A new non-monotone filter trust region algorithm for solving nonlinear systems of equalities and inequalities

Chao Gu,Hua Wang

doi:10.19139/soic.v3i3.100

Abstract

In this paper, we combine filter and non-monotone trust region algorithm for nonlinear systems of equalities and inequalities. The systems of equalities and inequalities are transformed into a continuous equality constrained optimization solved by the new algorithm. Filter method guarantees global convergence of the algorithm under appropriate assumptions. The second order correction step is used to overcome Maratos effect so that superlinearly local convergence is achieved. Preliminary numerical results are reported.

Highlights

Consider the following nonlinear systems of equalities and inequalities ci(x) = 0, i ∈ E (1a) ci(x) ≤ 0, i ∈ I (1b) where ci(x): Rn → R, i ∈ E ∪ I, ∩ I = ∅.Systems of nonlinear equalities and inequalities appear in a wide variety of problems
These systems play a central role in the model formulation design and analysis of numerical techniques employed in solving problems arising in optimization, complementarity, and variational inequalities
Filter methods were presented by Fletcher and Leyffer [6] for nonlinear programming, offering an alternative to merit functions, as a tool to guarantee global convergence of algorithms for nonlinear optimization

Summary

Introduction

Systems of nonlinear equalities and inequalities appear in a wide variety of problems These systems play a central role in the model formulation design and analysis of numerical techniques employed in solving problems arising in optimization, complementarity, and variational inequalities [12, 15, 18, 19, 20, 28, 30]. Filter methods were presented by Fletcher and Leyffer [6] for nonlinear programming, offering an alternative to merit functions, as a tool to guarantee global convergence of algorithms for nonlinear optimization. We present a new non-monotone filter trust region algorithm which is different from [22, 26, 27]. Let ∇Φ(x) denote the gradient of the function Φ(x), ∇c(x) denote the Jacobian of the constraint c(x)

A new non-monotone filter trust region algorithm

Global convergence

Local convergence

Numerical experiments