A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of … Read more
Solving mixed-integer nonlinear optimization problems (MINLPs) to global optimality is extremely challenging. An important step for enabling their solution consists in the design of convex relaxations of the feasible set. Known solution approaches based on spatial branch-and-bound become more effective the tighter the used relaxations are. Relaxations are commonly established by convex underestimators, where each … Read more
This paper offers a novel approach for computing globally optimal solutions to the pump scheduling problem in drinking water distribution networks. A tight integer linear relaxation of the original non-convex formulation is devised and solved by branch and bound where integer nodes are investigated through non-linear programming to check the satisfaction of the non-convex constraints … Read more
We consider quadratic optimization in variables (x,y), 0
While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD) constraint and only enforce PSD-ness on smaller k times k principal submatrices — we call this the sparse SDP relaxation. Surprisingly, … Read more
Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study sufficient conditions for a convex hull result that immediately implies that the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such … Read more
We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is … Read more
The problem of finding global minima of nonlinear discrete functions arises in many fields of practical matters. In recent years, methods based on discrete filled functions become popular as ways of solving these sort of problems. However, they rely on the steepest descent method for local searches. Here we present an approach that does not … Read more
A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the alphaBB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained … Read more
We investigate the unconstrained global optimization of functions with low effective dimensionality, that are constant along certain (unknown) linear subspaces. Extending the technique of random subspace embeddings in [Wang et al., Bayesian optimization in a billion dimensions via random embeddings. JAIR, 55(1): 361–387, 2016], we study a generic Random Embeddings for Global Optimization (REGO) framework … Read more