Convergence of a Solution Algorithm in Indefinite Quadratic Programming

TL;DR

This paper proves the R-linear convergence of a Proximal DC decomposition algorithm for indefinite quadratic programming with linear constraints, and shows local convergence to a unique solution under certain conditions, supported by numerical analysis.

Contribution

It establishes convergence properties of the Proximal DC algorithm for indefinite quadratic programming, including R-linear convergence and local convergence to a unique solution.

Findings

The algorithm converges R-linearly to a KKT point.

Sequences converge locally to a unique solution near it.

Numerical results analyze parameter influence and compare algorithms.

Abstract

It is proved that, for an indefinite quadratic programming problem under linear constraints, any iterative sequence generated by the Proximal DC decomposition algorithm $R$-linearly converges to a Karush-Kuhn-Tucker point, provided that the problem has a solution. Another major result of this paper says that DCA sequences generated by the algorithm converge to a locally unique solution of the problem if the initial points are taken from a suitably-chosen neighborhood of it. To deal with the implicitly defined iterative sequences, a local error bound for affine variational inequalities and novel techniques are used. Numerical results together with an analysis of the influence of the decomposition parameter, as well as a comparison between the Proximal DC decomposition algorithm and the Projection DC decomposition algorithm, are given in this paper. Our results complement a recent and important paper of Le Thi, Huynh, and Pham Dinh (J. Optim. Theory Appl. 179 (2018), 103-126).