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Data Driven Decisions @ The Ohio State University Industrial and Systems Engineering Advances in Stochastic Mixed Integer Programming Lecture at the INFORMS Optimization Section Conference in Miami, February 26, 2012 Suvrajeet Sen Data Driven Decisions Lab Integrated Systems Engineering Ohio State University

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Some Historical Remarks Classification of SMIP SMIP Models: Risk, Recourse, Resilience Structural Properties Decomposition: Benders’ and Beyond Illustrative Computational Results Overview of this Lecture

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Historical Remarks: IOS Age of the INFORMS Optimization Section is a) 0 < age 10 b) 10 < age 15 c) 15 < age 20 d) 20 < age 25 e) age > 25 INFORMS OS was founded at the Spring ORSA/TIMS Meeting in Los Angeles, April 1995.

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Historical Remarks: My assessment of SIP/SMIP Stochastic Integer Programming Integer Programming Linear Programming Stochastic Linear Programming Discrete Choice 1950s- Present 1960s - Present Uncertainty Major Hurdles Still Remain!!

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Why model uncertainty? For most people, that’s just reality In the real world: “Risk is everywhere” Certainty: “Risk is merely a 4-letter word” In the real world: “There is a market for information” Certainty: “Information has no value” In the real world: “Hind sight is 20/20” Certainty: Foresight is 20/20

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Some data: [1980 – 2000) … prior to 2000 annotated bibliography (Stougie and van der Vlerk) Theory: (7+) papers Simple Integer Recourse: 2 Structural Properties of Expected Recourse Function: 4 Complexity: (1+) General Purpose Algorithms: 17 papers Benders’-type methods: 5 Grobner-basis methods: 2 Convex Approximations for Simple Integer Recourse: 2 Other: 8 (Sampling with first-stage integer, disjunctive cuts) More Historical Remarks: “Walk Before You Can Run”

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Books/Surveys: 6 altogether Dissertations: 3-4 (1 prior to 2000 in North America) Habilitation: 1 Published surveys: 3 (includes hierarchical planning) Special Purpose Models/Algorithms: 25 papers Production Planning/Scheduling: 3 Network and Routing: 11 Location: 7 Other: 4 In the 12 years: more than 350 articles listed in http://mally.eco.rug.nl/index.html?BIBLIO/SIP.HTML More Historical Remarks: “Walk Before You Can Run” [1980-2000)

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Schultz, R (2003) “Stochastic Programming with Integer Variables,” Mathematica l Programming-B, 285-309 Stougie, L. and M.H. van der Vlerk (2005) “Approximation in Stochastic Integer Programming” Sen, S. (2005) “Stochastic Mixed-Integer Programming Algorithms,” Handbook of Discrete Optimization, (Aardal, Nemhauser, Weismantel, eds.) …. Some newer surveys are also available …. “Now we are running” (Survey Articles 2000-)

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… And you know we’re serious because of applications with realistic data Manufacturing Supply Chain: Two-stage Design (IBM, Intel) Biofuel Supply Chain: Multi-stage Design (Fan et al) Homeland Security – Defender/Attacker/Defender (Wood et al – NPS, Ordonez/Tambe, Smith) Electric Power – Unit Commitment (Birge/Takriti, Philpott, Guan/Zhang), Fuel Price Hedging (Sen et al) Military – Prioritizing Choices (Morton), UAV/MAV (Evers et al) Fighting Forest Fire (Ntaimo)

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SMIP Classification: A (B-C-D-E) Notation for SMIP Two Stage Stochastic Linear Programming Min c T x + E[f(x, ω)] Ax = b, x ≥ 0 where, f(x, ω) = Min g T y Wy ≥ r(ω) – T(ω)x y ≥ 0 Variations depend on where the randomness appears

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Stochastic MIP with First Stage Integers Min c T x + E[f(x, ω)] Ax ≥ b, x ∈ R n 1 × Z n 2 where, f(x, ω) = Min g T y Wy ≥ r(ω) – T(ω)x y ∈ R n 3 Z n denotes integer vectors of length n. With second-stage integers, extremely difficult!

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Stochastic Combinatorial Optimization Min c T x + E[f(x, ω)] Ax ≥ b, x ∈ B n 1 where,(0l18 f(x, ω) = Min g T y Wy ≥ r(ω) – T(ω) x y ∈ R n 2 × B n 3 Here B n denotes binary vectors of length n. Many different structures for SMIP!

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Describing SMIP Problems B = Set of stages with Binary Vars. C = Set of stages with Continuous Vars. D = Set of stages with Discrete Vars. (arbitrary integers, not just binary) E = Endogenous Uncertainty (Y/N) Louveaux has proposed a notation that covers all SP problems (e.g. notation includes whether random variables are cont/discrete) Above notation helps clarify domain of applicability of results/algorithms etc.

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Traditional Benders’ Decomposition SLP: B = { ∅ }, C={1,2}, D ={ ∅ } Wollmer, Norkin et al, Poojari/Mitra: B = {1}, C={1,2}, D ={1} Special Structure: Simple Integer Recourse: B = {2}, C={1}, D ={2} + structure of second stage Global Optimization and IP Ahmed, Tawarmalani, Sahinidis: B = {2}, C={1,2}, D ={2} ; + Fixed Tenders Grossman & Co. (E = Y)

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Disjunctive Programming for Two-Stage Caroe/Tind, Sherali/Fraticelli, Sen/Higle, Sen/Sherali: B = {1,2}, C={2}, D ={ ∅ } Ntaimo/Sen: B = {1,2}, C={1,2}, D ={ ∅ } Lagrangian-based Methods for Multi-stage Multi-stage SMIPs: Caroe/Schultz, Roemisch et al, Alonso-Ayuso et al, Lulli/Sen, Guan et al B = {1,2, … N}, C={1,2 … N}, D ={1,2 … N}b

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SMIP Models Modeling Risk Modeling Recourse Modeling Resilience Multi-stage Models Models not Covered (Chance Constraints with Discrete Distributions) Special Structured IP (Knapsack, Mixing etc.) See Prékopa, Dentcheva, Ruszczynski … Leudtke et al (2010), Küçükyavuz (2010), Saxena et al (2009), Shen et al (2010) Stochastic MIP Models: Risk, Recourse, and Resilience

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Risk in SMIP We have only stated models via “Expected Values” Is the reliance on “Expectation” a handicap? Of course! But many risk measures (e.g. down-side risk, mean absolute deviation, CVaR, etc.) can be re-formulated using expectation of a slightly modified, though mathematically similar function.

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SCO for Modeling Risk We have only stated models via “Expected Values” Is the reliance on “Expectation” a handicap? Of course! But many risk measures (e.g. down-side risk, mean absolute deviation, CVaR, etc.) can be re-formulated using expectation of a slightly modified, though mathematically similar function Important: Inequalities are indispensible for risk modeling

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SCO for Modeling Risk Example: Kahneman/Tversky “S” curve for risk- aversion can be linearized using 0-1 variables. Similar to non-convex piecewise linear programming. Each piece requires a binary (switch variable) r

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SCO for Modeling Recourse: Stochastic Server Location Problem

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SCO for Modeling Recourse: SSLP This SCO has two sets of decisions: 1. Choose server locations (e.g. bases) 2. Once demand nodes (e.g threats) appear, then assign servers to demand nodes

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SCO for Modeling Recourse: SSLP

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Our Stochastic Server Location Problem (SSLP) also includes some policy constraints: Policy that each customer will receive service from only one site has been established. Moreover, service site must be located within a prescribed zone ( z ). Max number to be located is v, with each zone having no more than w z servers. SCO for Modeling Recourse: SSLP

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The SSLP model objective: minimize Cost – Expected Revenue Potential (last term denotes Penalty for lost demand) Min Σ j c j x j – E[Σ ij q ij y ij (ω) + Σ j Q j Y j (ω)] subject to: constraints on supply-side, Σ j x j ≤ v, Σ j ∈ J(z) x j ≥ w z, ∀ z demand-side, Σ j y ij (ω) + Y j (ω) = ω i, ∀ i supply/demand: Σ i y ij (ω) – Y j (ω) ≤ u j x j, ∀ j Plus: All variables are binary SCO for Modeling Recourse: SSLP

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Modeling Resilience Logical conditions are as follows: y 0 jk ≤ 1 – x j y 1 jk ≤ x j

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Multi-Stage SMIP Models Non-anticipativity in the Two Stage Model (*) is the non-anticipativity constraint … all scenarios must agree on first-stage

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Two-stage: NA only on Root Node Multi-stage: Difficult, unless Ocotillo-type Trees

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Recursive Formulation using State Variables Challenge of Coniferous Trees

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SMIP with Recursive Formulation

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Most structural properties and algorithms for SMIP assume relatively complete and sufficiently expensive recourse. - ∞ < f(x,ω) < + ∞ with probability 1. Under the above assumption, the expected recourse function is real-valued and lower semi-continuous. Structural Properties

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Complexity of Two-Stage SMIP Two-stage stochastic programs with recourse having finitely many scenarios is #P-hard (The class #P asks for the count (i.e. “how many”, rather than “are there any”?) The proof reduces any graph reliability problem to a two-stage stochastic combinatorial optimization problem

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What Do we Need? Two Issues in Algorithm Design: - Cuts for Second Stage IP - Approximation of f (also convexification) A Potent Brew! Decomposition (SP) and Convexification (IP)

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Gomory Cuts for SIP Decomposition Hot off the Printer! First stage 0-1, Second-stage General Integer, Disjunctive Decomposition (D 2 ) First stage: 0-1 Second-stage: mixed 0-1 Disjunctive Decomposition with Branch-and-Cut (D 2 -BAC) First stage 0-1 Second-stage: mixed-integer Beyond Benders’ Decomposition: Second-stage IP

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Recall --- SLP Two Stage Stochastic Linear Programming Min c T x + E[f(x, ω)] Ax = b, x ≥ 0 where, f(x, ω) = Min g T y Wy ≥ r(ω) – T(ω)x y ≥ 0 Variations depend on where the randomness appears

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Recall --- Benders’ Decomposition or L-shaped Method Standard Benders’ Master (OR – 501) Where denote Non-negative Second-stage Dual Multipliers

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Caroe-Tind extension of Benders’ or L-shaped Method for Second-stage SIP Gomory Cuts to represent Subadditive Value Functions Where denote Non-decreasing Second- stage value function approximations

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Structure Similar to Benders’ And a More General Framework But … Need to overcome bottlenecks Subproblems are Integer Programs Master Problems are required to Optimize Non- convex functions

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Our Recommendation: maintain Benders’ piecewise linear approximations Notice the change below! Where denote Non-negative Second-stage Dual Multipliers Notice that RHS r has changed to and T to

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Our Suggestion: Solve Second stage using Updated LP approximations Each iteration will involve only LP solutions in the second-stage Solve LP relaxation TWICE Once solve with an Old Convexification Derive a Cut to Update the Convexification We will have First-stage is same as Benders’ original proposal Second-stage are LPs.

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But can this be achieved? Yes under certain assumptions! First stage pure binary (B = {1,2}) C =, D={2} Use Gomory Cuts (Gade, Küçükyavuz, Sen) If C= {2}, B = {2} Use Disjunctive Set Convexification (Sen and Higle) If C= {2}, D = {2} Use Disjunctive Value Approximations for Branch-and-Cut (Sen and Sherali) First stage general MILP (Global Optimization) {∅}{∅}

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Second Stage Set Convexification Original Constraints Valid Inequalities as Functions of x Parametric Gomory Cuts: Affine Parametric Disjunctive Cuts: Piecewise Linear Concave

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Parametric Gomory Cuts Finiteness with Lexicographic Dual Simplex (Gade, Küçükyavuz, Sen) ScenObjVarsConsGDD-SGDD-RB&B Nodes B&B + Gom Nodes 4-63.5022247 (13)7 (32)542 (6) 9-66.1747547 (39)6 (76)3068 (13) 36-67.3318221610 (183)6 (384)1.55E752 (50) 121-67.676077269 (526)6 (1032)7.60E613224 (167)

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Convexify π 0 (x,ω) by viewing its epigraph as a disjunctive set such as the one shown below. π 0 (x,ω) 0 1 First stage binary variable Parametric Disjunctive Cuts

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Convergence for Disjunctive Decomposition (Set Convexification) Assumptions Complete recourse All integer variables are 0-1 Maintain all cuts in W k Certain rules of order hold (a la lexicographic dual simplex in Gomory’s proof) Under these assumptions, the D 2 method results in a convergent algorithm (Sen and Higle).

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π 0 (x,ω) Value Approximations for Branch-and- Cut in Second Stage (Sen and Sherali) —There will be one piece per node of a truncated BAC tree in the second-stage —Disjunctive Programming lets us convexify the function (for each outcome )

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55 Illustrative Computational Results with D 2 and D 2 -BAC SCALABILITY: D 2 scales well with increase in number of scenarios (linear) D 2 does not scale well with increase in size of master program (x)

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56 Computational Results Cont… The D 2 Algorithm Solves some of the largest (0-1) instances Scalability - Linear in the number of scenarios D 2 CPU time for SSLP_10_50 with 100 scenarios

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Treating Cut Generation as a Specialized Two-Stage LP Computational Results (with Y. Yuan) InstanceD2 D2-BAC D2-BAC++ 5.25.50 1.64 0.700.36 5.25.1002.15 1.73 0.89 5.50.100 7.10 3.70 1.56 5.50.500 34.50 23.05 12.36 5.50.1000 140.4764.17 22.77 5.50.2000 603.37274.40 42.74 10.50.50 295.95373.98262.13 10.50.100 396.76 452.31 486.99 10.50.500 1902.2 2772.22 1313.38 10.50.1000 5410.1 5677.802139.47 10.50.2000 9055.29 > 10800 3916.47 15.45.5 110.34 232.30 211.79 15.45.10 1494.89222.41 153.41 15.45.15 7210.63 1988.26 803.56

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Conclusions Decomposition (SP) + Valid Inequalities (IP) provide a potent potion! But … Stochastic MIP still needs a lot of work Specially structured cuts (already at play in Chance Constrained SP) Multi-stage extensions (very rich area) Real-world Applications ….

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