This paper addresses derivative-free optimization problems where the variables lie implicitly in an unknown discrete closed set. The evaluation of the objective function follows a projection onto the discrete set, which is assumed dense rather than sparse. Such a mathematical setting is a rough representation of what is common in many real-life applications where, despite … Read more
We present a new derivative-free algorithm, ORBIT, for unconstrained local optimization of computationally expensive functions. A trust-region framework using interpolating Radial Basis Function (RBF) models is employed. The RBF models considered often allow ORBIT to interpolate nonlinear functions using fewer function evaluations than the polynomial models considered by present techniques. Approximation guarantees are obtained by … Read more
A numerical study of model-based methods for derivative-free optimization is presented. These methods typically include a geometry phase whose goal is to ensure the adequacy of the interpolation set. The paper studies the performance of an algorithm that dispenses with the geometry phase altogether (and therefore does not attempt to control the position of the … Read more
We propose data profiles as a tool for analyzing the performance of derivative-free optimization solvers when there are constraints on the computational budget. We use performance and data profiles, together with a convergence test that measures the decrease in function value, to analyze the performance of three solvers on sets of smooth, noisy, and piecewise-smooth … Read more
In this paper a numerical approach for the optimization of stirrer configurations is presented. The methodology is based on a flow solver, and a mathematical optimization tool, which are integrated into an automated procedure. The flow solver is based on the discretization of the incompressible Navier-Stokes equations by means of a fully conservative finite-volume method … Read more
In this survey article we give the basic description of the interpolation based derivative free optimization methods and their variants. We review the recent contributions dealing with the maintaining the geometry of the interpolation set, the management of the trust region radius and the stopping criteria. Derivative free algorithms developed for problems with some structure … Read more
It has been shown recently that the efficiency of direct search methods that use opportunistic polling in positive spanning directions can be improved significantly by reordering the poll directions according to descent indicators built from simplex gradients. The purpose of this paper is twofold. First, we analyze the properties of simplex gradients of nonsmooth functions … Read more
In this paper we prove global convergence for first and second-order stationarity points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of linear or quadratic models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds but, … Read more
Generating set search (GSS) is a family of direct search methods that encompasses generalized pattern search and related methods. We describe an algorithm for asynchronous linearly-constrained GSS, which has some complexities that make it different from both the asynchronous bound-constrained case as well as the synchronous linearly-constrained case. The algorithm has been implemented in the … Read more
In this paper a new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can … Read more