Perspective Reformulation and Applications

In this paper we survey recent work on the perspective reformulation approach that generates tight, tractable relaxations for convex mixed integer nonlinear programs (MINLP)s. This preprocessing technique is applicable to cases where the MINLP contains binary indicator variables that force continuous decision variables to take the value 0, or to belong to a convex set. … Read more

Code verification by static analysis: a mathematical programming approach

Automatic verification of computer code is of paramount importance in embedded systems supplying essential services. One of the most important verification techniques is static code analysis by abstract interpretation: the concrete semantics of a programming language (i.e.the values $\chi$ that variable symbols {\tt x} appearing in a program can take during its execution) are replaced … Read more

A note on Burer’s copositive representation of mixed-binary QPs

In an important paper, Burer recently showed how to reformulate general mixed-binary quadratic optimization problems (QPs) into copositive programs where a linear functional is minimized over a linearly constrained subset of the cone of completely positive matrices. In this note we interpret the implication from a topological point of view, showing that the Minkowski sum … Read more

Disjunctive cuts for non-convex MINLP

Mixed Integer Nonlinear Programming (MINLP) problems present two main challenges: the integrality of a subset of variables and nonconvex (nonlinear) objective function and constraints. Many exact solvers for MINLP are branch-and-bound algorithms that compute a lower bound on the optimal solution using a linear programming relaxation of the original problem. In order to solve these … Read more

Old Wine in a New Bottle: The MILP Road to MIQCP

This paper surveys results on the NP-hard mixed-integer quadratically constrained programming problem. The focus is strong convex relaxations and valid inequalities, which can become the basis of efficient global techniques. In particular, we discuss relaxations and inequalities arising from the algebraic description of the problem as well as from dynamic procedures based on disjunctive programming. … Read more

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.  The primary goal is … Read more

An algorithmic framework for MINLP with separable non-convexity

Global optimization algorithms, e.g., spatial branch-and-bound approaches like those implemented in codes such as BARON and COUENNE, have had substantial success in tackling complicated, but generally small scale, non-convex MINLPs (i.e., mixed-integer nonlinear programs having non-convex continuous relaxations). Because they are aimed at a rather general class of problems, the possibility remains that larger instances … Read more

On Solving Single-objective Fuzzy Integer Linear Fractional Programs

A suggested program with fuzzy linear fractional objective and integer decision variables (FILFP) is considered. The fuzzy coefficients are involved in the numerator of the linear objective function and can be characterized by trapezoidal fuzzy numbers. The purpose of this paper is to outline an algorithm available to solve (FILFP). In addition, an illustrative example … Read more

On convex relaxations of quadrilinear terms

The best known method to find exact or at least epsilon-approximate solutions to polynomial programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the … Read more

Stochastic binary problems with simple penalties for capacity constraints violations

This paper studies stochastic programs with first-stage binary variables and capacity constraints, using simple penalties for capacities violations. In particular, we take a closer look at the knapsack problem with weights and capacity following independent random variables and prove that the problem is weakly \NP-hard in general. We provide pseudo-polynomial algorithms for three special cases … Read more