In this paper a new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can … Read more
The major focus of this work is to compare several methods for computing the proximal point of a nonconvex function via numerical testing. To do this, we introduce two techniques for randomly generating challenging nonconvex test functions, as well as two very specific test functions which should be of future interest to Nonconvex Optimization Benchmarking. … Read more
This note presents an application of the smooth optimization technique of Nesterov for solving two-stage stochastic linear programs. It is shown that the original O(1/e) bound of Nesterov on the number of main iterations required to obtain an e-optimal solution is retained. CitationTechnical Report, School of Industrial & Systems Engineering, Georgia Institute of Technology, 2006.ArticleDownload … Read more
Different stationarity and regularity concepts for extended real-valued functions on metric spaces are considered in the paper. The properties are characterized in terms of certain local constants. A classification scheme for stationarity/regularity constants and corresponding concepts is proposed. The relations between different constants are established. CitationUniversity of Ballarat, School of Information Technology and Mathematical Sciences, … Read more
We describe a conic interior point decomposition approach for solving a large scale semidefinite programs (SDP) whose primal feasible set is bounded. The idea is to solve such an SDP using existing primal-dual interior point methods, in an iterative fashion between a {\em master problem} and a {\em subproblem}. In our case, the master problem … Read more
In this paper, we propose an interior-point method for large sparse minimax optimization. After a short introduction, where various barrier terms are discussed, the complete algorithm is introduced and some implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Thus nonconvex problems can be solved successfully. The results … Read more
In this paper, we propose an interior-point method for large sparse l_1 optimization. After a short introduction, the complete algorithm is introduced and some implementation details are given. We prove that this algorithm is globally convergent under standard mild assumptions. Thus nonconvex problems can be solved successfully. The results of computational experiments given in this … Read more
The present article summarizes the state of the art algorithms to compute the discrete Moreau envelope, and presents a new linear-time algorithm, named NEP for NonExpansive Proximal mapping. Numerical comparisons between the NEP and two existing algorithms: The Linear-time Legendre Transform (LLT) and the Parabolic Envelope (PE) algorithms are performed. Worst-case time complexity, convergence results, … Read more
The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with … Read more
This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and … Read more