Linear Reformulation of Probabilistically Constrained Optimization

Linear Reformulation of Probabilistically Constrained Optimization Problems Using Combinatorial Patterns Miguel A. Lejeune ∗ George Washington University

International Colloquium on Stochastic Modeling and Optimization Dedicated to Professor A. Prékopa ∗ Supported

by Army Research Office Grant #W911NF-09-1-0497

Colloquium – Dr. A. Prékopa (Dec 2009)

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Problem Formulation Probabilistically constrained programming problem min c T x subject to Ax ≥ b P Tj x ≥ ξj , j ∈ J ≥ p x ∈ R+ with ξ having a multivariate probability distribution with finite support Approach: Combinatorial pattern approach Rationale: Capturing impact of single variables and “interactions” between variables on constraint satisfiability Output: Linear reformulations Colloquium – Dr. A. Prékopa (Dec 2009)

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Example min x1 + 2x2 8 − x1 − 2x2 ≥ ξ1 subject to P ≥ 0.7 8x1 + 6x2 ≥ ξ2 x1 , x2 ≥ 0 k 1 2 3 4 5 6 7 8 9 10

ω1k 6 2 1 4 3 4 6 1 4 5

ω2k 3 3 4 5 6 8 8 9 9 10

F (ω k ) 0.2 0.1 0.1 0.3 0.3 0.5 0.7 0.2 0.7 0.8

with pk = 0.1, k = 1, . . . , 10 ∈ Ω. Colloquium – Dr. A. Prékopa (Dec 2009)

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Example Feasibility set is the union of the two following polyhedra: S1 = {(x1 , x2 ) ∈ R2+ : 8 − x1 − 2x2 ≥ 6, 8x1 + 6x2 ≥ 8} , S2 = {(x1 , x2 ) ∈ R2+ : 8 − x1 − 2x2 ≥ 4, 8x1 + 6x2 ≥ 9} , and is non-convex:

h 2 (x)

(4,9) (6,8)

h 1 (x)

Could also be disconnected (Henrion, 2002). Colloquium – Dr. A. Prékopa (Dec 2009)

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Applications Applications: Power management (Prékopa et al., 1980) Finance: cash matching (Henrion, 2001; Dentcheva et al., 2004), market risk (Pirvu, 2009) Reservoir management (Prékopa, Szántai, 1978) Water management (Takyi, Lence, 1999) Chemical distillation process (Henrion et al., 2001) Wireless communication (Chalise et al., 2007) Pollution (Gren, 2008) ´ Production-distribution problems (LE, Ruszczynski, 2007) Munitions prepositioning (Avital, 2005) Military supply chain operations (Kress et al., 2007) Medical service design (Noyan, 2009) Vaccination strategies (Tanner et al., 2008) etc. Colloquium – Dr. A. Prékopa (Dec 2009)

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Solution Methods p-efficiency concept (Prékopa, 1990): disjunctive problem: Identification of finite, unknown number of p-efficient points Enumerative algorithm (Prékopa, 1990; Prékopa, 1995; Prékopa et ´ al., 1997; Beraldi, Ruszczynski, 2002; LE, 2008) or optimization-based generation (LE, Noyan, 2009) Convexification (Dentcheva et al., 2001) Column generation algorithm (LE, Ruszczy´nski, 2007)

MIP approach List possible realizations of multivariate random vector Associate a binary variable with each scenario MIP formulation with cover constraint ´ Use of structural properties (Ruszczy nski, 2002; Cheon et al., 2006; Luedtke et al., 2009; Küçükyavuz, 2009)

Robust approach Derivation of convex approximations (Calafiore, Campi, 2005; Nemirovski, Shapiro, 2005, 2006) Colloquium – Dr. A. Prékopa (Dec 2009)

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Structure Combinatorial patterns defining sufficient conditions for P Tj x ≥ ξj , j ∈ J ≥ p Binarization of probability distribution F Generation of set of relevant realizations Representation of combination (F , p) of probability distribution F and probability level p as a partially defined Boolean function (pdBf) Extension as a disjunctive normal form - collection of patterns Linear programming generation of combinatorial patterns

Derivation of: Linear programming tight inner approximation Linear deterministic equivalent

Numerical results Extendibility and conclusion Colloquium – Dr. A. Prékopa (Dec 2009)

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Partitioning of Extended Set of Realizations Definition (p-Sufficient Realization) A realization k is p-sufficient if P(ξ ≤ ωk ) = F (ω k ) ≥ p and is p-insufficient if F (ω k ) < p.

Example ( F1−1 (0.7) = 4; F2−1 (0.7) = 8 ) k : ωjk ≥ Fj−1 (p), j ∈ J Let: Zj = {ωjk : Fj (ωjk ) ≥ p, k ∈ Ω}

Extended set of p-insufficient points

Z = Z1 × . . . × Zj × . . . × Z|J| Extended set of points: ΩE = Ω Z Colloquium – Dr. A. Prékopa (Dec 2009)

Extended set of p-sufficient points

M. Lejeune

k

ω1k

ω2k

Ik

1 2 3 4 5 6 8 11 7 9 10 12 13 14 15

6 2 1 4 3 4 1 5 6 4 5 4 5 6 6

3 3 4 5 6 8 9 8 8 9 10 10 9 9 10

0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 8 / 29

Binarization of Probability Distribution Mapping of numerical vector ω k into a binary one β k : k k , β21 , . . . , βijk , . . . , k ∈ ΩE ω k → β k = β11 Set C of cut points cij 1 if ωjk ≥ cij k βij = 0 otherwise

, i = 1, . . . , nj , j ∈ J

Consistent set C of cut points: preserves the disjointedness between ΩE− and ΩE+

Definition A sufficient-equivalent set of cut points C e C e = {cij : cij = ωjk , Fj (ωjk ) ≥ p, i = 1, . . . , nj , j ∈ J, k ∈ ΩE+ } is consistent Colloquium – Dr. A. Prékopa (Dec 2009)

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Representation of (F , p) as a pdBf Example (C e = {c11 = 4; c21 = 5; c31 = 6; c12 = 8; c22 = 9; c32 = 10} ) k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Numerical ω1k ω2k 6 3 2 3 1 4 4 5 3 6 4 8 6 8 1 9 4 9 5 10 5 8 4 10 5 9 6 9 6 10


¯− Binarized Images Ω B k k k β21 β31 β12 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1

k β11

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¯+ and Ω B k k β22 β32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1

Ik 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 10 / 29

¯ of Relevant Realizations Set Ω Example ß i2

ω12

ω 10

ω 15

ω9

ω 13

ω 14

10=(.,.,.,1,1,1)

9=(.,.,.,1,1,0)

ω8

8=(.,.,.,1,0,0)

ω6 ω2=ω3=ω5

ω4

4=(1,0,0,.,.,.)

k 6 7 9 10 11 12 13 14 15

Numerical ω1k ω2k 4 8 6 8 4 9 5 10 5 8 4 10 5 9 6 9 6 10


ω7 ω 11 ω1 5=(1,1,0,.,.,.)

6=(1,1,1,.,.,.)

ß i1

¯ − and Ω ¯+ Binarized Images Ω B B k k k k k k β11 β21 β31 β12 β22 β32 1 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 M. Lejeune

Ik 0 1 1 1 0 1 1 1 1 11 / 29

Extension ¯ +, Ω ¯ − ) representing (F , p) Binarization process: pdBf g(Ω B B ¯ +, Ω ¯ −) Objective: Simple and compact extension f of g( Ω B B

Definition (Boros et al., 1997) ¯ +, Ω ¯ − ): Ω ¯ +, Ω ¯ − ⊆ Bn. Let B = {0, 1} and pdBf g defined by ( Ω B B B B ¯ +, Ω ¯ − ) if: A function f : B n → B is called an extension of the pdBf g( Ω B B ¯ + (f ) and Ω ¯− ⊆ Ω ¯ + (f ) . ¯+ ⊆ Ω Ω B B B B Any Boolean function can be represented by a disjunctive normal form (DNF)


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¯ +, Ω ¯ −) DNF Extension for g(Ω B B

DNF: f = Term: t =

v ∈V

tv

¯ βij , with βij , β¯ij called literals

βij

ij∈Pt

ij∈Nt

Binary mapping of realization:

ωk

→

βk

=

Term covers a realization k: t(k) = 1 ⇔

k , . . . , βk , . . . β11 ij

ij∈P

βijk

¯k βij = 1

ij∈N

Degree of a term: number of literals: d = |P| + |N|

Theorem (LE, 2009) The DNF f =

v ∈V

tv , such that

¯− , ¯ + and f (k) = 0, k ∈ Ω f (k) = 1, k ∈ Ω B B ¯− ¯ +, Ω is an extension of g(Ω B a deterministic equivalent B ). It provides reformulation of the P Tj x ≥ ξj , j ∈ J ≥ p. Colloquium – Dr. A. Prékopa (Dec 2009)

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Pattern Properties DNF as a disjunction of p-patterns

Definition A p-pattern is a term t such that:

¯+ k ∈Ω B

t(k) ≥ 1 and

¯− k ∈Ω B

t(k) = 0 .

Corollary Consider a sufficient-equivalent set C e of cut points and the set of ¯ B: relevant realizations Ω ¯+ . ¯ − ⇒ t(k) = 1 for at least one k ∈ Ω t(k) = 0, k ∈ Ω B B Prime p-patterns are of (large) degree |J| ⇒ Rationale for optimization-based generation


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Approach Derive the “optimal” p-pattern: defines minimal conditions for P Tj x ≥ ξj , j ∈ J ≥ p: → deterministic equivalent No particular/stylized form Derive a p-pattern: defines sufficient conditions for P Tj x ≥ ξj , j ∈ J ≥ p → tight inner approximation Coverage criterion SUFFICIENT-EQUIVALENT SET OF CUT POINTS c i2 10

ω 12 ω 10 ω 15

(0,0,0,1,1,1)

ω9

9 8

ω 13 ω 14 ω7

ω

ω

6

11

4

5

6

c i1

Decision variables: ¯ + is covered by t y k : defines whether k ∈ Ω B uij : defines whether literal βij is included in t Colloquium – Dr. A. Prékopa (Dec 2009)

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Integer Programming Formulation IP1 for p-Patterns Theorem Any feasible solution (u, y) of IP1 k z = min y ¯+ k ∈Ω B

subject to

nj

βijk uij + |J|y k ≥ |J|,

¯+ k ∈Ω B

βijk uij ≤ |J| − 1,

¯− k ∈Ω B

j∈J i=1 nj j∈J i=1 nj

uij = 1,

j∈J

i=1

y k , uij ∈ {0, 1}, defines a prime p-pattern

t =

¯+ j ∈ J, i = 1, . . . , n j , k ∈ Ω B βij

uij =1 j∈J,i=1,...,nj

of degree |J| and (u ∗ , y∗ ) defines the p-pattern with maximal coverage. Colloquium – Dr. A. Prékopa (Dec 2009)

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Integer Programming Formulation IP2 for p-Patterns Theorem Any feasible solution (u, y) of IP2 z = min

yk

¯+ k ∈Ω B nj

subject to

βijk uij + y k = |J|,

¯+ k ∈Ω B


¯− k ∈Ω B

j∈J i=1 nj j∈J i=1 nj

uij = 1,

j∈J

uij ∈ {0, 1},

j ∈ J, i = 1, . . . , n j

i=1

¯+ k ∈Ω B

k

0 ≤ y ≤ |J|, defines a prime p-pattern with degree |J|:

t =

βij .

uij =1 j∈J,i=1,...,nj Colloquium – Dr. A. Prékopa (Dec 2009)

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Patterns with IP1 and IP2 Example With (both) IP1 and IP2, we obtain: u ∗11 = u∗22 = 1 , z∗ = 1. Thus, t = β11 β22

k 6 11 7 9 10 12 13 14 15

Numerical ω1k ω2k 4 8 5 8 6 8 4 9 5 10 4 10 5 9 6 9 6 10


¯− Binarized Images Ω B k k k β21 β31 β12 1 0 0 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1

k β11

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¯+ and Ω B k k β22 β32 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 1 1

Ik 0 0 1 1 1 1 1 1 1

Coverage ∗ yk – – 1 0 0 0 0 0 0 18 / 29

Linear Programming Formulation LP1 for p-Patterns Theorem Any feasible solution (u, y) of LP1 z = min subject to

nj

yk

¯+ k ∈Ω B

βijk uij + |J|y k ≥ |J|,

¯+ k ∈Ω B

j∈J i=1 nj


¯− k ∈Ω B

j∈J i=1 nj

uij = 1,

j∈J

i=1

¯+ j ∈ J, i = 1, . . . , nj , k ∈ Ω B

0 ≤ y k , uij ≤ 1, defines a p-pattern t =

βij .

uij >0 j∈J,i=1,...,nj


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Prime p-Pattern with Linear Programming Formulation Objective function: relaxed number of non-covered p-sufficient realizations p-pattern not necessarily prime

Example (u11 = 0, u21 = 0.5, u31 = 0.5, u12 = 0.5, u22 = 0.5, u32 = 0) is feasible for LP1 and results in the pattern t = β21 β31 β12 β22

Corollary A prime p-pattern t =

βij , with

¯i j =1 u j j∈J,i=1,...,nj

¯ij = argmax uij > 0, j ∈ J

,

i

¯ij u

1 0

if i = ¯ij otherwise

can immediately be derived from any feasible solution (u, y) of the linear programming problem. Colloquium – Dr. A. Prékopa (Dec 2009)

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Pricing Literals Pricing literals so that t defines (close to) minimal conditions Define parameters b ij associated with literals βij intra-component j pricing: include in t the literals imposing the least demanding conditions. If possible, preferable to include βi j than βij if i > i . Thus: bij > bi j , i > i , j ∈J

Example ß i2

ω12

ω 10

ω 15

ω9

ω 13

ω 14

10=(.,.,.,1,1,1)

9=(.,.,.,1,1,0)

ω8

8=(.,.,.,1,0,0)

ω6 ω2=ω3=ω5

ω7 ω 11

ω4

4=(1,0,0,.,.,.)

ω1 5=(1,1,0,.,.,.)

6=(1,1,1,.,.,.)

ß i1

inter-component pricing: value of bij , i = 1, . . . nj , associated with j, is an increasing function of the cost associated with component j. Colloquium – Dr. A. Prékopa (Dec 2009)

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Linear Programming Formulation LP2 for p-Patterns bij : price of including literal βij in t

Theorem Any feasible solution (u) of LP2 z = min

nj

bij uij

j∈J i=1

subject to

nj


¯− k ∈Ω B

j∈J i=1 nj

uij = 1,

j∈J

i=1

0 ≤ uij ≤ 1, defines a p-pattern t =

j ∈ J, i = 1, . . . , nj βij .

uij >0 j∈J,i=1,...,nj

Only includes n continuous variables and n + |Ω − B | constraints ∗ Optimal solution of (u) of LP2 defining “least costly” p-pattern Colloquium – Dr. A. Prékopa (Dec 2009)

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LP Inner Approximation with p-Pattern Consider a p-pattern t =

βij , with P the set of literals included in t.

ij∈P

t(k) = 1 ⇔

βijk = 1 ⇔ ωjk ≥ cij , ij ∈ P

ij∈P

Theorem The LP problem min c T x subject to Ax ≥ b Tj x ≥ cij , ij ∈ P x ∈ R+ is an inner approximation of the probabilistic problem. Evaluate tightness Colloquium – Dr. A. Prékopa (Dec 2009)

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Test Set Supply chain-distribution problem

ckj xkj min k ∈K j∈J

subject to

xkj ≤ Mk ,

k ∈K

j∈J

P(

k ∈K

xkj ≤ Vkj ,

k ∈ K,j ∈ J

xkj ≥ ξj , j ∈ J) ≥ p x ≥0

32 types of instances differentiating along tuple (|J|, |Ω|, p): |J|=10, 20 |Ω|=5000,10000,20000,50000 p=0.875, 0.9, 0.925, 0.95 8 instances per type MATLAB - AMPL - CPLEX 11.1 Colloquium – Dr. A. Prékopa (Dec 2009)

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Inner Approximation: Speed and Tightness I

|J| 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

Instance Type |Ω| p 5000 0.875 5000 0.9 5000 0.925 5000 0.95 10000 0.875 10000 0.9 10000 0.925 10000 0.95 20000 0.875 20000 0.9 20000 0.925 20000 0.95 50000 0.875 50000 0.9 50000 0.925 50000 0.95

IP1 Time 1.59 0.11 0.03 0.03 1.62 0.30 0.03 0.03 1.60 0.28 0.03 0.03 1.58 0.39 0.03 0.03


Gap 1.316% 0.458% 0.734% 0.325% 0.986% 0.396% 0.778% 0.001% 0.292% 1.532% 0.352% 0.775% 0.231% 1.241% 0.381% 0.918%

IP2 Time 0.41 0.06 0.03 0.03 0.39 0.12 0.02 0.02 0.41 0.09 0.03 0.03 0.41 0.14 0.03 0.03

Gap 1.449% 0.253% 0.971% 0.346% 0.960% 0.432% 0.939% 0.001% 0.487% 1.509% 0.369% 1.031% 0.472% 1.525% 0.815% 0.846%

M. Lejeune

LP1 Time 0.03 0.02 0.03 0.01 0.02 0.03 0.04 0.04 0.01 0.03 0.01 0.03 0.02 0.03 0.01 0.03

Gap 1.698% 0.485% 0.856% 0.457% 1.442% 0.421% 0.967% 0.001% 0.541% 2.360% 0.796% 1.029% 0.237% 2.254% 1.259% 2.638%

LP2 Time 0.03 0.02 0.03 0.02 0.02 0.03 0.03 0.03 0.02 0.03 0.02 0.03 0.02 0.03 0.02 0.03

Gap 0.802% 0.000% 0.560% 0.116% 1.292% 0.868% 0.000% 0.001% 1.631% 1.341% 0.004% 0.002% 0.000% 1.033% 0.137% 0.853%

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Inner Approximation: Speed and Tightness II |J| 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 1 2 3 4

Instance Type |Ω| p 5000 0.875 5000 0.9 5000 0.925 5000 0.95 10000 0.875 10000 0.9 10000 0.925 10000 0.95 20000 0.875 20000 0.9 20000 0.925 20000 0.95 50000 0.875 50000 0.9 50000 0.925 50000 0.95

IP1 Time 14.56 3.94 0.05 0.02 15.70 11.61 0.05 0.03 26.54 12.70 0.02 0.03 28.96 21.17 0.05 0.03

Gap 0.796% 0.610% 0.260% 0.001% 0.924% 0.401% 0.205% 0.332% 0.904% 0.404% 0.175% 0.142% 0.862% 0.666% 0.459% 0.181%

IP2 Time 2.43 1.31 0.05 0.03 4.70 3.69 0.05 0.02 4.27 2.28 0.00 0.02 4.98 3.58 0.05 0.05

Gap 0.801% 0.608% 0.257% 0.000% 0.922% 0.406% 0.207% 0.340% 0.896% 0.408% 0.175% 0.137% 0.862% 0.678% 0.479% 0.189%

LP1 Time 0.22 0.03 0.04 0.02 0.26 0.05 0.02 0.06 0.13 0.04 0.01 0.03 1.78 1.36 0.01 0.01

Gap 1.234% 0.896% 0.364% 0.011% 1.264% 0.856% 0.745% 0.867% 1.762% 2.697% 1.671% 2.012% 3.002% 3.216% 2.891% 2.421%

LP2 Time 0.24 0.03 0.03 0.02 0.30 0.03 0.02 0.03 0.13 0.05 0.02 0.03 1.62 1.44 0.02 0.02

Gap 0.800% 0.459% 0.000% 0.126% 0.925% 0.001% 0.069% 0.068% 0.896% 0.878% 0.000% 0.137% 0.862% 0.956% 0.275% 0.252%

LP formulations the fastest ( < 2 sec) IP formulations also fast (