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A continuous location problem in which a firm wants to set up a single new facility in a competitive environment is considered. Other facilities offering the same product or service already exist in the area. Both the location and the quality of the new facility are to be found so as to maximize the profit obtained by the firm. This is a hard-to-solve global optimization problem. An evolutionary algorithm called Universal Evolutionary Global Optimizer (UEGO) seems to be the best procedure to cope with it, but the algorithm needs several hours of CPU time for solving large instances. In this paper, four parallelizations of UEGO are presented. They all are coarse-grain methods which differ in their migratory policies. A computational study is carried out to compare the performance of the parallel algorithms. The results show that one of the parallelizations always gives the best objective function value and has an almost linear speed-up for up to 16 processing elements for large instances. [ABSTRACT FROM AUTHOR]

In this paper, we suggest a new version of the Gauss-Newton method for solving a system of non-linear equations which combines the idea of sharp merit function with the idea of quadratic regularization. For this scheme, we prove general convergence results and, under a natural non-degeneracy assumption, local quadratic convergence. We analyze the behavior of this scheme on a natural problem class for which we get global and local worst-case complexity bounds. The implementation of each step of the scheme can be done by standard convex optimization technique. [ABSTRACT FROM AUTHOR]

Direct search optimization algorithms are becoming an important alternative to well-established gradient based methods. Due to the fact that a single cost function evaluation may take a substantial amount of time, optimization can be a lengthy process. In order to shorten the run time one often resorts to parallel algorithms. Asynchronous algorithms are particularly efficient since they have no synchronization points. This paper is an attempt to establish a convergence theory for a class of such parallel direct search algorithms. The notion of a search direction generator (SDG) is introduced. An algorithmic framework for parallel distributed optimization methods based on SDGs is presented along with the corresponding convergence theory. The theory almost completely decouples the stepsize control from the sufficient descent requirement, which is necessary for the finite termination of the algorithm's inner loop. The proposed framework has several attributes considered very favourable in loosely coupled parallel systems (e.g. clusters of workstations), such as fault tolerance and scalability. The framework is illustrated by optimizing a set of test problems on a cluster of workstations. In all tested cases, a speedup was obtained that increased with the increasing number of workstations. Fault tolerance and scalability of the framework were also demonstrated by removing and adding workstations to the cluster while an optimization run was in progress. [ABSTRACT FROM AUTHOR]

We investigate the Extended Proximal Auxiliary Problem Principle (EPAPP) [A. Kaplan and R. Tichatschke, Extended Auxiliary Problem Principle using Bregman distances, Optimization 53 (2004), pp. 603-623, A. Kaplan and R. Tichatschke, Bregman functions and auxiliary problem principle, Optim. Methods Softw. 23, 1 (2008), pp. 95-107] for solving variational inequalities whose operator - here called the main operator - is the sum of a maximal monotone and a continuous operator. As usual for proximal methods using Bregman distances, Kalpan and Tichatschke required that the main operator is paramonotone. There are two main purposes of the present article. First, we establish the convergence of the EPAPP method without the use of paramonotonicity by replacing the latter assumption by other (partly rather natural) ones. An example showing that the main operator does not even have to be monotone is also given. Second, we discuss the EPAPP method with a stopping criterion of fixed-relative-error type by Solodov and Svaiter [An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions, Math. Oper. Res. 25 (2000), pp. 214-230]. Kaplan and Tichatschke used a stopping criterion of summable-error type was used to admit an inexact solution of the corresponding auxiliary problems. To our knowledge, such a stopping criterion has not been applied to an extended APP yet; also in this situation, the paramonotonicity hypothesis is avoided. Independent of the stopping criterion under consideration, we admit an outer approximation of the set-valued component of the main operator. Owing to the use of Bregman-like functions to construct the symmetric components of the auxiliary operators, an interior-point effect is provided, meaning that - with a certain precaution - the auxiliary problems can be treated as unconstrained ones. [ABSTRACT FROM AUTHOR]

We give a lower semicontinuous type regularity condition and a closedness type one which turn out to be necessary and sufficient for the fulfilment of two different formulae involving the ε-subdifferential of a perturbation function, respectively. These regularity conditions prove to be sufficient also for having formulae for the classical subdifferential of a perturbation function. Some recently published results concerning ε-subdifferentials are rediscovered as special cases. [ABSTRACT FROM AUTHOR]

In support set expansion sensitivity analysis, one concerns to find the range of parameter variation where the perturbed problem has an optimal solution with the support set that includes the support set of the given optimal solution of the unperturbed problem. In this article, we consider the perturbed convex quadratic optimization problem and present a method to identify the support set expansion sets for this problem. [ABSTRACT FROM AUTHOR]

Generally, global convergence of the conjugate gradient methods for unconstrained optimization problems needs Wolfe line search or strong Wolfe line search. In this article, we propose a cautious DY conjugate gradient method and prove that this method with Armijo line search converges globally if the objective function has Lipschitz continuous gradients. We also present some preliminary numerical results to show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This article concerns a new class of optimization-related problems called equilibrium problems with equilibrium constraints (EPEC). One may treat them as two-level hierarchical problems, which involve equilibria at both lower and upper levels. Such problems naturally appear in various applications providing an equilibrium counterpart (at the upper level) of mathematical programs with equilibrium constraints (MPEC). We develop a unified approach to both EPECs and MPECs from the viewpoint of multiobjective optimization subject to equilibrium constraints. The problems of this type are intrinsically nonsmooth and require the use of generalized differentiation for their analysis and applications. This article presents necessary optimality conditions for EPECs in finite-dimensional spaces based on advanced generalized differential tools of variational analysis. The optimality conditions are derived in normal form under certain qualification requirements, which can be regarded as proper analogs of the classical Mangasarian-Fromovitz constraint qualification in the general settings under consideration. [ABSTRACT FROM AUTHOR]

In this article, we present a new approach of hybrid simulated annealing method for minimizing multimodel functions called the simulated annealing heuristic pattern search (SAHPS) method. Two subsidiary methods are proposed to achieve the final form of the global search method, SAHPS. First, we introduce the approximate descent direction (ADD) method, which is a derivative-free procedure with high ability of producing a descent direction. Then, the ADD method is combined with a pattern search method with direction pruning to construct the heuristic pattern search (HPS) method. The last method is hybridized with simulated annealing (SA) to obtain the SAHPS method. The experimental results through well-known test functions are shown to demonstrate the efficiency of the proposed method SAHPS. [ABSTRACT FROM AUTHOR]