A penalty-free method for equality constrained optimization

Zhongwen Chen,,School Of Mathematics Science, Soochow University, Suzhou, 215006 ,Yujie Jiao,Songqiang Qiu

doi:10.3934/jimo.2013.9.391

Zhongwen Chen, ,School Of Mathematics Science, Soochow University, Suzhou, 215006 + Show 2 more

Open Access

https://doi.org/10.3934/jimo.2013.9.391

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Abstract

A penalty-free method is introduced for solving nonlinear programming with nonlinear equality constraints. This method does not use any penalty function, nor a filter. It uses trust region technique to compute trial steps. By comparing the measures of feasibility and optimality, the algorithm either tries to reduce the value of objective function by solving a normal subproblem and a tangential subproblem or tries to improve feasibility by solving a normal subproblem only. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. Under usual assumptions, we prove that the given algorithm is globally convergent to first order stationary points. Preliminary numerical results on CUTEr problems are reported.

Highlights

We consider the following equality constrained optimization problem min f (x) s. t. c(x) = 0, (1)where f : Rn → R, c : Rn → Rm are twice continuously differentiable functions.Here, we propose a new algorithm based on trust region for solving (1) whose main feature is that it does not use any penalty function, nor a filter.Trust region method is an important class of methods for (1), see, e.g., [9] and the references therein
We propose a new algorithm based on trust region for solving (1) whose main feature is that it does not use any penalty function, nor a filter
Qiu et al [16] proposed another new penalty-free-type algorithm based on trust region techniques for (1), which uses a feasibility safeguarding value, a progressively decreasing limit, like the “trust funnel”, on the permitted constraint violation of the successive iterates to drive global convergence

Summary

Introduction

Qiu et al [16] proposed another new penalty-free-type algorithm based on trust region techniques for (1), which uses a feasibility safeguarding value, a progressively decreasing limit, like the “trust funnel”, on the permitted constraint violation of the successive iterates to drive global convergence. A tangential step dtk is obtained by solving the following tangential subproblem min m(xk + dnk + dt), s. The algorithm solves (2) and (3) successively to obtain a normal step dnk and a tangential step dtk.

Results

Conclusion