Accelerating the DC algorithm for smooth functions

Francisco J Aragón Artacho,Phan T Vuong,Ronan M T Fleming

doi:10.1007/s10107-017-1180-1

Francisco J Aragón Artacho, Phan T Vuong + Show 1 more

Open Access

https://doi.org/10.1007/s10107-017-1180-1

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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 Łojasiewicz 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 Łojasiewicz property. Numerical tests on various biochemical models clearly show that our algorithms outperform 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.

Highlights

Many problems arising in science and engineering applications require the development of algorithms to minimise a nonconvex function
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

Summary

Introduction

Many problems arising in science and engineering applications require the development of algorithms to minimise a nonconvex function. The key difference between their method and ours is the starting point used for the line search: in our algorithms we use the point generated by DCA, instead of using the previous iteration. We show that the problem of finding a steady state of these networks, which plays a crucial role in the modelling of biochemical reaction systems, can be reformulated as a minimisation problem involving DC functions This is the main motivation and starting point of our work: when one applies DCA to find a steady state of these systems, the rate of convergence is usually quite slow.

Preliminaries

Boosted DC Algorithms

A DC problem in biochemistry

Numerical experiments

Findings

Concluding remarks