Convergence Of Optimal Values Research Articles

A Novel Smoothing Approach for Linear Objective Optimizations Subject to Fuzzy Relation Inequalities With Addition-Min Composition

This article mainly investigates a smoothing approach for solving linear objective optimizations subject to fuzzy relational inequalities with addition-min composition. Since their constraints are concave, it is almost impossible to convert such a problem into a linear programming, which has a convex feasible domain. In this article, we construct a second-order continuously differentiable function for smoothing the constraints. Based on the approximation, a new smoothing approach is proposed for solving the problem. It is proved that any cluster of the generated solution sequence is an optimal solution. The convergence of optimal values is also proved. Some numerical experiments are presented for illustrating the performance and the effectiveness of the new approach. The numerical results show that, compared to the given smoothing method, the new approach usually reaches a more accurate solution both in sense of feasibility and optimality.

Existence and Asymptotic Behavior of Second-Order Difference Equation with Tikhonov Regularization

We study existence and asymptotic behavior of a bounded solution of the following Tikhonov regularized second-order difference equation ():where, A is a multivalued maximal monotone operator defined on a Hilbert space and are positive real parameters. We first prove under condition existence of unique bounded solution of (). For asymptotic behavior, we use a suitable assumption on cn and to prove strong convergence of un to the element of minimal norm of Some applications are thereafter discussed with respect to minimization and saddle-point problems. Specially, we study the rate of convergence of optimal values in convex minimization and convex-concave problems. We end the paper by concluding remarks and noticing some research perspectives.

Stochastic mathematical programs with equilibrium constraints, modelling and sample average approximation

In this article, we discuss the sample average approximation (SAA) method applied to a class of stochastic mathematical programs with variational (equilibrium) constraints. To this end, we briefly investigate the structure of both–the lower level equilibrium solution and objective integrand. We show almost sure convergence of optimal values, optimal solutions (both local and global) and generalized Karush–Kuhn–Tucker points of the SAA program to their true counterparts. We also study uniform exponential convergence of the sample average approximations, and as a consequence derive estimates of the sample size required to solve the true problem with a given accuracy. Finally, we present some preliminary numerical test results.

Fitting parametric sets to probability distributions

We consider the problem of approximation of distributions by sets. Given a probability P on a metric space (S,d) and a class A of subsets of S, find an approximative set A∈ A that minimizes the mean discrepancy of a random point X ∼ P from the set A: 
 W(A,P) = ∫Sφ(d(x,A))P(dx) → min A∈ A.
 We are especially interested in the case of parametric approximative sets, A = {A(Θ) : Θ∈ T }, where the aim is to find a value of the parameter Θ∈ T which minimizes 
 W(Θ,P) = W(A(Θ,P))→ min Θ∈ T .
 Current article is an extension of Käärik and Pärna (2003) to the case where approximating sets A(Θ) are unions of k parametric sets of the same type. We will prove the convergence of optimal values of loss functions and the existence of optimal sets in some functional spaces.

An Approach to the Optimization and Approximation of Solutions of any Operator Inclusion

In this paper we shall present the optimization problem in a set of solutions of operator inclusion with the maximal monotone operator. Then we treat optimization problem by Galerkin method and we prove convergence of optimal values of approximated optimization problems to the one for the original problem. Finally, we apply the results to give a simple example.

Discrete approximation of linear two–stage stochastic programming problem

This paper describes a possibility for approximate solution of stochastic programming problems with complete recourse. We replace the static form of linear problem in Lp-space by a sequence of discretized problems in finite-dimensional spaces. We present conditions that guarantee the convergence of optimal values of discretized problems to the optimal value of the initial problem.