Efficient Separation of RLT Cuts for Implicit and Explicit Bilinear Products

The reformulation-linearization technique (RLT) is a prominent approach to constructing tight linear relaxations of non-convex continuous and mixed-integer optimization problems. The goal of this paper is to extend the applicability and improve the performance of RLT for bilinear product relations. First, a method for detecting bilinear product relations implicitly contained in mixed-integer linear programs is developed based on analyzing linear constraints with binary variables, thus enabling the application of bilinear RLT to a new class of problems. Our second contribution addresses the high computational cost of RLT cut separation, which presents one of the major difficulties in applying RLT efficiently in practice. We propose a new RLT cutting plane separation algorithm which identifies combinations of linear constraints and bound factors that are expected to yield an inequality that is violated by the current relaxation solution. A detailed computational study based on implementations in two solvers evaluates the performance impact of the proposed methods.Comment: 16 pages, 0 figures, submitted to the 24th Conference on Integer Programming and Combinatorial Optimizatio

Topics in Mixed Integer Nonlinear Optimization

Mixed integer nonlinear optimization has many applications ranging from machine learning to power systems. However, these problems are very challenging to solve to global optimality due to the inherent non-convexity. This typically leads the problem to be NP-hard. Moreover, in many applications, there are time and resource limitations for solving real-world problems, and the sheer size of real instances can make solving them challenging. In this thesis, we focus on important elements of nonconvex optimization - including mixed integer linear programming and nonlinear programming, where both theoretical analyses and computational experiments are presented. In the first chapter we look at Mixed Integer Quadratic Programming (MIQP), the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. We utilize the augmented Lagrangian dual (ALD), which augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomial bound on the weight on the penalty term to obtain a zero gap. In the second chapter we apply the technique of lifting to bilinear programming, a special case of quadratic constrained quadratic programming. We first show that, for sets described by one bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality. To reduce computational burden, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bilinear sets. We then study a separable bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We study the lifting function of the bilinear cover inequality and lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bilinear sets with box constraints. In the third chapter we continue our research around separable bilinear programming. We first prove that the semidefinite programming relaxation provides no benefit over the McCormick relaxation for separable bilinear optimization problems. We then design a simple randomized separation heuristic for lifted bilinear cover inequalities. In our computational experiments, we separate many rounds of these inequalities starting from the McCormick relaxation of bilinear instances where each constraint is a separable bilinear constraint set. Our main result is to demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of the gap closed, when these inequalities are added at the root node compared to when these inequalities are not added. In the fourth chapter we look at Mixed Integer Linear Programming (MILP) that arises in operational applications. Many routinely-solved MILPs are extremely challenging not only from a worst-case complexity perspective, but also because of the necessity to obtain good solutions within limited time. An example is the Security-Constrained Unit Commitment (SCUC) problem, solved daily to clear the day-ahead electricity markets. We develop ML-based methods for improving branch-and-bound variable selection rules that exploit key features of such operational problems: similar decisions are generated within the same day and across different days, based on the same power network. Utilizing similarity between instances and within an instance, we build one separate ML model per variable or per group of similar variables for learning to predict the strong branching score. The approach is able to produce branch-and-bound trees which gap closed only slightly worse than that of trees obtained by strong branching, while it outperforms previous machine learning schemes.Ph.D

Reformulations of mathematical programming problems as linear complementarity problems

A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are (i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables, (ii.) Minimum Linear Complementarity Problem (MLCP) which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized, (iii.) Second Minimum Linear Complementarity Problem (SMLCP) which is an MLCP but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value. A number of well known mathematical programming problems, namely quadratic programming (convex, nonconvex, pseudoconvex nonconvex), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems

Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition

In this paper, we study chance constrained mixed integer program with consideration of recourse decisions and their incurred cost, developed on a finite discrete scenario set. Through studying a non-traditional bilinear mixed integer formulation, we derive its linear counterparts and show that they could be stronger than existing linear formulations. We also develop a variant of Jensen's inequality that extends the one for stochastic program. To solve this challenging problem, we present a variant of Benders decomposition method in bilinear form, which actually provides an easy-to-use algorithm framework for further improvements, along with a few enhancement strategies based on structural properties or Jensen's inequality. Computational study shows that the presented Benders decomposition method, jointly with appropriate enhancement techniques, outperforms a commercial solver by an order of magnitude on solving chance constrained program or detecting its infeasibility

Facets of a mixed-integer bilinear covering set with bounds on variables

We derive a closed form description of the convex hull of mixed-integer bilinear covering set with bounds on the integer variables. This convex hull description is determined by considering some orthogonal disjunctive sets defined in a certain way. This description does not introduce any new variables, but consists of exponentially many inequalities. An extended formulation with a few extra variables and much smaller number of constraints is presented. We also derive a linear time separation algorithm for finding the facet defining inequalities of this convex hull. We study the effectiveness of the new inequalities and the extended formulation using some examples

Recursive McCormick Linearization of Multilinear Programs

Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLP). One example is the family of Recursive McCormick Linearization (RML) strategies, where bilinear products are substituted for artificial variables, which deliver a relaxation of the original problem when introduced together with concave and convex envelopes. In this article, we introduce the first systematic approach for identifying RMLs, in which we focus on the identification of linear relaxation with a small number of artificial variables and with strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation for the identification of minimum-size RMLs; we show that this problem is NP-hard in general, whereas a special case is fixed-parameter tractable. Moreover, we explore structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best possible relaxation bound is optimal. Our numerical results on a collection of benchmarks indicate that our algorithms outperform the RML strategy implemented in state-of-the-art global optimization solvers.Comment: 22 pages, 11 figures, Under Revie

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York