Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

Abstract

In this work, we propose an algorithmic approach to improve mixed-integer models that are originally formulated as convex Generalized Disjunctive Programs (GDP). The algorithm seeks to obtain an improved continuous relaxation of the MILP/MINLP reformulation of the GDP, while limiting the growth in the problem size. There are three main stages that form the basis of the algorithm. The first one is a pre-solve, consequence of the logic nature of GDP, which allows us to reduce the problem size, find good relaxation bounds and identify properties that help us determine where to apply a basic step. The second stage is the iterative application of basic steps, selecting where to apply them, and monitoring the improvement of the formulation. Finally, we use a hybrid reformulation of GDP that seeks to exploit both of the advantages attributed to the two common GDP-to-MILP/MINLP transformations, the Big-M and Hull reformulation. We illustrate the application of this algorithm with several examples. The results show the improvement in the problem formulations by generating models with improved relaxed solutions and relatively small growth in the number of continuous variables and constraints. The algorithm generally leads to reduction in the solution times.

Improved Big-M reformulation for generalized disjunctive programs

“…We note that the general concept of this reformulation was presented before [26]. More recently, a similar reformulation was presented for linear problems in the context of MILP formulation techniques [27].…”

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

“…There has been work suggesting which basic steps to apply [55,81,82], and to algorithmically Figure 5: Illustration of (HR) (a) before, and (b) after the application of a basic step improve formulations while limiting the problem size growth [83].…”

Computational strategies for improved MINLP algorithms

“…The test instances are 19 benchmark MINLP models from Trespalacios and Grossmann [31] and minlplib [32] .…”

“…For example, when solving the convex MINLP problem CLAY55M with DICOPT [31] , the number of iterations and CPU time can be very large as shown in Figure 1, due to the fact that there are many infeasible subproblems.…”