In this paper we present and analyze a finitely-convergent disjunctive cutting plane algorithm to obtain an \(\epsilon\)-optimal solution or detect infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained in a manner similar to Saxena, Bonami, and Lee [Math. Prog. 130: 359–413, 2011] and Fampa and Lee [J. Global Optim. … 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
We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield … Read more
The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens … Read more
In this paper we introduce a technique to produce tighter cutting planes for mixed-integer non-linear programs. Usually, a cutting plane is generated to cut off a specific infeasible point. The underlying idea is to use the infeasible point to restrict the feasible region in order to obtain a tighter domain. To ensure validity, we require … Read more
We study the problem of minimizing a convex function on the integer lattice when the function cannot be evaluated at noninteger points. We propose a new underestimator that does not require access to (sub)gradients of the objective but, rather, uses secant linear functions that interpolate the objective function at previously evaluated points. These linear mappings … Read more
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over … Read more
We apply a state-of-the-art, local derivative-free solver, Py-BOBYQA, to global optimization problems, and propose an algorithmic improvement that is beneficial in this context. Our numerical findings are illustrated on a commonly-used test set of global optimization problems and associated noisy variants, and on hyperparameter tuning for a machine learning test set. As Py-BOBYQA is a … Read more
In this paper, we consider the nonconvex quadratic optimization problem with a single quadratic constraint. First we give a theoretical characterization of the local non-global minimizers. Then we extend the recent characterization of the global minimizer via a generalized eigenvalue problem to the local non-global minimizers. Finally, we use these results to derive an efficient … Read more
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program … Read more