Filter Trust Region Method for Nonlinear Semi-Infinite Programming Problem

Ke Su,Lele Ren,Chun Xu

doi:10.1155/2018/3921592

Abstract

We present a filter trust region method for nonlinear semi-infinite programming. Based on the discretization technique and motivated by the multiobjective programming, we transform the semi-infinite problem into a finite one. Together with the filter technique, we propose a modified method that avoids the merit function. Compared with the existing methods, our method is more flexible and easier to implement. Under some mild conditions, the convergent properties are proved. Moreover, the numerical results are reported in the end.

Highlights

Consider nonlinear semi-infinite programming problem (SIP) as follows: SIP : min f (x) s.t. g (x, w) ≤ 0, w ∈ Ω, x∈D⊆Rn (1)where D is a compact set, Ω is a closed set, f : Rn 󳨀→ R is a twice continuously differentiable function, and g : Rn ×Ω 󳨀→ R is continuous and differentiable about variables x and w
Discretization methods are based on nonlinear programming problems which are obtained by discretization of the original problem and incorporate some grid-refinement strategy [4, 5]
Without the merit function, we adopt the filter technique to decide whether a new iteration point is acceptable or not

Summary

Introduction

Consider nonlinear semi-infinite programming problem (SIP) as follows: SIP : min f (x) s.t. g (x, w) ≤ 0, w ∈ Ω, x∈D⊆Rn (1). Our new method might be useful in the discretization context for semi-infinite programming since it drastically decreases the number of constraints. Hettich and Honstede present an iterative method which attempts to find a solution satisfying optimality conditions of the original problem. This method is only locally convergent and is very restrictive for practical use [6]. Coope and Watson have proposed a Sequential Quadratic Programming (SQP) method that utilizes an exact L1 penalty function and global convergence obtained [8]. A globally convergent SQP method for SIP is proposed which utilizes an exact L∞ penalty function and trust region methods [9].

Description of Algorithm

Convergent Properties

Numerical Results

Conclusion