Nowadays, solving nonsmooth (not necessarily differentiable) optimization problems plays a very important role in many areas of industrial applications. Most of the algorithms developed so far deal only with nonsmooth...
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Nowadays, solving nonsmooth (not necessarily differentiable) optimization problems plays a very important role in many areas of industrial applications. Most of the algorithms developed so far deal only with nonsmooth convex functions. In this paper, we propose a new algorithm for solving nonsmooth optimization problems that are not assumed to be convex. The algorithm combines the traditional cutting plane method with some features of bundle methods, and the search direction calculation of feasible direction interior point algorithm (Herskovits, J. Optim. Theory Appl. 99(1):121-146, 1998). The algorithm to be presented generates a sequence of interior points to the epigraph of the objective function. The accumulation points of this sequence are solutions to the original problem. We prove the global convergence of the method for locally Lipschitz continuous functions and give some preliminary results from numerical experiments.
The minimum sum-of-squares clustering problem is considered. The mathematical modeling of this problem leads to a min-sum-min formulation which, in addition to its intrinsic bi-level nature, has the significant charac...
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The minimum sum-of-squares clustering problem is considered. The mathematical modeling of this problem leads to a min-sum-min formulation which, in addition to its intrinsic bi-level nature, has the significant characteristic of being strongly nondifferentiable. To overcome these difficulties, the resolution, method proposed adopts a smoothing strategy using a special C-infinity differentiable class function. The final solution is obtained by solving a sequence of low dimension differentiable unconstrained optimization subproblems which gradually approach the original problem. The use of this technique, called hyperbolic smoothing, allows the main difficulties presented by the original problem to be overcome. A simplified algorithm containing only the essentials of the method is presented. For the purpose of illustrating both the reliability and the efficiency of the method, a set of computational experiments was performed, making use of traditional test problems described in the literature (C) 2009 Elsevier Ltd. All rights reserved.
Typically, practical optimization problems involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such problems are restricted to certain meaningful intervals. In this article, ...
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Typically, practical optimization problems involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such problems are restricted to certain meaningful intervals. In this article, we propose an efficient adaptive limited memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. The method combines the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, while the constraint handling is based on the projected gradient method and the dual subspace minimization. The preliminary numerical experiments to be presented confirm the usability of the method.
Practical optimization problems often involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such large problems are restricted to certain meaningful intervals. In the article [...
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Practical optimization problems often involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such large problems are restricted to certain meaningful intervals. In the article [N. Karmitsa and M.M. Makela, Adaptive limited memory bundle method for bound constrained large-scale nonsmooth optimization, Optimization (to appear)], we described an efficient limited-memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. Although this method works very well in numerical experiments, it suffers from one theoretical drawback, namely, that it is not necessarily globally convergent. In this article, a new variant of the method is proposed, and its global convergence for locally Lipschitz continuous functions is proved.
Practical optimization problems often involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such large problems are restricted to certain meaningful intervals. In the article [...
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Practical optimization problems often involve nonsmooth functions of hundreds or thousands of variables. As a rule, the variables in such large problems are restricted to certain meaningful intervals. In the article [N. Karmitsa and M.M. Makela, Adaptive limited memory bundle method for bound constrained large-scale nonsmooth optimization, Optimization (to appear)], we described an efficient limited-memory bundle method for large-scale nonsmooth, possibly nonconvex, bound constrained optimization. Although this method works very well in numerical experiments, it suffers from one theoretical drawback, namely, that it is not necessarily globally convergent. In this article, a new variant of the method is proposed, and its global convergence for locally Lipschitz continuous functions is proved.
We apply the optimality conditions of nondifferentiable minimax programming to formulate a general second-order Mond-Weir dual to the nondifferentiable minimax programming involving second-order pseudo-quasi Type I fu...
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We apply the optimality conditions of nondifferentiable minimax programming to formulate a general second-order Mond-Weir dual to the nondifferentiable minimax programming involving second-order pseudo-quasi Type I functions. We establish also weak, strong, and strict converse duality theorems.
A pair of Wolfe type second-order symmetric dual programs involving nondifferentiable functions is considered and appropriate duality theorems are established under eta(1)-bonvexity/eta(2)-boncavity. Several known res...
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A pair of Wolfe type second-order symmetric dual programs involving nondifferentiable functions is considered and appropriate duality theorems are established under eta(1)-bonvexity/eta(2)-boncavity. Several known results including that of Mond and Gulati et al. are obtained as special cases. Published by Elsevier Inc.
In this paper, we introduce nondifferentiable multiobjective fractional programming problems with cone constraints over arbitrary closed convex cones, where every component of the objective function contains a term in...
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In this paper, we introduce nondifferentiable multiobjective fractional programming problems with cone constraints over arbitrary closed convex cones, where every component of the objective function contains a term involving the support function of a compact convex set. For this problem, Wolfe and Mond-Weir type duals are proposed. We establish weak and strong duality theorems for a weakly efficient solution under suitable (V, rho)-invexity assumptions. As special cases of our duality relations, we give some known duality results.
A pair of Mond-Weir type second order symmetric nondifferentiable multiobjective programs is formulated. Weak, strong and converse duality theorems are established under eta-pseudobonvexity assumptions. Special cases ...
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A pair of Mond-Weir type second order symmetric nondifferentiable multiobjective programs is formulated. Weak, strong and converse duality theorems are established under eta-pseudobonvexity assumptions. Special cases are discussed to show that this paper extends some work appeared in this area. (c) 2004 Elsevier Inc. All rights reserved.
Usual symmetric duality results are proved forWolfe and Mond -Weir type nondifferentiable nonlinear symmetric dual programs under Fconvexity F-concavity and F-pseudoconvexity F-pseudoconcavity assumptions. These duali...
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Usual symmetric duality results are proved forWolfe and Mond -Weir type nondifferentiable nonlinear symmetric dual programs under Fconvexity F-concavity and F-pseudoconvexity F-pseudoconcavity assumptions. These duality results are then used to formulate Wolfe and MondWeir type nondifferentiable minimax mixed integer dual programs and symmetric duality theorems are established. Moreover, nondifferentiable fractional symmetric dual programs are studied by using the above programs.
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