ELSEVIER Computers and Chemical Engineering 24 (2000) 1361-1366

Computers & Chemical Engineering

www.elsevier.com/locate/compchemeng

Generalized branch-and-cut framework for mixed-integer nonlinear optimization problems

Padmanaban Kesavan, Paul I. Barton *

Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, M.4 02139, USA

Ab~ra~

Branch and bound (BB) is the primary deterministic approach that has been applied successfully to solve mixed-integer nonlinear programming (MINLPs) problems in which the participating functions are nonconvex. Recently, a decomposition algorithm was proposed to solve nonconvex MINLPs. In this work, a generalized branch and cut (GBC) algorithm is proposed and it is shown that both decomposition and BB algorithms are specific instances of the GBC algorithm with a certain set of heuristics. This provides a unified framework for comparing BB and decomposition algorithms. Finally, a set of heuristics which may be potentially more efficient computationally compared to all currently available deterministic algorithms is presented. © 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Branch-and-cut algorithms; Mixed-integer nonlinear optimization; Decomposition heuristics

1. Introduction

Affixed-integer optimization problems belong to one of the most difficult classes of problems with respect to the computational complexity involved. All determinis- tic algorithms currently available to solve mixed-integer problems of generic type are of exponential complexity and no polynomial time algorithms are known at present to solve these problems. Two classes of expo- nential time algorithms, namely branch and bound (BB) (Ryoo & Sahinidis, 1995; Adjiman, Schweiger & Floudas, 1998b) and decomposition algorithms (Kesa- van, Allgor & Barton, 1999; Kesavan & Barton, 1999a), have been proposed and applied successfully to solve mixed-integer nonlinear problems (MINLPs) in which the participating functions are nonconvex. BB al- gorithms are developed on the basis of a divide and conquer strategy. The basic idea of BB algorithms is to use a binary tree to represent all feasible combinations of the discrete variables. At different levels of the BB tree, the feasible region is divided into subdomains and valid lower and upper bounds are generated systemati- cally (Floudas, 1995). Decomposition algorithms are

* Corresponding author. Tel.: + 1-617-2536526; fax: + 1-617- 2539695.

E-mail address: pib@mit.edu (PT Barton)

based on solving a sequence of subproblems to generate lower bounds, valid outer approximations and upper bounds. The sequence of nondecreasing lower bounds is generated by solving a so called relaxed master problem which is easier to solve (with respect to the existence of algorithms that yield the global optimum) compared to the original problem. The upper bound is obtained by solving the original problem with the in- teger variables fixed to the current solution of the relaxed master problem.

The Outer approximation decomposition algorithm (Duran & Grossmann, 1986; Fletcher & Leyffer, 1994) for convex MINLPs derives linearizations of the non- linear constraints about the solution of the upper bounding problem. In the case of nonconvex MINLPs these linearizations are not valid since they cut off portions of the feasible space (Kocis & Grossmann, 1988) and the algorithm may converge to a suboptimal solution. A rigorous decomposition algorithm for non- convex MINLPs was proposed recently (Kesavan et al., 1999; Kesavan & Barton, 1999a). Valid linearizations are derived in this algorithm by solving an additional problem (convex NLP) corresponding to each integer combination. The BB algorithms for nonconvex MINLPs (Ryoo & Sahinidis, 1995; Adjiman et al., 1998b) generate lower bounds at each node of the BB tree by relaxing the integer variables to be continuous

0098-1354/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved. PII: S0098-1 3 5 4 ( 0 0 ) 0 0 4 2 1 - X

1362 P. Kesavan, P.L Barton/Computers and Chemical Engineering 24 (2000) 1361-1366

(which win yield a nonconvex NLP) and solving a convex NLP obtained by constructing valid underesti- mators for the nonconvex terms in the nonconvex NLP. The upper bound is usually obtained by solving the nonconvex NLP for a local solution. In Section 4 a high level description of the GBC algorithm for mixed- integer problems is presented. In Sections 5 and 6 the decomposition algorithm (Kesavan et al., 1999; Kesa- van & Barton, 1999a) is shown to be a specific instance of the GBC algorithm and an instance of the GBC algorithm with a set of heuristics which may potentially more efficient (on average) is discussed.

2. Generalized branch and cut algorithm

The class of MINLPs considered in this section con- form to the following formulation:

min f(x, y) x,y

s.t. gx(x, y) < 0

gz(X, y ) <7 0

x6 U c R ~

y ~ U c Y c R q (1)

where the scalar valued function f : X × Y ~ R and the vector valued function g~:X x Y--*R" are nonconvex, and the vector valued function gz:X × Y ~ R p is convex. X is a nonempty, compact, convex set defined by X = {x:x~R n, Ax <7 c}. U is a discrete set defined by U= {y:yeY, integer} where Y is a convex polytope. Y= [0, 1] q is assumed for the purposes of discussion. Equality constraints can be represented by a pair of inequalities which then conform to the form defined by Eq. (1). The problem as defined by Eq. (1) will be referred to as P hereafter.

The GBC algorithm to find the global solution of nonconvex MINLPs is based on the derivation of the following subproblems.

2.1. Lower bounding convex M I N L P

The solution of which yields a valid lower bound to the global solution of the nonconvex MINLP. Let L~(x, y) and L2(x, y) represent convex underestimators for f (x , y) and gl(x, y), respectively. The lower bound- ing convex MINLP (referred to as P1 hereafter) is:

min L6x , y) x.y

s.t. L2(x, y) <_ 0

g2(x, y) < 0

x e X c R n

y ~ U = {0, 1} q (2)

2.2. Relaxed problem

The global solution of which represents a valid lower bound to that subset of the integer combinations (cor- responding to the leaf nodes of the BB tree) not yet explored by the algorithm. Several alternatives exist to derive the relaxed problem. For example, the relaxed problem can be obtained by relaxing the integer vari- ables to be continuous (will yield a nonconvex NLP) in the original problem. In this case, if the global solution of the relaxed problem satisfies the integrality con- straints, then the solution corresponds to the global solution of the original nonconvex MINLP. Alterna- tively, the relaxed problem can be derived by construct- ing a lower bounding convex MINLP, then deriving the outer approximation (Duran & Grossmann, 1986; Fletcher & Leyffer, 1994) master problem of the convex MINLP, followed by deletion of some of the con- straints of the master problem. The relaxed problem thus obtained is a MILP with a finite number of constraints.

3. Lower bounding problem (LBP)

A continuous optimization problem which can be solved to guaranteed global optimality. The solution of the LBP will yield a valid lower bound to the solution of the original problem at the current node and all its descendant nodes of the BB tree, and an upper bound to the solution of the original problem under certain conditions. For example, let the relaxed problem be obtained by relaxing the integer variables to be continu- ous. Then, the LBP at any node of the BB tree is obtained by fixing some subset of the integer variables in the relaxed problem and relaxing its complement to be continuous. The LBP solved at any node is a non- convex NLP which can be solved by BB (Ryoo & Sahinidis, 1995; Adjiman, Androulakis, Floudas & Neumaier, 1998a). In this case, if an integer feasible solution to the LBP is obtained at any node, then the solution represents a valid lower bound to the solution of the original problem at the current and all its descendant nodes, and an upper bound to the global solution of the original problem. If the relaxed problem is a MILP, then the LBP solved at any node is a LP (obtained by fixing some subset of the integer variables in the relaxed problem). An integer feasible solution at any node is only a lower bound in this case. A LBP may also be obtained by constructing convex underesti- mators for the nonconvex terms of the nonconvex NLP at the current node (which will then yield a convex NLP). In this case also, the solution of the LBP repre- sents only a valid lower bound irrespective of whether it satisfies the integrality constraints or not.

P. Kesavan, P.L Barton~Computers and Chemical Engineering 24 (2000) 1361-1366 1363

3.1. Upper bounding problem/primal problem

The solution of which represents a valid upper bound to the global solution of the original problem. If the relaxed problem is an MILP, then the upper bounding problem can be obtained by fixing the integer variables in P. If the relaxed problem is obtained by relaxation of the integer variables to be continuous in the original problem, then an integer feasible solution at any node of the BB tree represents a valid upper bound to the original problem. If the relaxed problem is a nonconvex NLP and the LBP is a convex NLP, then the upper bound can be obtained by solving for a local solution of the nonconvex NLP, obtained by fixing the integer variables to the solution of the LBP.

3.2. Primal bounding problem

Which is a convex NLP, the solution of which pro- vides a valid and tighter lower bound to the Primal problem for each fixed binary realization y than that provided by the current LBP. The primal bounding problem is obtained by fixing the integer variables in P1. Valid linearizations or cuts to be augmented to the relaxed problem can be derived about the solution of the primal bounding problem.

The distinction between a BC algorithm and a BB algorithm used in the rest of the paper is the derivation and addition of cuts (constraints) to the relaxed prob- lem by the BC algorithm which are valid for all nodes of the BB tree. These will improve the quality of lower bounds obtained at future nodes visited in the BB tree.

4. High level description of the GBC algorithm

A BB tree is considered (which may be only integer variables, only continuous variables or both depending on the heuristic). The BB tree is initialized with the root node.

LOOP 1. Node selection

1.1. Check termination criteria. If satisfied, EXIT. 1.2. Choose an active node n,. based on a heuristic.

2. ShouM the lower bounding problem be solved? If yes, solve the lower bounding problem and update the lower bound.

3. ShouM the upper bounding problem be solved? If yes, solve the upper bounding problem and update the upper bound.

4. Should cuts be generated? If yes, derive cuts valid for the entire BB tree.

5. Branching versus refathoming. Should the problem be resolved with the new relaxed problem (aug- mented with cuts)? If yes, refathom the BB tree with the new relaxed problem and the current UBD.

6. Branching step. If the current node is not a leaf node, choose a branching variable (based on a heuristic), create children nodes and add them to current set of active nodes. Delete n~ from the current set of active nodes.

END LOOP

There are several decisions which need to be made in the GBC algorithm. These are listed below: 1. Node selection. The active node to be considered

may be selected in several different ways. An exam- ple is to choose the node with the lowest lower bound.

2. Lower and upper bounding problems. Several choices exist for the problems to be solved for the lower and upper bounds as described previously. Decisions also have to be made as to whether to solve for the lower bound or the upper bound or both at a particular node.

3. Cut generation. There are several alternatives to generate cuts and the nodes at which the cuts are derived. For example, cuts may be derived from the solution of the Primal Bounding problem and/or lower bounding problems (e.g. Gomory cuts), an integer cut (Balas & Jeroslow, 1972) excluding the integer combination corresponding to the current node, or by solving an additional problem (Stubbs & Mehrotra, 1999).

4. Branching and refathoming. The children nodes may be created in several different ways (branching rule). Instead of branching, the nodes may be refathomed starting at any node (not necessarily the root node) once the cuts are generated.

The conditions under which each of the six steps in the GBC algorithm are executed also have to be specified. In general, there is no rigorous procedure to make these choices for the GBC algorithm. In other words, the choices made define the set of heuristics employed.

5. Heuristics employed by the decomposition algorithm to solve nonconvex MINLPs

A decomposition algorithm to solve P has been pro- posed recently (Kesavan et al., 1999; Kesavan & Bar- ton, 1999a). The set of heuristics inherent in these algorithm are listed below: 1. Relaxed problem. A MILP (the relaxed master prob-

lem of the decomposition algorithm (Kesavan et al., 1999; Kesavan & Barton, 1999a)), the solution of which represents a valid lower bound to the solution of that subset of U not yet explored by the algorithm

2. Lower bounding problem. A L P obtained by fixing a subset of the integer variables (corresponding to a node in the BB tree) in the relaxed problem and relaxing its complement to be continuous.

1364 P. Kesavan, P.L Barton/Computers and Chemical Engineering 24 (2000) 1361-1366

3. Upper bounding problem. A nonconvex NLP, ob- tained by fixing all the integer variables in P. Solve (globally) the NLPs corresponding to the integer combinations for which the solution of the primal bounding problem is less than or equal to the solution of the relaxed problem. The NLPs are solved successively starting with the integer combi- nation corresponding to the minimum of the primal bounding solutions, until the minimum of the pri- mal bounding solution is greater than the solution of the relaxed problem.

4. Node and branching variable selection. Use the heuristic as employed by the decomposition al- gorithm (Kesavan et al., 1999; Kesavan & Barton, 1999a) to solve the relaxed master problem.

5. Cut generation. Derive cuts at the node correspond- ing to the solution of the MILP. The cuts are derived by linearization of nonlinear constraints and the objective function of P1 about the solution of the primal bounding problem (referred to as NLPB hereafter). If the NLPB is not feasible, derive cuts by solving a feasibility problem (Fletcher & Leyffer, 1994).

6. Branching versus refathoming. Branch and create child nodes until the solution of the relaxed problem is obtained. Augment the relaxed problem with the cuts derived (valid for all nodes of the BB tree) and resolve the relaxed problem (i.e. refathom all the nodes starting at the root node of the BB tree) with the current UBD. The nodes are refathomed start- ing at the root node as soon as cuts are derived rather than branching and continuing the unfath- omed active nodes.

Remark 1. The upper bounding problem (nonconvex NLP) is the most computationally intensive problem in the algorithm. Step 3 above ensures that the minimum number of primal~upper bounding problems are solved to find the global solution of P (Kesavan et al., 1999; Kesavan & Barton, 1999a). Although all the decisions mentioned above correspond to a set of heuristics in the GBC framework, the number of primal problems solved is not a heuristic.

6. BC algorithm for solving nonconvex MINLPs

Let UBDPB represent the minimum of the solutions of the Primal bounding problems solved. Initialize: 1. Choose an integer combination y~ and U B D - +

oo, U B D P B = +oo, l = 0 , S °=O, T~I=O, U°=O, k = l .

2. Solve NLBP(y 1) or feasibility problem if NLBP(y l) is unfeasible, and let the solution be x ~. Update, (Slk=S~ -1 and T k = T k - l w { k } ) i f NLBP(y 1) is feasible or (S k = S k - lw{k} and T k = T k- l ) if NLBP(y 1) is not feasible.

3. IF (NLBP(y 1) is feasible and Ll(xl, y 1) < UBDPB) THENupdate current best point by setting x* = x 1, y , =y l , k* = k and UBDPB = Ll(x 1, yl).

4. Derive the lower bounding problem (LBP), an MILP, by linearizing the objective and constraints (active and/or inactive) of P1 about (x 1, y 1).

Let the LBP to be solved be given as:

min fl x,y,fl s.t. A x + B y + e f l - d < _ O

xeX, ye U (3)

where A, B, and c, d are conformable matrices and vectors, respectively. Let L~ represent the current set of valid inequalities for LBP, and the linear inequalities defining U, contains at least the constraints, Ax + By + cfl - d< O, xeX, ye Y. Let Fo, F1 --- {1, 2 . . . . . q} be the set of y's that have been fixed at 0 and 1 respectively. Define K(~3, Fo, F0 as,

{x,y, f l : A x + B y + c f l - d < O , y i = 0 for ieFo,

Yi= 1 for ieF1, xeX, ye Y} (4)

and let LP(L~, Fo, F1) denote the LP,

rain fl x,y,fl x y ticK(D, Fo, FO (5)

The active nodes of the enumeration tree are repre- sented by a list p of ordered pairs (F0, F1). Let UB represent the solution of the current LBP (a new LBP is obtained whenever (3 is updated by augmenting addi- tional cuts/constraints to the LBP) and let lbd,., iep represent a valid lower bound to the solution of the LP at any node. For example, lb~ for any node can be set equal to the solution of the ancestor node.

LOOP I. Refathoming. Set p = {(Fo = O, F1 = 0)}, UB = + oo,

lbdrootnoae = -- o0. 2. Node selection. If (UB >_ UBDPB or UB >_ UBD)

and (p = 0 or min lbd i > UB), go to Step 6. If (p = 0 i~p

min/~p lbdi >_ UB), go to Step 5. Otherwise, choose an ordered pair (F o, Fl)ep based on a node selection heuristic and remove it from p.

3. Lower bounding step. Solve LP(L~, Fo,F1). If the problem is not feasible go to Step 2. Otherwise, let the solution be yk with corresponding objective value ilk. If flk>_UB, go to Step 2. I f y k = {0, 1}, update UB = ilk, yV =yk and go to Step 2.

4. Branching. Choose a branching variable yj, je{1,2 . . . . . q} and 0 < y j < l (according to a

P. Kesavan, P.L Barton/Computers and Chemical Engineering 24 (2000) 1361-1366 1 3 6 5

branching rule) and generate the subproblems corre- sponding to (F0w {j }, F,) and (Fo, F1 u {j }), set lbdi for the subproblems to flk and add them to p. Go to Step 2.

5. Cut generation. Set k = k + 1. Solve NLPB(y v) or the feasibility problem if NLPB(y v) is not feasible, and let the solution be x v. If NLPB(y v) is feasible, and L~ (x ~, y v) < UBDPB, update UBDPB = Ll(x~,y~), x * = x ~, y*=y~, k*=k. Linearize the objective and constraints (active and/or inactive) about (x ~,y~) and add these constraints to U. Set (S~ = S~ -~ and T~ = T1 k-~ w {k}) if NLPB(y v) is feasible or (S~=S~- lw{k} and T1 k = T ~ - l ) if NLPB(y ~) is in feasible. Go to Step 1.

6. Upper bounding step. If (UBDPB > UBD) and (UB > UBD), EXIT. 6.1. Set l = l + 1, solve the NLP(y*), and let the

solution be x~. Update U~ = U~- ~ w {k* }. If (NLP(y*) is feasible and f (x ~, y*) < UBD) then update current best point by setting, x* = xV v* - v* UBD =f(Xp, y*). Update the con- p , 1 p - - .Y

straint fl < UBD in U. 6.2. If T~\U~ 50, update UBDPB=

k l min(Ll(xm, ym)), meTl\U1 (UBDPB corre- sponds to the primal bounding solution of (xS, yS)). Update x * = x ~, y , = y S and k*=s. Otherwise, set UBDPB = + ~ . Go to Step 2.

END LOOP

On exit the solution of P is given by UBD, x*, y~. This algorithm is equivalent to that described in Kesa- van and Barton (1999a), but this restatement empha- sizes it as a BC algorithm. A set of heuristics analogous to the above, except that the nodes are not refathomed starting at the root node once a cut is generated can also be employed in the GBC algorithm. It has been shown (Kesavan & Barton, 1999b) that this heuristic is computationally superior on average for convex MINLPs. Since the main difference between the al- gorithms for solving convex and nonconvex MINLPs with a MILP relaxed problem is that the latter solves an additional primal bounding problem to derive valid linearizations, the no refathoming heuristic can be ex- pected to perform better computationally (on average) as compared to the decomposition heuristic for solving nonconvex MINLPs. It is also easy to verify that the BB algorithms proposed previously (Ryoo & Sahinidis, 1995; Adjiman et al., 1998b) are all specific instances of GBC algorithm with a certain set of heuristics.

linearizations and upper bound constraint (i.e. fl < UBD) are discussed below.

Cuts can be derived by lift and project (Balas, Ceria & Cornu~jols, 1996a; Balas, Ceria, Cornu6jols & Na- traj, 1996b; Nemhauser & Wolsey, 1990; Padberg & Rinaldi, 1991) from the nonintegral solution of the relaxed problem at any node and can be augmented to ~3. Recently, valid cuts based on the convex hull of the MINLP have been proposed (Stubbs & Mehrotra, 1999) to solve convex MINLPs. The relaxed problem employed in this case was obtained by relaxing the integer variables to be continuous (convex NLP) in the convex MINLP. The cuts are derived at the nodes with a non integral solution of the relaxed problem. Addi- tional problems called separation problems (convex NLPs) are solved with respect to all or some subset of integral variables which have nonintegral solution at the current node.

It can be inferred from Lemma 1 of Kesavan and Barton (1999b), using the observation that the convex hull of P is a subset of P1, the cuts derived from the convex hull of P1 (Stubbs & Mehrotra, 1999) will be tighter (i.e. will cut off a larger portion of the feasible space of MILP that is not in the convex hull of the P1) compared to that derived from the lift and project cuts (Balas et al. 1996a,b; Nemhauser & Wolsey 1990; Pad- berg & Rinaldi 1991). However the derivation of cuts from the convex hull of the P1 requires solving addi- tional convex NLPs at the nodes corresponding to nonintegral solutions, whereas the lift and project cuts require solving additional LPs. Decisions regarding the type of cuts and the number of rounds of cuts have to be based on heuristics.

If the solution of the LBP at any node is not feasible, then an integer cut (Balas & Jeroslow, 1972) can be added to the relaxed problem as suggested by Duran and Grossmann (1986).

Finally, it should be noted that the cuts when aug- mented to the relaxed problem will improve the quality of lower bounds obtained at the nodes of the BB tree. However, if all the cuts that are derived at every node of the BB tree are augmented to the relaxed problem, then the size of the LBP to be solved at the nodes of the BB tree will increase. In particular, since the number of nodes grows exponentially, the number of valid cuts derived can also grow exponentially. It is proposed that empirical constraint dropping strategies similar to that for BC algorithm for MILPs (Balas et al., 1996a) be developed to add only a subset of cuts that have been derived to the relaxed problem.

7. Valid cuts for the GBC algorithm

The valid cuts for the GBC algorithm that can be augmented to the relaxed problem in addition to the

8. Conclusions

In this paper, a generalized branch and cut (GBC) algorithm has been presented to solve mixed-integer

1366 P. Kesavan, P.L Barton/Computers and Chemical Engineering 24 (2000) 1361-1366

optimization problems. This provides a unified frame- work for comparing all currently available deterministic algorithms to solve these problems. All current decom- position and branch and bound algorithms for noncon- vex MINLPs which employ implicit enumeration to solve subproblems are specific instances of the GBC algorithm with a certain set of heuristics. A set of heuristics which may be potentially more efficient com- putationally (on average) than all the current al- gorithms is also discussed.

Acknowledgements

This work was supported by the National Science Foundation under grant CTS-9703623.

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