A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ vertices are not known when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$ and $s\ge 3$, and show that the perimeters and the widths obtained … Read more
We consider the bound-constrained global optimization of functions with low effective dimensionality, that are constant along an (unknown) linear subspace and only vary over the effective (complement) subspace. We aim to implicitly explore the intrinsic low dimensionality of the constrained landscape using feasible random embeddings, in order to understand and improve the scalability of algorithms … Read more
Hammerstein-Wiener models constitute a significant class of block-structured dynamic models, as they approximate process nonlinearities on the basis of input-output data without requiring identification of a full nonlinear process model. Optimization problems with Hammerstein-Wiener models embedded are nonconvex, and thus local optimization methods may obtain suboptimal solutions. In this work, we develop a deterministic global … Read more
In this paper, a new randomized solver (called VRDFON) for noisy unconstrained derivative-free optimization (DFO) problems is discussed. Complexity result in the presence of noise for nonconvex functions is studied. Two effective ingredients of VRDFON are an improved derivative-free line search algorithm with many heuristic enhancements and quadratic models in adaptively determined subspaces. Numerical results … Read more
In this paper we introduce RAPOSa, a global optimization solver specifically designed for (continuous) polynomial programming problems with box-constrained variables. Written entirely in C++, RAPOSa is based on the Reformulation-Linearization Technique developed by Sherali and Tuncbilek (1992) and subsequently improved in Sherali et al. (2012a), Sherali et al. (2012b) and Dalkiran and Sherali (2013). We … Read more
It is well known that the convex hull of {(x,y,xy)}, where (x,y) is constrained to lie in a box, is given by the Reformulation-Linearization Technique (RLT) constraints. Belotti et al. (2010) and Miller et al. (2011) showed that if there are additional upper and/or lower bounds on the product z=xy, then the convex hull can … Read more
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
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
\(\) 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 … 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