−2ε2 +1)L > (1.5− ε1)L =C2 ·L ,

so that the proposed control deviation bound is not fulfilled.


Finally, if ε1 > 0.5, thus C2 < 1, we can construct a similar example for which control i1 is already forced on the first block and thecontrol deviation is again greater than C2 ·L .

2. The MD time constraints are equivalent to MU time constraints with CU =CD for a problem with only two controls nω = 2. Prop. 2provides an example for this case, where θ ≥ 1

2 (CU + ∆) holds. This example can be also applied for more than two controls bysetting the relaxed control values ai,b to zero, for i > 2. Then, the difference to the upper bound 3

4CD + 32 ∆ from Prop. 3 is the one

stated in the assertion.ut

Prop. 5 tells us the DNFR scheme with C2 = 1.5 and χD = 1 can not be improved. The following example motivates why we havechosen the set of the MD time forbidden control in such a way that the active control can only be changed after a duration of CD/2 atthe earliest. If it is already possible to switch after one interval ∆ j , DNFR may construct greedy solutions with a large control deviationat long MD times CD. The following example shows the reason for this.

Example 1 Let nω = 3 and a rounding threshold C2 ≥ 2nω−32nω−2 be given. We alter DNFR in the following way: instead over blocks we

iterate forward over all intervals, but we keep for a given MD time C1 =CD the threshold C2 ·L for forced, future forced and admissibleactivation. In order to construct feasible solutions for (CIA-D) we extend the definition of I D

b by letting all controls to be MD timeforbidden that are inactive but were active in the previous period of length CD. Next, we are going to supply exemplary relaxed valuesfor this modified DNFR scheme with large control deviation. We first introduce recursively the indices

ji := min

j ∈ [N] | ∑jl= ji−1

∆l >C2L, i = 1,2,3,

where j0 := 1. Let the relaxed values be given as follows

(ai, j)i∈[nω ], j∈[N] =

1 · · ·j1︷︸︸︷1 0 · · ·

j2︷︸︸︷0 0 · · ·

j3︷︸︸︷0 · · ·

0 · · · 0 1 · · · 1 0 · · · 0 · · ·0 · · · 0 0 · · · 0 1 · · · 1 · · ·

,

Then, the modified DNFR may construct the following solution:

(wi, j)i∈[nω ], j∈[N] =

1 0 0 · · ·j4︷︸︸︷0 · · ·

0 1 0 · · · 0 · · ·0 0 1 · · · 1 · · ·

,

where j4 := min j ∈ [N] | ∑l∈[ j] ∆l <CD +∆1 is the last index before the MD phase of i1 ends. At first, i1 is earliest future forced, butnot anymore after being active on j = 1. Then, i2 is activated on the second interval before i3 is the earliest future forced control andneeds to stay active until j4, since other controls are MD time forbidden. We notice that with small ∆1 it can hold j4 ≤ j2 and therefore

|θi3, j4 |= |j4

∑l=3

(0−1)∆l | ≤ |CD + ∆ −∆2| ≤CD + ∆ −∆ .

If we compare the term on the right inequality side with the bound from Prop. 3, i.e., 2nω−32nω−2 (CD + ∆) = 3

4 (CD + ∆), we conclude thatthe latter is smaller only if CD < 4∆ − ∆ . Since ∆ can be arbitrarily small and we assumed CD big compared with the grid length, themodified DNFR scheme would provide no improving bounds. Similiar “greedy” examples can be constructed for nω > 3 and blocklengths greater than ∆ j .

Some comments on these sharpness properties are in order.

Remark 8 (Quality of bound for (CIA−D)) The MD time configuration of DNFR, i.e., χD = 1, yields smaller upper bounds comparedwith the DNFR algorithm with MU time configuration, i.e. χD = 0 and C1 =CD, only for instances with more than three controls and alarge MD time CD compared with the grid length ∆ . In fact, we conjecture the upper bound for any nω to be θ max = 1

2CD + ∆ , thereforeonly slightly greater than the one for nω = 2. With this threshold, there would be no forced control until the first MD time forbiddencontrol appears and we postulate that active controls which become forced without activation during the next CD time duration may stayactive without other controls becoming forced. Of course, this argumentation justifies no proof - Prop. 5 together with Example 1 statesthat a generic solution fulfilling this bound can not be found via the DNFR scheme and it is presumably hard, if not even impossible, toconstruct it by another polynomial time algorithm.

Remark 9 (Quality of bound for (CIA−UD)) The integrality gap bound for (CIA−UD) as stated in Prop. 3 is tight for CU ≥CD bythe result of Prop. 2. For CU <CD, the bound is not necessarily sharp, but it is again difficult to prove tight bounds due to the problem’scombinatorial structure.

Remark 10 If we deal with MDT C1 that begin and end exactly on the grid points, the upper bounds become 2nω−32nω−2CU for (CIA−U),

34CD for (CIA−D), and accordingly reduced for (CIA−UD).


C Bounds for DSUR with Minimum Down Time and nω > 3

Remark 11 (Rounding gap for DNFR without MU times) Consider the assumptions of Prop. 4, but with nω > 3. Then, there are examplesresulting in a rounding gap θ(wDSUR) of at least

CD/2+(nω −2) ∆ ≤ θ(wDSUR).

Proof We construct example values a in a similar way as in the proof of Prop. 4 for the case nω = 3. First, we assume an instance withMD time and number of intervals chosen sufficiently large in the sense of

MD ≥ 2(nω −1), N = (nω −1)(MD +1)+ dMD/2e−1.

Algorithm 3 creates with these prerequisites a matrix of a-values, which reuses the essential two ideas of the example in the proof ofProp. 4.

Algorithm 3: Construction algorithm for a from proof to Rem.11.1 Initialize a = 0.2 for k = 0 . . .(nω −3) do3 for j = 1+ kMD . . .kMD +2(nω − k) do4 Set ai, j = 1, with i≡ j− kMD ( mod nω ).5 end6 for j = kMD +2(nω − k)+1 . . .(k+1)(MD +1) do7 Set ai, j = 1, with i = nω − k.8 end9 end

10 Set a1, j = 1, with j = (nω −2)(MD +1)+1.11 for j = (nω −2)(MD +1)+2 . . .(nω −2)(MD +1)+ bMD/2c+1 do12 Set a2, j = 1.13 end14 if MD is odd then15 Set a1, j = a2, j = 0.5, with j = (nω −2)(MD +1)+ dMD/2e+1.16 end17 for j = (nω −2)(MD +1)+ dMD/2e+2 . . .(nω −1)(MD +1)+ dMD/2e−1 do18 Set a1, j = 1.19 end20 return: Relaxed control values a;

Firstly, for all controls 3≤ i≤ nω sequences of intervals with length MD are created on which the controls i = 1,2 already accumu-late large positive θ -values due to their MD time (lines 2-9). Secondly, the values for the first two controls are then set in a way so thatDSUR activates control i = 1 based on the large summed-up relaxed values a1, j on the next MD intervals (lines 10-19). Thus, DSURconstructs the following solution wDSUR for a as given in Alg. 3

wDSURi, j =

1 if i = 1 ∧ j ∈ 1+ k ·MD | k ∈ [nω −3]0∪1+(nω −2) ·MD + k | k ∈ [MD +nω −3+ dMD/2e]0,1 else if i = 2 ∧ j ∈ 2+ k ·MD | k ∈ [nω −3]0,1 else if i≥ 3 ∧ j ∈ i+ k ·MD | k ∈ [nω − i]0∪i+(nω − i) ·MD + k | k ∈ [MD +1− i],0 else.

Finally, the rounding gap θi, j for control i = 2 and on interval j = (nω −1)(MD +1)+ dMD/2e−1− (D−1) is again the one stated inthe assertion. ut

Acknowledgements This project has received funding from the European Research Council (ERC) under the European Union’s Hori-zon 2020 research and innovation programme (grant agreement No 647573) and from Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) – 314838170, GRK 2297 MathCoRe and SPP 1962.

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