We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM … Read more
A direct-search derivative-free Matlab optimizer for bound-constrained problems is described, whose remarkable features are its ability to handle a mix of continuous and discrete variables, a versatile interface as well as a novel self-training option. Its performance compares favourably with that of NOMAD, a state-of-the art package. It is also applicable to multilevel equilibrium- or … Read more
Applications in engineering frequently require the adjustment of certain parameters. While the mathematical laws that determine these parameters often are well understood, due to time limitations in every day industrial life, it is typically not feasible to derive an explicit computational procedure for adjusting the parameters based on some given measurement data. This paper aims … Read more
We investigate the application of the nonmonotone spectral projected gradient (SPG) method to a region-based variational model for image segmentation. We consider a “discretize-then-optimize” approach and solve the resulting nonlinear optimization problem by an alternating minimization procedure that exploits the SPG2 algorithm by Birgin, Martìnez and Raydan (SIAM J. Optim., 10(4), 2000). We provide a … Read more
We describe the design of a C++ vector-manipulation substrate that allows first-order optimization algorithms to be expressed in a concise and readable manner, yet still achieve high performance in parallel computing environments. We use standard object-oriented techniques of encapsulation and operator overloading, combined with a novel “symbolic temporaries” delayed-evaluation system that greatly reduces the overhead … Read more
We propose a trust-region method for nonlinear semidefinite programs with box-constraints. The penalty barrier method can handle this problem, but the size of variable matrices available in practical time is restricted to be less than 500. We develop a trust-region method based on the approach of Coleman and Li (1996) that utilizes the distance to … Read more
We consider the problem of optimizing an unknown function given as an oracle over a mixed-integer box-constrained set. We assume that the oracle is expensive to evaluate, so that estimating partial derivatives by finite differences is impractical. In the literature, this is typically called a black-box optimization problem with costly evaluation. This paper describes the … Read more
In this paper we introduce a new method for solving box-constrained mixed-integer polynomial problems to global optimality. The approach, a specialized branch-and-bound algorithm, is based on the computation of lower bounds provided by the minimization of separable underestimators of the polynomial objective function. The underestimators are the novelty of the approach because the standard approaches … Read more
The aim of this paper is to extend the applicability of the incomplete oblique projections method (IOP) previously introduced by the authors for solving inconsistent linear systems to the box constrained case. The new algorithm employs incomplete projections onto the set of solutions of the augmented system Ax-r= b, together with the box constraints, based … Read more
We propose a new framework for the optimization of computationally expensive black box problems, where neither closed-form expressions nor derivatives of the objective functions are available. The proposed framework consists of two procedures. The first constructs a global metamodel to approximate the underlying black box function and explores an unvisited area to search for a … Read more