A Spline Smoothing Newton Method for Semi-Infinite Minimax Problems

Li Dong,Yu Xiao,Bo Yu

doi:10.1155/2014/852074

Abstract

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

Highlights

In this paper, we consider the following semi-infinite minimax problems:(P) min max ψ (x, y), x∈Rn y∈Y (1)where ψ : Rn × Rm → R
The spline smoothing technique uses a smooth spline function instead of thousands of component functions and acts as an active set technique, so only few components in the max function are computed at each iteration
Numerical tests show that the new method is very efficient for semiinfinite minimax problems with complicated component functions

Summary

Introduction

We consider the following semi-infinite minimax problems:. where ψ : Rn × Rm → R. We consider the following semi-infinite minimax problems:. We assume (as in Assumption 3.4.1 in [1]) that both ψ(⋅, ⋅) and ∇xψ(⋅, ⋅) are Lipschtiz continuous on bounded sets Such a semi-infinite minimax problem P is an exciting part of mathematical programming. Computation cost is increased; efficiency of the discretization method is affected To overcome these problems, Polak et al proposed algorithms with smoothing techniques for solving finite and semiinfinite minimax problems (see [15, 16]). By Theorem 2, φN(xN) − φ(x) → 0 as N → ∞, and, since by assumption of tN → 0, as N → ∞, it follows from (14) that γqN,tN(xN) − φN(xN) → 0 as N → ∞, which completes our proof

Spline Smoothing Newton Method and Its Convergence

Numerical Experiment

Conclusion