Nicolas W. Sawaya

An algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

A cutting plane method for solving linear generalized disjunctive programming problems

Raman and Grossmann [Raman, R., & Grossmann, I.E. (1994). Modeling and computational techniques for logic based integer programming. Computers and Chemical Engineering, 18(7), 563–578] and Lee and Grossmann [Lee, S., & Grossmann, I.E. (2000). New algorithms for nonlinear generalized disjunctive programming. Computers and Chemical Engineering, 24, 2125–2141] have developed a reformulation of Generalized Disjunctive Programming (GDP) problems that is based on determining the convex hull of each disjunction. Although the relaxation of the reformulated problem using this method will often produce a significantly tighter lower bound when compared with the traditional big-M reformulation, the limitation of this method is that the representation of the convex hull of every disjunction requires the introduction of new disaggregated variables and additional constraints that can greatly increase the size of the problem. In order to circumvent this issue, a cutting plane method that can be applied to linear GDP problems is proposed in this paper. The method relies on converting the GDP problem into an equivalent big-M reformulation that is successively strengthened by cuts generated from an LP or QP separation problem. The sequence of problems is repeatedly solved, either until the optimal integer solution is found, or else until there is no improvement within a specified tolerance, in which case one switches to a branch and bound method. The strip-packing, retrofit planning and zero-wait job-shop scheduling problems are presented as examples to illustrate the performance of the proposed cutting plane method.

A discretization-based approach for the optimization of the multiperiod blend scheduling problem

Abstract In this paper, we introduce a new generalized multiperiod scheduling version of the pooling problem to represent time varying blending systems. A general nonconvex MINLP formulation of the problem is presented. The primary difficulties in solving this optimization problem are the presence of bilinear terms, as well as binary decision variables required to impose operational constraints. An illustrative example is presented to provide unique insight into the difficulties faced by conventional MINLP approaches to this problem, specifically in finding feasible solutions. Based on recent work, a new radix-based discretization scheme is developed with which the problem can be reformulated approximately as an MILP, which is incorporated in a heuristic procedure and in two rigorous global optimization methods, and requires much less computational time than existing global optimization solvers. Detailed computational results of each approach are presented on a set of examples, including a comparison with other global optimization solvers.

A hierarchy of relaxations for linear generalized disjunctive programming

Generalized disjunctive programming (GDP), originally developed by Raman and Grossmann (1994), is an extension of the well-known disjunctive programming paradigm developed by Balas in the mid 70s in his seminal technical report (Balas, 1974). This mathematical representation of discrete-continuous optimization problems, which represents an alternative to the mixed-integer program (MIP), led to the development of customized algorithms that successfully exploited the underlying logical structure of the problem. The underlying theory of these methods, however, borrowed only in a limited way from the theories of disjunctive programming, and the unique insights from Balas’ work have not been fully exploited.

Computational implementation of non-linear convex hull reformulation

Abstract Lee and Grossmann [Lee, S., & Grossmann, I. E. (2000). New algorithms for nonlinear generalized disjunctive programming. Computers and Chemical Engineering , 24 , 2125–2141] have developed a reformulation for nonlinear Generalized Disjunctive Programming (GDP) problems that obtains from the intersection of the convex hulls of every disjunction. In order to computationally implement this method, it is necessary to reformulate the problem in such a way so as to avoid division by zero in the nonlinear inequalities present amongst the convex hull constraints, while preserving the convex nature of the problem. To accomplish this, we propose to replace the original set of nonlinear constraints by two different sets of convex constraints that circumvent the aforementioned problem and that approximate the original set of constraints exactly at their limit. Furthermore, we compare the two sets of approximating constraints against each other and give rigorous theoretical conditions under which one is superior to the other. Finally, we illustrate the efficiency of both approximations on a variety of numerical examples draw from the Chemical Engineering and Operations Research literature.

A Novel Global Optimization Approach to the Multiperiod Blending Problem

Abstract In this paper, we introduce a generalized multiperiod version of the pooling problem to represent time varying blending systems, and also propose novel approaches to solve these problems to global optimality. The primary difficulties in solving this optimization problem are the presence of bilinear terms inherent in blending operations, as well as binary decision variables required to impose the operational constraints over multiple time periods. A general nonconvex MINLP formulation is presented that is used to globally optimize small systems, but quickly becomes intractable as problem size increases. A novel approximation of specified precision for the nonconvex bilinear terms is developed (a radix-based discretization scheme), with which the problem can be reformulated as an MILP. Solving this new formulation requires much less computational time than when the MINLP model is solved directly with a global optimization solver such as BARON. This then allows for the global optimization of larger blending systems. A comparison of the two formulations is presented, along with detailed computational results of each approach.