Computational Optimization and Applications

Computational Optimization and Applications A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Algorithms and Software --Manuscript Draft-- Manuscript Number: Full Title: A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Algorithms and Software Article Type: Manuscript Keywords: Stochastic programming; L-shaped method; Scenario Decomposition; Software benchmark Corresponding Author: I. Grossmann Carnegie-Mellon University Pittsburgh,, PA UNITED STATES Corresponding Author Secondary Information: Corresponding Author's Institution: Carnegie-Mellon University Corresponding Author's Secondary Institution: First Author: Juan J Torres First Author Secondary Information: Order of Authors: Juan J Torres Can Li Robert M Apap I. Grossmann Order of Authors Secondary Information: Funding Information: Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Manuscript Click here to access/download;Manuscript;TwoStage.pdf Click here to view linked References Noname manuscript No. (will be inserted by the editor) 1 2 3 4 5 A Review on the Performance of Linear and 6 Mixed Integer Two-Stage Stochastic Programming 7 Algorithms and Software 8 9 10 Juan J. Torres · Can Li · Robert M. 11 Apap · Ignacio E. Grossmann 12 13 14 15 16 Received: date / Accepted: date 17 18 19 Abstract This paper presents a tutorial on the state-of-the-art methodologies 20 for the solution of two-stage (mixed-integer) linear stochastic programs and 21 provides a list of software designed for this purpose. The methodologies are 22 classified according to the decomposition alternatives and the types of the vari- 23 ables in the problem. We review the fundamentals of Benders Decomposition, 24 Dual Decomposition and Progressive Hedging, as well as possible improve- 25 26 ments and variants. We also present extensive numerical results to underline 27 the properties and performance of each algorithm using software implemen- 28 tations including DECIS, FORTSP, PySP, and DSP. Finally, we discuss the 29 strengths and weaknesses of each methodology and propose future research 30 directions. 31 32 Keywords Stochastic programming · L-shaped method · Scenario Decom- 33 position · Software benchmark 34 35 36 37 Juan J. Torres UniversitéParis 13, Sorbonne Paris Cité,LIPN, CNRS, (UMR 7030), France 38 E-mail: torresfi[email protected] 39 Can Li 40 Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pitts- 41 burgh, PA 15213, USA 42 E-mail: [email protected] 43 Robert M. Apap 44 Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pitts- 45 burgh, PA 15213, USA 46 Ignacio E. Grossmann 47 Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pitts- 48 burgh, PA 15213, USA 49 E-mail: [email protected] 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 2 Juan J. Torres et al. 1 1 Introduction 2 3 In the modeling and optimization of real-world problems, there is usually a 4 level of uncertainty associated with the input parameters and their future 5 outcomes. Stochastic Programming (SP) models have been widely studied to 6 7 solve optimization problems under uncertainty over the past decades [5, 31]. 8 SP is acknowledged for providing superior results than the corresponding de- 9 terministic model with nominal values for the uncertain parameters, which can 10 lead to suboptimal or infeasible solutions. SP applications in process systems 11 engineering include manufacturing networks and supply chain optimization 12 [17, 37], production scheduling [63], synthesis of process networks [57]. 13 Two-stage stochastic programming is a commonly applied framework for 14 cases where parameter uncertainties are decision-independent (exogenous un- 15 certainties). In stochastic models, uncertain parameters are explicitly repre- 16 sented by a set of scenarios. Each scenario corresponds to one possible re- 17 alization of the uncertain parameters according to a discretized probability 18 distribution. The goal of such schemes is to optimize the expected value of the 19 objective function over the full set of scenarios, subject to the implementation 20 of common decisions at the beginning of the planning horizon. 21 Stochastic programs are often difficult to solve due to their large size 22 and complexity that grows with the number of scenarios. To overcome those 23 24 problems, decomposition algorithms such as Benders decomposition [59], La- 25 grangean decomposition [19], and Progressive Hedging [50], have been devel- 26 oped to solve linear programming (LP) and mixed-integer linear programming 27 (MILP) stochastic problems. Moreover, several modeling systems and opti- 28 mization platforms have included extensions for an adequate algebraic repre- 29 sentation of stochastic problems, as offered by major software vendors such as 30 GAMS, LINDO, XpressMP, AIMMS, and Maximal. 31 In recent years, diverse commercial and open-source applications have been 32 developed specifically to represent and solve multistage SP problems. Some of 33 them include capabilities to read and build stochastic MPS (SMPS) files, the 34 standard exchange format for SP applications. However, despite advances in 35 the field and proven benefits, SP has not been widely used in industrial appli- 36 cations. In this paper, we review the current state-of-art of available methods 37 and software for the solution of two-stage stochastic programming problems 38 and evaluate their performance, using large-scale test libraries in SMPS for- 39 mat. 40 41 The remainder of this paper is organized as follows. In Section 2, we explain 42 the mathematical formulation of (mixed-integer) linear stochastic problems. 43 Section 3 describes the classical L-shaped algorithm. Section 4 summarizes en- 44 hancement strategies to improve the performance of Benders decomposition. 45 Section 5 describes scenario decomposition methods and algorithmic modifica- 46 tions. In section 6, we present the software packages for Benders decomposition 47 and show some computational results. In section 7, we describe algorithmic 48 innovations in software packages for dual decomposition and show some com- 49 putational results. Finally, in Section 8 we summarize our conclusions. 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Title Suppressed Due to Excessive Length 3 1 2 Problem Statement 2 3 We consider a two-stage stochastic mixed-integer linear problem (P) in the 4 following form: 5 T X T (P ) min TC = c x + τ!d! y! (1a) 6 x;y 7 !2Ω 8 s:t: Ax ≤ b (1b) 9 x 2 X; X = x : x 2 f0; 1g 8i 2 I ; x ≥ 0 8i 2 InI (1c) 10 i 1 i 1 11 W!y! ≤ h! − T!x 8! 2 Ω (1d) 12 y 2 Y 8! 2 Ω (1e) 13 ! 14 where x denotes the `here and now' decisions, taken at the beginning of the 15 planning horizon before the uncertainties unfold, and Ω is the set of scenarios. 16 Vector y! represents the recourse or corrective actions (wait and see), applied 17 after the realization of the uncertainty. Matrix A 2 Rm1×n1 and vector b 2 18 m1 R represent the first-stage constraints. Matrices T! and W!, and vector 19 m2 m2×n1 h! 2 R , represent the second-stage problem. Matrices T! 2 R and 20 m2×n2 W! 2 R are called technological and recourse matrices, respectively. Let 21 I = f1; 2; ··· ; n g be the index set of all first-stage variables. Set I ⊆ I is the 22 1 1 subset of indices for binary first-stage variables. Let J = f1; 2; ··· ; n g be the 23 2 index set of all second-stage variables. If the second-stage variables are mixed- 24 25 integer, Y = y : yj 2 f0; 1g; 8j 2 J1; yj ≥ 0 8j 2 JnJ1 , where set J1 ⊆ J is the subset of indices for binary second-stage variables. If all the second- 26 27 stage variables are continuous, set J1 = ; and Y = y : yj ≥ 0 8j 2 J . The 28 objective function (TC) minimizes the total expected cost with the scenario 29 probability τ!, and the cost vectors c and d!. Equation (1) is often referred 30 to as the deterministic equivalent, or extensive form of the SP. 31 Formulation (P) can be rewritten in an equivalent form (PNAC) with 32 nonanticipativity constraints (NACs), where the first-stage variables are no 33 longer shared, and each scenario represents an instance of a deterministic 34 problem with a specific realization outcome [9, 52]. 35 36 X T T (P NAC) min TC = τ!(c x! + d! y!) (2a) x! ;y! 37 !2Ω 38 X 39 s:t: H!x! = 0 (2b) 40 !2Ω 41 (x!; y!) 2 G! 8! 2 Ω (2c) 42 43 In equation (2c) , G! represents the feasible region for scenario !, which is 44 defined by constraints (1b)-(1e). Nonanticipativity constraints (2b) are added 45 to ensure that the first-stage decisions are the same across all scenarios. Nonan- 46 ticipativity constraints are represented by suitable sequence of matrices H! 2 47 Rn1·(jΩ|−1)×n1 . One example of such constraints is the following: 48 49 x! = x!−1 8! = 2; 3; :::; jΩj (3) 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 4 Juan J. Torres et al. 1 Given the mathematical structure of the deterministic equivalent formu- 2 lations (P) and (PNAC), (mixed-integer) linear stochastic problems can be 3 solved using decomposition methods derived from duality theory [38]. Such 4 methods split the deterministic equivalent into a master problem and a se- 5 6 ries of smaller subproblems to decentralize the overall computational burden. 7 Decomposition methodologies are classified in two groups: (i) time-stage or 8 vertical decomposition which includes Benders decomposition and variants, 9 and, (ii) scenario-based or horizontal decomposition.

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