Accelerating the DC algorithm for smooth functions

TL;DR

This paper presents two accelerated algorithms for minimizing smooth difference of convex functions, combining DCA with line search to improve convergence speed and efficiency in biochemical network applications.

Contribution

The authors introduce novel algorithms that accelerate the classical DC algorithm using a line search based on descent directions, with proven convergence and rate analysis.

Findings

Algorithms outperform DCA, being over four times faster on average.

Numerical tests confirm global convergence to biochemical network steady states.

Applicable to real analytic functions satisfying the Lojasiewicz property.

Abstract

We introduce two new algorithms to minimise smooth difference of convex (DC) functions that accelerate the convergence of the classical DC algorithm (DCA). We prove that the point computed by DCA can be used to define a descent direction for the objective function evaluated at this point. Our algorithms are based on a combination of DCA together with a line search step that uses this descent direction. Convergence of the algorithms is proved and the rate of convergence is analysed under the Lojasiewicz property of the objective function. We apply our algorithms to a class of smooth DC programs arising in the study of biochemical reaction networks, where the objective function is real analytic and thus satisfies the Lojasiewicz property. Numerical tests on various biochemical models clearly show that our algorithms outperforms DCA, being on average more than four times faster in both computational time and the number of iterations. Numerical experiments show that the algorithms are globally convergent to a non-equilibrium steady state of various biochemical networks, with only chemically consistent restrictions on the network topology.