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
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
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
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
We extend recent work on nonlinear optimal control problems with integer restrictions on some of the control functions (mixed-integer optimal control problems, MIOCP) in two ways. We improve a theorem that states that the solution of a relaxed and convexified problem can be approximated with arbitrary precision by a solution fulfilling the integer requirements. Unlike … Read more
Quadratic assignment problems (QAPs) are among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings … Read more
If a mathematical program (be it linear or nonlinear) has many symmetric optima, solving it via Branch-and-Bound techniques often yields search trees of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for automatically finding the formulation group of any given Mixed-Integer Nonlinear Program, and reformulating the problem so … Read more
In model-based nonlinear optimal control switching decisions that can be optimized often play an important role. Prominent examples of such hybrid systems are gear switches for transport vehicles or valves in chemical engineering. Optimization algorithms need to take the discrete nature of the variables that model these switching decisions into account. Unnecessarily, for many applications … Read more
A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables $Y_{ij}$ to represent each of the products $x_i x_j$ of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can … Read more
In this paper, we develop a convexification tool that enables the construction of convex hulls for orthogonal disjunctive sets using convex extensions and disjunctive programming techniques. A distinguishing feature of our technique is that, unlike most applications of disjunctive programming, it does not require the introduction of new variables in the relaxation. We develop and … Read more