It is known that the Clarke generalized directional derivative is nonnegative along the limit directions generated by directional direct-search methods at a limit point of certain subsequences of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point. In this paper we generalize this result for non-Lipschitzian functions using Rockafellar generalized … Read more
In this paper we address the global optimization of functions subject to bound and linear constraints without using derivatives of the objective function. We investigate the use of derivative-free models based on radial basis functions (RBFs) in the search step of direct-search methods of directional type. We also study the application of algorithms based on … Read more
Portfolio optimisation for a Fund of Hedge Funds (“FoHF”) has to address the asymmetric, non-Gaussian nature of the underlying returns distributions. Furthermore, the objective functions and constraints are not necessarily convex or even smooth. Therefore traditional portfolio optimisation methods such as mean-variance optimisation are not appropriate for such problems and global search optimisation algorithms could … Read more
We propose a Generating Search Set method for solving the oblique l_1 Procrustes problem. Implementative details, algorithmic choices and theoretical properties of the method are discussed. The results of some numerical experiments are reported. Citation in Applied and Industrial Mathematics in Italy III – Proceedings of the 9th Conference SIMAI, De Bernardis et. Al. (eds), … Read more
The goal of this paper is to show that the use of minimum Frobenius norm quadratic models can improve the performance of direct-search methods. The approach taken here is to maintain the structure of directional direct-search methods, organized around a search and a poll step, and to use the set of previously evaluated points generated … Read more
PSwarm was developed originally for the global optimization of functions without derivatives and where the variables are within upper and lower bounds. The underlying algorithm used is a pattern search method, more specifically a coordinate search method, which guarantees convergence to stationary points from arbitrary starting points. In the (optional) search step of coordinate search, … Read more
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 consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally Lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra. A generating set search method is proposed, which is named “structured” because the structure of the set of nondifferentiability near the current iterate is exploited to … Read more
We generalize the Nelder-Mead simplex and LTMADS algorithms and, the frame based methods for function minimization to Riemannian manifolds. Examples are given for functions defined on the special orthogonal Lie group $\mathcal{SO}(n)$ and the Grassmann manifold $\mathcal{G}(n,k)$. Our main examples are applying the generalized LTMADS algorithm to equality constrained optimization problems and, to the Whitney … Read more
We present a general procedure for handling equality constraints in optimization problems that is of particular use in direct search methods. First we will provide the necessary background in differential geometry. In particular, we will see what a Riemannian manifold is, what a tangent space is, how to move over a manifold and how to … Read more