# A two-phase gradient method for quadratic programming problems with a single linear constraint and bounds on the variables

## TL;DR

This paper introduces a two-phase gradient method for quadratic programming with a single linear constraint and bounds, combining identification and unconstrained minimization phases to efficiently find stationary points or optimal solutions.

## Contribution

The proposed algorithm extends the GPCG framework to more general problems and introduces a novel stopping criterion based on optimality and bindingness measures.

## Key findings

- Algorithm converges to a stationary point if the objective is bounded.
- Finite convergence to the optimal solution for strictly convex problems.
- Numerical experiments demonstrate the method's effectiveness.

## Abstract

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Mor\'e and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportioning, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.