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This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem. [ABSTRACT FROM PUBLISHER]

Abstract: Last few years have witnessed the development of various multi-objective evolutionary algorithms since they allow the generation of the overall Pareto front for multi-objective optimizations. With the problems in the real-world becoming more and more complex, however, no reported work in the literature focuses on the high-dimensional objective optimizations (HOPs). In this paper, we propose an evolutionary algorithm named HOEA (high-dimensional objective evolutionary algorithm) for HOPs. By adopting the concept of nonlinear definition for optimizing object, HOPs can be solved by HOEA, while the well-known multi-objective evolutionary algorithms work poorly on HOPs. We further analyze the nonlinear dynamic properties of HOEA on the basis of martingale theoretical framework. The theoretical results indicate that this new algorithm is indeed capable of achieving convergence. We also conduct experiments on HOEA with two representative test instances. The experimental results either confirm our theoretical results or show that the proposed algorithm is efficient and effective for HOPs. [Copyright &y& Elsevier]

Nonmonotone line search approach is a new technique for solving optimization problems. It relaxes the line search range and finds a larger step-size at each iteration, so as to possibly avoid local minimizer and run away from narrow curved valley. It is helpful to find the global minimizer of optimization problems. In this paper we develop a new modification of matrix-free nonmonotone Armijo line search and analyze the global convergence and convergence rate of the resulting method. We also address several approaches to estimate the Lipschitz constant of the gradient of objective functions that would be used in line search algorithms. Numerical results show that this new modification of Armijo line search is efficient for solving large scale unconstrained optimization problems. [ABSTRACT FROM AUTHOR]

Noise is present in many real-world continuous optimization problems. Stochastic search algorithms such as Evolution Strategies (ESs) have been proposed as effective search methods in such contexts. In this paper, we provide a mathematical analysis of the convergence of a (1+1)-ES on unimodal spherical objective functions in the presence of noise. We prove for a multiplicative noise model that for a positive expected value of the noisy objective function, convergence or divergence happens depending on the infimum of the support of the noise. Moreover, we investigate convergence rates and show that log-linear convergence is preserved in presence of noise. This result is a strong theoretical foundation of the robustness of ESs with respect to noise. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an affine scaling optimal path approach in association with nonmonotonic interior backtracking line search technique for nonlinear optimization subject to linear constraints. We shall employ the eigensystem decomposition and affine scaling mapping to form affine scaling optimal curvilinear path very easily. By using interior backtracking line search technique, each iterate switches to trial step of strict interior feasibility. The nonmonotone criterion is used to speed up the convergence progress in the contours of objective function with large curvature. Theoretical analysis is given which prove that the proposed algorithm is globally convergent and has a local superlinear convergence rate under some reasonable conditions. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm. [Copyright &y& Elsevier]

Line search methods are traditional and successful methods for solving unconstrained optimization problems. Its convergence has attracted more attention in recent years. In this paper we analyze the general results on convergence of line search methods with seven line search rules. It is clarified that the search direction plays a main role in these methods and that step-size guarantees the global convergence in some cases. It is also proved that many line search methods have same convergence property. These convergence results can enable us to design powerful, effective, and stable algorithms in practice. Finally, a class of special line search methods is investigated. [Copyright &y& Elsevier]