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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]

The DFP algorithm of unconstrained optimization possesses excellent properties of convergence for convex functions. However, a convergence theory of the DFP algorithm without the convexity assumption has not yet been established. This paper gives the following result: If the objective function is suitably smooth, and if the DFP algorithm produces a convergent point sequence, then the limit point of the sequence is a critical point of the objective function. Also, some open questions are mentioned. [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]

In this paper, we introduce and study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By applying the auxiliary principle technique due to Glowinski-Lions-Tremolières (Ref 1) and the minimax inequality due to Ding-Tan (Ref 2), we establish existence and uniqueness theorems for solutions of generalized nonlinear variational-like inequalities; also, we suggest two general algorithms and prove the convergence of the iterative sequences generated by the algorithms. Our results extend, improve, and unify several known results due to Cohen, Ding, Bose, Parida-Sahoo-Kumar, Dien, Ding-Tarafdar, Noor, and others. [ABSTRACT FROM AUTHOR]

Stochastic search methods for global optimization and multi-objective optimization are widely used in practice, especially on problems with black-box objective and constraint functions. Although there are many theoretical results on the convergence of stochastic search methods, relatively few deal with black-box constraints and multiple black-box objectives and previous convergence analyses require feasible iterates. Moreover, some of the convergence conditions are difficult to verify for practical stochastic algorithms, and some of the theoretical results only apply to specific algorithms. First, this article presents some technical conditions that guarantee the convergence of a general class of adaptive stochastic algorithms for constrained black-box global optimization that do not require iterates to be always feasible and applies them to practical algorithms, including an evolutionary algorithm. The conditions are only required for a subsequence of the iterations and provide a recipe for making any algorithm converge to the global minimum in a probabilistic sense. Second, it uses the results for constrained optimization to derive convergence results for stochastic search methods for constrained multi-objective optimization. [ABSTRACT FROM AUTHOR]

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]

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]

We study the properties of the augmented Lagrangian function for nonlinear semidefinite programming. It is shown that, under a set of sufficient conditions, the augmented Lagrangian algorithm is locally convergent when the penalty parameter is larger than a certain threshold. An error estimate of the solution, depending on the penalty parameter, is also established. [ABSTRACT FROM AUTHOR]

Abstract: Research on a new metaheuristic for optimization is often initially focused on proof-of-concept applications. It is only after experimental work has shown the practical interest of the method that researchers try to deepen their understanding of the method''s functioning not only through more and more sophisticated experiments but also by means of an effort to build a theory. Tackling questions such as “how and why the method works’’ is important, because finding an answer may help in improving its applicability. Ant colony optimization, which was introduced in the early 1990s as a novel technique for solving hard combinatorial optimization problems, finds itself currently at this point of its life cycle. With this article we provide a survey on theoretical results on ant colony optimization. First, we review some convergence results. Then we discuss relations between ant colony optimization algorithms and other approximate methods for optimization. Finally, we focus on some research efforts directed at gaining a deeper understanding of the behavior of ant colony optimization algorithms. Throughout the paper we identify some open questions with a certain interest of being solved in the near future. [Copyright &y& Elsevier]

In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality. [ABSTRACT FROM AUTHOR]

A continuous approach using NCP function for approximating the solution of the max-cut problem is proposed. The max-cut problem is relaxed into an equivalent nonlinearly constrained continuous optimization problem and a feasible direction method without line searches is presented for generating an optimal solution of the relaxed continuous Optimization problem. The convergence of the algorithm is proved. Numerical experiments and comparisons on some max-cut test problems show that we can get the satisfactory solution of max-cut problems with less computation time. Furthermore, this is the first time that the feasible direction method is combined with NCP function for solving max-cut problem, and similar idea can be generalized to other combinatorial Optimization problems. [ABSTRACT FROM AUTHOR]

Evolutionary algorithms (EAs) are widely employed to solve a broad range of optimization problems. Even though they work in an algorithmically simple manner, it is not always easy to understand what is going on during a particular optimization run. It is especially desirable to gain further insight into the state and course of the algorithm if the optimization does not yield the expected results or if we are not sure whether the result achieved is really the best result possible. During an optimization run an EA produces a vast amount of data. The extraction of useful information is a nontrivial task. In this article, we review visualization methods used to extract this useful information. We also demonstrate the application of visualization techniques and explain how they help us to understand the course and state of the EA. This extra information gained by the use of visualization techniques is often the difference between a good result and a very good result. In complex real-world applications, merely achieving a good result often means that the approach has failed. On the other hand, a success means large gains in productivity or safety or a decrease in costs. [ABSTRACT FROM AUTHOR]