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In practical applications of optimization it is common to have several conflicting objective functions to optimize. Frequently, these functions are subject to noise or can be of black-box type, preventing the use of derivative-based techniques.We propose a novel multiobjective derivative-free methodology, calling it direct multisearch (DMS), which does not aggregate any of the objective functions. Our framework is inspired by the search/poll paradigm of direct-search methods of directional type and uses the concept of Pareto dominance to maintain a list of nondominated points (from which the new iterates or poll centers are chosen). The aim of our method is to generate as many points in the Pareto front as possible from the polling procedure itself, while keeping the whole framework general enough to accommodate other disseminating strategies, in particular when using the (here also) optional search step. DMS generalizes to multiobjective optimization (MOO) all direct-search methods of directional type.We prove under the common assumptions used in direct search for single optimization that at least one limit point of the sequence of iterates generated by DMS lies in (a stationary form of) the Pareto front. However, extensive computational experience has shown that our methodology has an impressive capability of generating the whole Pareto front, even without using a search step.Two by-products of this paper are (i) the development of a collection of test problems for MOO and (ii) the extension of performance and data profiles to MOO, allowing a comparison of several solvers on a large set of test problems, in terms of their efficiency and robustness to determine Pareto fronts.

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In this paper we develop, analyze, and test a new algorithm for the global minimization of a function subject to simple bounds without the use of derivatives. The underlying algorithm 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 phase of pattern search we apply a particle swarm scheme to globally explore the possible nonconvexity of the objective function. Our extensive numerical experiments showed that the resulting algorithm is highly competitive with other global optimization methods also based on function values.

Abstract. In this paper we prove global convergence for first and second-order stationary points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of quadratic (or linear) models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds but, apart from that, the analysis is independent of the sampling techniques.A number of new issues are addressed, including global convergence when acceptance of iterates is based on simple decrease of the objective function, trust-region radius maintenance at the criticality step, and global convergence for second-order critical points.Key words. Trust-Region Methods, Derivative-Free Optimization, Nonlinear Optimization, Global Convergence. AMS subject classifications. 65D05, 90C30, 90C561. Introduction. Trust-region methods are a well studied class of algorithms for the solution of nonlinear programming problems [2,8]. These methods have a number of attractive features. The fact that they are intrinsically based on quadratic models makes them particularly attractive to deal with curvature information. Their robustness is partially associated with the regularization effect of minimizing quadratic models over regions of predetermined size. Extensive research on solving trust-region subproblems and related numerical issues has led to efficient implementations and commercial codes. On the other hand, the convergence theory of trust-region methods is both comprehensive and elegant in the sense that it covers many problem classes and particularizes from one problem class to a subclass in a natural way. Many extensions have been developed and analyzed to deal with different algorithmic adaptations or problem features (see [2]).One problem feature which frequently appears in computational science and engineering is the unavailability of derivative information, which can occur in several forms and degrees. Trust-region methods have been designed since the beginning of their development to deal with the absence of second-order derivatives and to incorporate quasi-Newton techniques. However, the design and analysis of rigorous trust-region methods for derivative-free optimization, when both first and secondorder derivatives are unavailable and hard to approximate directly, is a relatively recent topic [1,3,7,12].In this paper we address trust-region methods for unconstrained derivative-free optimization. These methods maintain linear or quadratic models which are based only on the objective function values computed at sample points. The corresponding models can be constructed by means of polynomial interpolation or regression or by any other approximation technique. The approach taken in this paper abstracts from