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A regularization method for ill-posed bilevel optimization problems

Maitine Bergounioux, Mounir Haddou (2006)

RAIRO - Operations Research

We present a regularization method to approach a solution of the pessimistic formulation of ill-posed bilevel problems. This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and responses. We prove existence of approximated solutions, give convergence result using Hoffman-like assumptions. We end with objective value error estimates.

A review on the ant colony optimization metaheuristic: basis, models and new trends.

Oscar Cordón, Francisco Herrera, Thomas Stützle (2002)

Mathware and Soft Computing

Ant Colony Optimization (ACO) is a recent metaheuristic method that is inspired by the behavior of real ant colonies. In this paper, we review the underlying ideas of this approach that lead from the biological inspiration to the ACO metaheuristic, which gives a set of rules of how to apply ACO algorithms to challenging combinatorial problems. We present some of the algorithms that were developed under this framework, give an overview of current applications, and analyze the relationship between...

A second order η -approximation method for constrained optimization problems involving second order invex functions

Tadeusz Antczak (2009)

Applications of Mathematics

A new approach for obtaining the second order sufficient conditions for nonlinear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η -approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η -approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear...

A second-order stochastic dominance portfolio efficiency measure

Miloš Kopa, Petr Chovanec (2008)

Kybernetika

In this paper, we introduce a new linear programming second-order stochastic dominance (SSD) portfolio efficiency test for portfolios with scenario approach for distribution of outcomes and a new SSD portfolio inefficiency measure. The test utilizes the relationship between CVaR and dual second-order stochastic dominance, and contrary to tests in Post [Post] and Kuosmanen [Kuosmanen], our test detects a dominating portfolio which is SSD efficient. We derive also a necessary condition for SSD efficiency...

A self-adaptive trust region method for the extended linear complementarity problems

Zhensheng Yu, Qiang Li (2009)

Applications of Mathematics

By using some NCP functions, we reformulate the extended linear complementarity problem as a nonsmooth equation. Then we propose a self-adaptive trust region algorithm for solving this nonsmooth equation. The novelty of this method is that the trust region radius is controlled by the objective function value which can be adjusted automatically according to the algorithm. The global convergence is obtained under mild conditions and the local superlinear convergence rate is also established under...

A self-scaling memoryless BFGS based conjugate gradient method using multi-step secant condition for unconstrained minimization

Yongjin Kim, Yunchol Jong, Yong Kim (2024)

Applications of Mathematics

Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. Based on the self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (SSML-BFGS) method, new conjugate gradient algorithms CG-DESCENT and CGOPT have been proposed by W. Hager, H. Zhang (2005) and Y. Dai, C. Kou (2013), respectively. It is noted that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. Therefore,...

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