Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional … Read more
In this work we explicit a derivative-free trust-region algorithm for unconstrained optimization based on the paper (Computational Optimization and Applications 53: 527-555, 2012) proposed by Powell. The objective function is approximated by quadratic models obtained by polynomial interpolation. The number of points of the interpolation set is fixed. In each iteration only one interpolation point … Read more
In this work, we look into new derivative-free methods to solve equality-constrained optimization problems. Of particular interest, are the trust-region techniques, which have been investigated for the unconstrained and bound-constrained cases. For solving equality-constrained optimization problems, we introduce a derivative-free adaptation of the trust-funnel method combined with a self-correcting geometry scheme and present some encouraging … Read more
In this paper, we propose new linesearch-based methods for nonsmooth optimization problems when first-order information on the problem functions is not available. In the first part, we describe a general framework for bound-constrained problems and analyze its convergence towards stationary points, using the Clarke-Jahn directional derivative. In the second part, we consider inequality constrained optimization … Read more
In this paper it is proposed to equip direct-search methods with a general procedure to minimize an objective function, possibly non-smooth, without using derivatives and subject to constraints on the variables. One aims at considering constraints, most likely nonlinear or non-smooth, for which the derivatives of the corresponding functions are also unavailable. The novelty of … Read more
In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in … Read more
In this paper we prove that the broad class of direct-search methods of directional type, based on imposing sufficient decrease to accept new iterates, exhibits the same global rate or worst case complexity bound of the gradient method for the unconstrained minimization of a convex and smooth function. More precisely, it will be shown that … Read more
The mesh adaptive direct search (MADS) class of algorithms is designed for nonsmooth optimization, where the objective function and constraints are typically computed by launching a time-consuming computer simulation. Each iteration of a MADS algorithm attempts to improve the current best-known solution by launching the simulation at a finite number of trial points. Common implementations … Read more
A number of variants of the classical Markowitz mean-variance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the portfolio, in an attempt to improve its performance and … Read more
For smooth objective functions it has been shown that the worst case cost of direct-search methods is of the same order as the one of steepest descent, when measured in number of iterations to achieve a certain threshold of stationarity. Motivated by the lack of such a result in the non-smooth case, we propose, analyze, … Read more