A feasible direction algorithm for nonlinear second-order cone programs

ABSTRACT

In this work, we present a new feasible direction algorithm for solving smooth nonlinear second-order cone programs. These consist of minimizing a nonlinear smooth objective function subject to some nonlinear second-order cone constraints. Given an interior point to the feasible set defined by the conic constraints, the algorithm generates a feasible sequence with monotone decreasing values of the objective function. Under mild assumptions, we prove the global convergence of the algorithm to KKT points. Finally, we present some computational results applied to several instances of randomly generated benchmark problems and robust support vector machine classification.

KEYWORDS:

• Feasible direction
• Second-order cone programming
• Interior-point methods

AMS SUBJECT CLASSIFICATIONS:

• 90C30
• 49M15
• 90C51

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. If A and B are in ${\mathcal{S}}_{+}^n$ then $AB\in {\mathcal{S}}_{+}^n$ iff AB is symmetric.

Additional information

Funding

The first author was supported by the Uruguayan Councils Agencia Nacional de Investigación e Innovación (ANII) and Comisión Sectorial de Investigación Científica (CSIC). The second author was supported by Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) grant number 1130905. The third author was supported by FONDECYT grant number 1160894.