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On computation of C-stationary points for equilibrium problems with linear complementarity constraints via homotopy method

Michal Červinka (2010)

Kybernetika

In the paper we consider EPCCs with convex quadratic objective functions and one set of complementarity constraints. For this class of problems we propose a possible generalization of the homotopy method for finding stationary points of MPCCs. We analyze the difficulties which arise from this generalization. Numerical results illustrate the performance for randomly generated test problems.

On continuous convergence and epi-convergence of random functions. Part I: Theory and relations

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Continuous convergence and epi-convergence of sequences of random functions are crucial assumptions if mathematical programming problems are approximated on the basis of estimates or via sampling. The paper investigates “almost surely” and “in probability” versions of these convergence notions in more detail. Part I of the paper presents definitions and theoretical results and Part II is focused on sufficient conditions which apply to many models for statistical estimation and stochastic optimization....

On continuous convergence and epi-convergence of random functions. Part II: Sufficient conditions and applications

Silvia Vogel, Petr Lachout (2003)

Kybernetika

Part II of the paper aims at providing conditions which may serve as a bridge between existing stability assertions and asymptotic results in probability theory and statistics. Special emphasis is put on functions that are expectations with respect to random probability measures. Discontinuous integrands are also taken into account. The results are illustrated applying them to functions that represent probabilities.

On Newton's polygons, Gröbner bases and series expansions of perturbed polynomial programs

Konstantin Avrachenkov, Vladimir Ejov, Jerzy A. Filar (2006)

Banach Center Publications

In this note we consider a perturbed mathematical programming problem where both the objective and the constraint functions are polynomial in all underlying decision variables and in the perturbation parameter ε. Recently, the theory of Gröbner bases was used to show that solutions of the system of first order optimality conditions can be represented as Puiseux series in ε in a neighbourhood of ε = 0. In this paper we show that the determination of the branching order and the order of the pole (if...

On robustness of set-valued maps and marginal value functions

Armin Hoffmann, Abebe Geletu (2005)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The ideas of robust sets, robust functions and robustness of general set-valued maps were introduced by Chew and Zheng [7,26], and further developed by Shi, Zheng, Zhuang [18,19,20], Phú, Hoffmann and Hichert [8,9,10,17] to weaken up the semi-continuity requirements of certain global optimization algorithms. The robust analysis, along with the measure theory, has well served as the basis for the integral global optimization method (IGOM) (Chew and Zheng [7]). Hence, we have attempted to extend the...

On Rohn's relative sensitivity coefficient of the optimal value for a linear-fractional program

Ştefan Iulius Ţigan, Ştefan Iulius, Ioan M. Stancu-Minasian (2000)

Mathematica Bohemica

In this note we consider a linear-fractional programming problem with equality linear constraints. Following Rohn, we define a generalized relative sensitivity coefficient measuring the sensitivity of the optimal value for a linear program and a linear-fractional minimization problem with respect to the perturbations in the problem data. By using an extension of Rohn's result for the linear programming case, we obtain, via Charnes-Cooper variable change, the relative sensitivity coefficient for...

On second–order Taylor expansion of critical values

Stephan Bütikofer, Diethard Klatte, Bernd Kummer (2010)

Kybernetika

Studying a critical value function ϕ in parametric nonlinear programming, we recall conditions guaranteeing that ϕ is a C 1 , 1 function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of D ϕ . Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of first- and second-order directional differentiability of the optimal value function in parametric optimization....

On the central path for nonlinear semidefinite programming

L. M. Grana Drummond, Alfredo Noel Iusem, B. F. Svaiter (2010)

RAIRO - Operations Research

In this paper we study the well definedness of the central path associated to a given nonlinear (convex) semidefinite programming problem. Under standard assumptions, we establish that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point.The monotonic behavior of the logarithmic barrier and the objective function...

On the complexity of determining tolerances for ε-optimal solutions to min-max combinatorial optimization problems

Diptesh Ghosh, Gerard Sierksma (2003)

Applicationes Mathematicae

This paper studies the complexity of sensitivity analysis for optimal and ε-optimal solutions to general 0-1 combinatorial optimization problems with min-max objectives. Van Hoesel and Wagelmans [9] have studied the complexity of sensitivity analysis of optimal and ε-optimal solutions to min-sum problems, and Ramaswamy et al. [17] the complexity of sensitivity analysis of optimal solutions to min-max problems. We show that under some mild assumptions the sensitivity analysis of ε-optimal solutions...

On the Existence of Optimal Solutions for Infinite Horizon Optimal Control Problems: Nonconvex and Multicriteria Problems

Dean A. Carlson (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questa Nota si continua la discussione iniziata in [4] dell'esistenza di soluzioni ottimali per problemi di ottimo controllo in [ 0 + ] . Si definiscono problemi generalizzati, e si ottengono estensioni di risultati già presentati in [4]. Si dimostrano anche varie relazioni tra le soluzioni ottimali dei problemi generalizzati e i problemi originali e non convessi di ottimo controllo. Alla fine si considerano problemi lineari nelle variabili di stato anche nel caso di costi funzionali a valori vettoriali...

On the separation of parametric convex polyhedral sets with application in MOLP

Milan Hladík (2010)

Applications of Mathematics

We investigate diverse separation properties of two convex polyhedral sets for the case when there are parameters in one row of the constraint matrix. In particular, we deal with the existence, description and stability properties of the separating hyperplanes of such convex polyhedral sets. We present several examples carried out on PC. We are also interested in supporting separation (separating hyperplanes support both the convex polyhedral sets at given faces) and permanent separation (a hyperplane...

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