Scenario-based cuts for structured two-stage stochastic and distributionally robust p-order conic mixed integer programs

Abstract

In this paper, we derive (partial) convex hull for deterministic multi-constraint polyhedral conic mixed integer sets with multiple integer variables using conic mixed integer rounding (CMIR) cut-generation procedure of Atamtürk and Narayanan (Math Prog 122:1-20, 2008), thereby extending their result for a simple polyhedral conic mixed integer set with single constraint and one integer variable. We then introduce two-stage stochastic p-order conic mixed integer programs (denoted by TSS-CMIPs) in which the second stage problems have sum of l p -norms in the objective function along with integer variables. First, we present sufficient conditions under which the addition of scenario-based nonlinear cuts in the extensive formulation of TSS-CMIPs is sufficient to relax the integrality restrictions on the second stage integer variables without impacting the integrality of the optimal solution of the TSS-CMIP. We utilize scenario-based CMIR cuts for TSS-CMIPs and their distributionally robust generalizations with structured CMIPs in the second stage, and prove that these cuts provide conic/linear programming equivalent or approximation for the second stage CMIPs. We also perform extensive computational experiments by solving stochastic and distributionally robust capacitated facility location problem and randomly generated structured TSS-CMIPs with polyhedral CMIPs and second-order CMIPs in the second stage, i.e. p = 1 and p = 2, respectively. We observe that there is a significant reduction in the total time taken to solve these problems after adding the scenario-based cuts. KeywordsTwo-stage stochastic p-order conic mixed integer program • Scenario-based cutting planes • Two-stage distributionally robust program • (Partial) convex hull • Conic mixed integer rounding • Multi-module capacitated facility location B Manish Bansal

Applications of Stochastic Mixed-Integer Second-Order Cone Optimization

“…We refer the reader to two papers that deal with important special cases of or related to the SMISOCP problem (3,4): The first paper is by Luo and Mehrotra [54] which extends the work of Sen and Sherali [51] for SMILP and proposes a decomposition method for (3,4) in which x ∈ {0, 1} p × R n−p in the first-stage problem in (3,4) and y ∈ Z q × R m−q in the second-stage problem in (3,4). The second paper is by Bansal and Zhang [55] which proposes scenario-based cuts for (3,4) in which n = p and k = r in the first-stage problem in (3,4) and the l 2 -norms are generalized to l p -norms in the second-stage problem in (3,4).…”

“…(28) Because the constraints in the first-stage problem (27) are linear with integer variables and without the use of second-order cone relaxations, a solution for Model (27,28) is now possible. One way to approach this model is by applying scenario-based cuts proposed in [55].…”

“…It is important to indicate that all of the optimization models that have been developed in this study are intractable for an extensive form formulation. Therefore, solutions for such application models are now possible by applying either the decomposition algorithm developed recently in [54] or the scenario-based cuts proposed recently in [55]. Developing other and more specialized algorithms to solve the generic two-stage stochastic mixed-integer second-order cone programming problem is an important subject of study.…”

A Finitely Convergent Cutting Plane, and a Bender's Decomposition Algorithm for Mixed-Integer Convex and Two-Stage Convex Programs using Cutting Planes

“…Recent work has also provided insights into developing tighter formulations by identifying globally valid parametric inequalities (see [23], and references therein). For two-stage stochastic mixedinteger conic optimization, Bansal and Zhang [24] have developed nonlinear sparse cuts for tightening the second-stage formulation of a class of two-stage stochastic p-order conic mixed-integer programs by extending the results of [25] on convexifying a simple polyhedral conic mixed-integer set.…”

A Decomposition Method for Distributionally-Robust Two-stage Stochastic Mixed-integer Cone Programs

“…Recent work has also provided insights into developing tighter formulations by identifying globally valid parametric inequalities (see [12], and references therein). Specifically, Bansal and Zhang [13] have developed nonlinear sparse cuts for tightening the second-stage formulation of a class of two-stage stochastic p-order conic mixed-integer programs by extending the results of [14] on convexifying a simple polyhedral conic mixed integer set.…”