# The Boosted DC Algorithm for linearly constrained DC programming

## TL;DR

This paper extends the Boosted DC Algorithm (BDCA) to linearly constrained DC programs, proving convergence properties and demonstrating improved performance over DCA through numerical experiments on various challenging problems.

## Contribution

The paper introduces an extension of BDCA for linearly constrained DC problems, with convergence proofs and empirical performance improvements.

## Key findings

- Cluster points are KKT points under Slater condition.
- Sequences are bounded and converge R-linearly for quadratic objectives.
- BDCA outperforms DCA in numerical tests on multiple problems.

## Abstract

The Boosted Difference of Convex functions Algorithm (BDCA) has been recently introduced to accelerate the performance of the classical Difference of Convex functions Algorithm (DCA). This acceleration is achieved thanks to an extrapolation step from the point computed by DCA via a line search procedure. In this work, we propose an extension of BDCA that can be applied to difference of convex functions programs with linear constraints, and prove that every cluster point of the sequence generated by this algorithm is a Karush--Kuhn-Tucker point of the problem if the feasible set has a Slater point. When the objective function is quadratic, we prove that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Finally, we present some numerical experiments where we compare the performance of DCA and BDCA on some challenging problems: to test the copositivity of a given matrix, to solve one-norm and infinity-norm trust-region subproblems, and to solve piecewise quadratic problems with box constraints. Our numerical results demonstrate that this new extension of BDCA outperforms DCA.