An iterative method using boundary distance for box-constrained nonlinear semidefinite programs

TL;DR

This paper introduces an iterative method for solving large-scale nonlinear semidefinite programs with box constraints, leveraging boundary distances to improve efficiency and convergence.

Contribution

The proposed method uses boundary distance-based search directions and quadratic form second derivatives, reducing computational costs for large problems.

Findings

Successfully solves problems with variable matrices larger than 5,000.

Faster than feasible direction methods for strongly nonlinear objectives.

Shows convergence properties based on search direction structures.

Abstract

We propose an iterative method for nonlinear semidefinite programs with box constraints. The search direction in the proposed method utilizes the distance from the current point to the boundary of a feasible set. The computation of the search direction exploits the second derivative of the objective function only in a quadratic form, and this property saves the computation cost compared to an evaluation of the whole entries of the second derivative. We compute a step length in an interval determined by a radius and we update the radius using a quadratic approximation function. In this paper, we also discuss convergence properties of the proposed method based on structures of the search direction. Numerical tests show that the proposed method solves problems in which the size of a variable matrix is larger than 5,000 and that it is faster than a feasible direction method for objective functions with strong nonlinearity.