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University of Pittsburgh, 2008 – Pittsburgh, PA
Mixed Integer Programming Models for
Non-Separable Piecewise Linear Cost
Functions
Juan Pablo Vielma
H. Milton Stewart School of Industrial and Systems Engineering
Georgia Institute of Technology
Joint work with Shabbir Ahmed and George Nemhauser.
is a piecewise linear
function (PLF) and is any compact set.
Convex = Linear Programming. Non-Convex = NP Hard.
Specialized algorithms (Tomlin 1981, ..., de Farias et al.
2008 ) or Mixed Integer Programming Models (12+ papers). /26
Piecewise Linear Optimization
2
min f0(x)
s.t.
fi(x) ≤0 ∀i ∈ I
x ∈X ⊂ Rn
/26
Mixed Integer Models for PLFs
Existing studies are for separable functions:
Contributions (Vielma et al. 2008a,b):
First models with a logarithmic # of binary variables.
Theoretical and computational comparison:
multivariate (non-separable) and lower
semicontinuous functions in a unifying framework.3
/26
Outline
Applications of Piecewise Linear Functions.
Modeling Piecewise Linear Functions.
Logarithmic Formulations.
Comparison of Formulations.
Extension to Lower Semicontinuous Functions.
Final Remarks.
4
/26
Applications of Piecewise Linear Functions
Economies of Scale: Concave
Single and multi-commodity network flow.
Applications in telecommunications, transportation,
and logistics.
(Balakrishnan and Graves 1989, ..., Croxton, et al. 2007).5
0 1 3 50
2
3.5
4
/26
Applications of Piecewise Linear Functions
Fixed Charges and Discounts
1.Fixed Costs in Logistics.
2.Discounts (e.g. Auctions: Sandholm,
et al. 2006, CombineNet).
3.Discounts in fixed charges (Lowe
1984).6
x
y
0
2
3
0 1 20
1
2
3
4
1. 2. 3.
0 1 30
1
2
3
Gas Network Optimization
(Martin et al. 2006)./26
Applications of Piecewise Linear Functions
Non-Linear and PDE Constraints
7
A∂ρ
∂t+ ρ0
∂q
∂x= 0,
∂p
∂x= −λ
|v|v
2Dρ .
Pipe
Demand Points
Pipes/Valves/
Compressors
Connections
Source
Gas Network Optimization
(Martin et al. 2006)./26
Applications of Piecewise Linear Functions
Non-Linear and PDE Constraints
7
A∂ρ
∂t+ ρ0
∂q
∂x= 0,
∂p
∂x= −λ
|v|v
2Dρ .
Pipe
Demand Points
Pipes/Valves/
Compressors
Connections
Source
Gas Network Optimization
(Martin et al. 2006)./26
Applications of Piecewise Linear Functions
Non-Linear and PDE Constraints
7
A∂ρ
∂t+ ρ0
∂q
∂x= 0,
∂p
∂x= −λ
|v|v
2Dρ .
Pipe
Demand Points
Pipes/Valves/
Compressors
Connections
Source
/26
Applications of Piecewise Linear Functions
Numerically Exact Global Optimization
Process engineering (Bergamini et al. 2005,
2008, Computers and Chemical Eng.)
Wetland restoration (Stralberg et al. 2009).8
/26
Applications of Piecewise Linear Functions
Numerically Exact Global Optimization
Process engineering (Bergamini et al. 2005,
2008, Computers and Chemical Eng.)
Wetland restoration (Stralberg et al. 2009).8
0.5 1.0 1.5 2.0 2.5 3.0
0.2
0.4
0.6
0.8
1.0
0 1 2 4 5
f(4) = 5
0
f(0) = 10
f(1) = 32
f(2) = 40
f(5) = 15
/26
Modeling Piecewise Linear Functions
Piecewise Linear Functions: Definition
9
Definition 1. Piecewise Linear f : D ⊂Rn →R:
f(x) :={mP x+ cP x∈ P ∀P ∈P.
for finite family of polytopes P such that D =⋃
P∈PP
f(x,y)
y
x ∈ P
/26
Modeling Piecewise Linear Functions
Piecewise Linear Functions: Definition
9
Definition 1. Piecewise Linear f : D ⊂Rn →R:
f(x) :={mP x+ cP x∈ P ∀P ∈P.
for finite family of polytopes P such that D =⋃
P∈PP
f(x,y)
y
x ∈ P
/26
Modeling Piecewise Linear Functions
Piecewise Linear Functions: Definition
9
Definition 1. Piecewise Linear f : D ⊂Rn →R:
f(x) :={mP x+ cP x∈ P ∀P ∈P.
for finite family of polytopes P such that D =⋃
P∈PP
/26
Modeling Piecewise Linear Functions
Modeling Function = Epigraph
Example: 10
0 1 2 4 5
f(4) = 5
0
f(0) = 10
f(1) = 32
f(2) = 40
f(5) = 15
(a) f .
0 1 2 4 5
5
0
10
32
40
15
(b) epi(f).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
λP1,
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
λP1,
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
x = 0λ0 + 2λ2 + 4λ4
z ≥ 1λ0 + 3λ2 + 0λ4
1 = λ0 + λ2 + λ4, λ0, λ2, λ4 ≥ 0
λ0 ≤ yP1, λ2 ≤ yP1
+ yP2, λ4 ≤ yP2
1 = yP1+ yP2
, yP1, yP2
∈ {0, 1}
λP1,
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
x = 0λ0 + 2λ2 + 4λ4
z ≥ 1λ0 + 3λ2 + 0λ4
1 = λ0 + λ2 + λ4, λ0, λ2, λ4 ≥ 0
λ0 ≤ yP1, λ2 ≤ yP1
+ yP2, λ4 ≤ yP2
1 = yP1+ yP2
, yP1, yP2
∈ {0, 1}
λP1,
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
x = 0λ0 + 2λ2 + 4λ4
z ≥ 1λ0 + 3λ2 + 0λ4
1 = λ0 + λ2 + λ4, λ0, λ2, λ4 ≥ 0
λ0 ≤ yP1, λ2 ≤ yP1
+ yP2, λ4 ≤ yP2
1 = yP1+ yP2
, yP1, yP2
∈ {0, 1}
λP1,
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Univariate
11
0
1
3
=
0 2 4
1
3
0 2
∪
0
3
2 4
are SOS2
x = 0λ0 + 2λ2 + 4λ4
z ≥ 1λ0 + 3λ2 + 0λ4
1 = λ0 + λ2 + λ4, λ0, λ2, λ4 ≥ 0
λ0 ≤ yP1, λ2 ≤ yP1
+ yP2, λ4 ≤ yP2
1 = yP1+ yP2
, yP1, yP2
∈ {0, 1}
λP1,
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
“Original
Constraints”
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
“Extra
Constraints”
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
SOS2 only for univariate
12
“Extra
Constraints”
Nonzero variables are associated to vertices of a
single polytope.
∑v∈V(P)
λvv = x,∑
v∈V(P)
λv (mP v + cP )≤ z
λv ≥ 0 ∀v ∈ V(P) :=
⋃P∈P
V (P ),∑
v∈V(P)
λv = 1
λv ≤∑
{P∈P :v∈V (P )}
yP ∀v ∈ V(P),∑
P∈P
yP = 1, yP ∈ {0,1} ∀P ∈P
Univariate (Dantzig, 1960) ... Multivariate (Lee and
Wilson (2001).
/26
Modeling Piecewise Linear Functions
Convex Combination (CC): Multivariate
12
“Extra
Constraints”
/26
Modeling Piecewise Linear Functions
Existing Models are Linear on
Other models: Multiple Choice
(MC), Incremental (Inc),
Disaggregated Convex
Combination (DCC).
Number of binary variables and
combinatorial “extra” constraints
are linear in .
For multivariate on a
grid .
Logarithmic sized formulations?13
f(x,y)
y
x
0 1 20
1
2
SOS1-2 (Beale and Tomlin 1970):
SOS1: At most one variable is nonzero.
SOS2: Only 2 adjacent variables are nonzero.
! (0,1,1/2,0,0) ! (0,1,0,1/2,0)
, allowed sets .
SOS1:
SOS2:
CC:/26
Logarithmic Formulations
SOS1, SOS2 and CC constraints.
14
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Injective function:
Variables:
Idea:
0 0
1 0
0 1
1 1
/26
Logarithmic Formulations
Logarithmic Formulation for SOS1
15
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
0 0
1 0
0 1
1 1
Injective function:
Variables:
Idea:
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
B : {0, . . . ,m− 1}→ {0,1}⌈log2 m⌉
w ∈ {0,1}⌈log2 m⌉
Where is ?!
In general:
Gray Code.
λ2
/26
Logarithmic Formulations
Logarithmic Formulation for SOS2
16
0 0
1 0
1 1
0 1
B(i) and B(i+1)
differ in one component
/26
Logarithmic Formulations
Independent Branching: Dichotomies
17
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
/26
Logarithmic Formulations
Independent Branching: Dichotomies
17
0 1 20
1
2λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
λ0
λ1
λ2
λ3
λ4
y
x0 1 2 3 40
1
2
3
4
T
= 4 /26
Logarithmic Formulations
Independent Branching for 2 var CC
Select Triangle by forbidding vertices.
2 stages:
Select Square by SOS2 on each variable.
Select 1 triangle from each square.
18
y
x0 1 2 3 40
1
2
3
4
T
= 4 /26
Logarithmic Formulations
Independent Branching for 2 var CC
Select Triangle by forbidding vertices.
2 stages:
Select Square by SOS2 on each variable.
Select 1 triangle from each square.
18
y
x0 1 2 3 40
1
2
3
4
T
= 4 /26
Logarithmic Formulations
Independent Branching for 2 var CC
Select Triangle by forbidding vertices.
2 stages:
Select Square by SOS2 on each variable.
Select 1 triangle from each square.
18
0 1 2 3 40
1
2
3
4 L̄ = {(r, s) ∈ J :
r even and s odd}
= {square vertices}
R̄ = {(r, s) ∈ J :
r odd and s even}
= {diamond vertices}
/26
Comparison of Formulations
Strength of LP Relaxations
Sharp Models: LP = lower convex envelope.
All popular models are sharp.
Locally Ideal: LP = Integral (All but CC, even Log).
Locally ideal implies Sharp.19
(a) epi(f). (b) conv(epi(f)).
LP relaxation
/26
Comparison of Formulations
Strength of LP Relaxations
Sharp Models: LP = lower convex envelope.
All popular models are sharp.
Locally Ideal: LP = Integral (All but CC, even Log).
Locally ideal implies Sharp.19
(a) epi(f). (b) conv(epi(f)).
LP relaxation
Instances
Transportation problems (10x10 & 5x2).
Univariate: Concave Separable Objective.
Multivariate: 2-commodity.
Functions: affine in k segments or k x k
grid triangulation (100 instances per k).
Solver: CPLEX 11 on 2.4Ghz machine.
Logarithmic versions of CC = Log,
DCC=DLog. /26
Comparison of Formulations
Computational Results
20
0 1 20
1
2
/26
Comparison of Formulations
Univariate Case (Separable)
21
1
10
100
1000
10000
4 8 16 32
Average Time Solve [s]
Number of Segments
DCCCCMC
DLogLog
/26
Comparison of Formulations
Univariate Case (Separable)
21
1
10
100
1000
10000
4 8 16 32
Average Time Solve [s]
Number of Segments
DCCCCMC
DLogLog
/26
Comparison of Formulations
Univariate Case (Separable)
21
1
10
100
1000
10000
4 8 16 32
Average Time Solve [s]
Number of Segments
DCCCCMC
DLogLog
/26
Comparison of Formulations
Univariate Case (Separable)
21
1
10
100
1000
10000
4 8 16 32
Average Time Solve [s]
Number of Segments
DCCCCMC
DLogLog
/26
Comparison of Formulations
Multivariate Case (Non-Separable)
22
1
10
100
1000
10000
4x4 8x8 16x16
Average Time Solve [s]
Grid Size
DCCCCMC
DLogLog
≤
0 2 4 50
1
4
3
2
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
≤
0 2 4 50
1
4
3
2
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
≤
0 2 4 50
1
4
3
2
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
} { ∈ } {
f(x, y) :=
3 (x, y) ∈ (0, 1]2
2 (x, y) ∈ {(x, y) ∈ 2 : x = 0, y > 0}
2 (x, y) ∈ {(x, y) ∈ 2 : y = 0, x > 0}
0 (x, y) ∈ {(0, 0)}.
Figure 8(b) is slightly more complicated and its domain is ˜ = con
x
y
0
2
3
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈ n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
} { ∈ } {
f(x, y) :=
3 (x, y) ∈ (0, 1]2
2 (x, y) ∈ {(x, y) ∈ 2 : x = 0, y > 0}
2 (x, y) ∈ {(x, y) ∈ 2 : y = 0, x > 0}
0 (x, y) ∈ {(0, 0)}.
Figure 8(b) is slightly more complicated and its domain is ˜ = con
x
y
0
2
3
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈ n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
} { ∈ } {
f(x, y) :=
3 (x, y) ∈ (0, 1]2
2 (x, y) ∈ {(x, y) ∈ 2 : x = 0, y > 0}
2 (x, y) ∈ {(x, y) ∈ 2 : y = 0, x > 0}
0 (x, y) ∈ {(0, 0)}.
Figure 8(b) is slightly more complicated and its domain is ˜ = con
x
y
0
2
3
/26
Lower Semicontinuous Functions
Lower Semicontinuous PLFs
23
f(x) :=
{
mP x + cP x ∈ P ∀P ∈ P
.P = {x ∈ n
: aix ≤ bi ∀i ∈ {1, . . . , p},
aix < bi ∀i ∈ {p, . . . ,m}}
Finite family of
copolytopes
} { ∈ } {
f(x, y) :=
3 (x, y) ∈ (0, 1]2
2 (x, y) ∈ {(x, y) ∈ 2 : x = 0, y > 0}
2 (x, y) ∈ {(x, y) ∈ 2 : y = 0, x > 0}
0 (x, y) ∈ {(0, 0)}.
Figure 8(b) is slightly more complicated and its domain is ˜ = con
x
y
0
2
3
/26
Lower Semicontinuous Functions
Lower Semicontinuous Models
Direct from Disjunctive Programming (Jeroslow and
Lowe)
“Extreme point” = DCC.
Traditional = Multiple Choice (MC).
Other models can be adapted to special types of
discontinuities (e.g. simple fixed charges).
MC, DCC, DLog are locally ideal and sharp.
Computations: 2-commodity FC discount function.
24
x
y
/26
Comparison of Formulations
Multivariate Lower Semicontinuous
25
1
10
100
1000
10000
4x4 8x8 16x16 32x32
Average Time Solve [s]
Grid Size
DCCMC
DLog
/26
Comparison of Formulations
Multivariate Lower Semicontinuous
25
1
10
100
1000
10000
4x4 8x8 16x16 32x32
Average Time Solve [s]
Grid Size
DCCMC
DLog
/26
Comparison of Formulations
Multivariate Lower Semicontinuous
25
1
10
100
1000
10000
4x4 8x8 16x16 32x32
Average Time Solve [s]
Grid Size
DCCMC
DLog
/26
Final Remarks
Final Remarks
Unifying theoretical framework: allows for
multivariate non-separable and lower
semicontinuous functions.
First logarithmic formulations: Theoretically
strong and provides significant
computational advantage for large .
Revive forgotten formulations and
functions: MC and fixed charge
discount function.26
x
y