Fenchel decomposition for stochastic mixed-integer programming

Abstract

This paper introduces a new cutting plane method for two-stage stochastic mixed-integer programming (SMIP) called Fenchel decomposition (FD). FD uses a class of valid inequalities termed, FD cuts, which are derived based on Fenchel cutting planes from integer programming. First, we derive FD cuts based on both the first and second-stage variables, and devise an FD algorithm for SMIP and establish finite convergence for binary first-stage. Second, we derive FD cuts based on the second-stage variables only and use an idea from disjunctive programming to lift the cuts to the higher dimension space including the first-stage variables. We then devise an alternative algorithm (FD-L algorithm) based on the lifted FD cuts. Finally, we report on computational results based on several test instances from the literature. The results are promising and show that both algorithms can outperform a standard direct solver and a disjunctive decomposition algorithm on some large-scale instances. Furthermore, the FD-L algorithm provides better performance than the FD algorithm in general.

Pure Cutting‐Plane Algorithms and their Convergence

“…However, pure cutting-plane algorithms have important applications in the solution of two-stage stochastic MIPs. We refer the reader to the articles by Sen and Higle [35], for a decomposition algorithm based on lift-and-project cuts for two-stage stochastic mixed-binary programs, Gade et al [36], for decomposition algorithms using Gomory cuts for two-stage stochastic pure integer programs, and Ntaimo [37], for a decomposition algorithm for two-stage stochastic mixed-binary programs based on the so-called Fenchel cuts. (A pure cuttingplane algorithm based on Fenchel cuts [38] is shown to be finitely convergent for deterministic MIPs in [39].…”

Stage- and scenario-wise Fenchel decomposition for stochastic mixed 0-1 programs with special structure

“…The basic stage-wise FD algorithm is given in Ntaimo [22]. Here we extend the algorithm by using the L-shaped algorithm to solve the LP relaxation of SIP2 and then carefully choosing a starting solution for the FD algorithm to yield better results.…”

“…In this paper, we study both stage-and scenario-wise decomposition for SIP2 using Fenchel decomposition (FD) for SIP. FD is a cutting plane approach that was originally developed for SIP2 under the stage-wise decomposition setting [22]. In this work we extend this approach to the scenario-wise decomposition setting and derive a new class of Fenchel cutting planes called, scenario FD cuts.…”

“…This work also compares the use of Fenchel cuts to Lagrangian cuts in finding good relaxation bounds for their problem. Fenchel cuts were derived for two-stage SIPs under a stage-wise decomposition setting in Ntaimo [22]. Considering x as a first-stage decision variable and y as the second-stage decision variable, two forms of cuts called Fenchel decomposition (FD) cuts are derived: one based on the (x,y) space and the other derived based on the y space and then lifted to the (x,y) space.…”

A Two-Phase Combined Gradient-Tunneling Based Algorithm for Constrained Integer Programming Problems

“…A heuristic approach to find the global optimum of complex nonlinear optimization problems is based on evolutionary and genetic algorithms; several works based on this approach can be found in the review papers [26,40]. Recent algorithmic and methodological developments for linear and non-linear integer programming and mixed problems can be found in [29,52,27,25,31,1,4,3,23,49,5]. Additionally, several commercial solvers for linear and non-linear integer programming problems are available.…”