Strict quasi-concavity and the differential barrier property of gauges in linear programming

Abstract

Concave gauge functions were introduced to give an analytical representation of cones. In particular, they give a simple and a practical representation of the positive orthant. There is a wide choice of concave gauge functions with interesting properties, representing the same cone. Besides the fact that a concave gauge cannot be identically zero on a cone(), it may be continuous, differentiable and even on its interior. The purpose of the present paper is to present another approach to penalizing the positivity constraints of a linear programme using an arbitrary strictly quasi-concave gauge representation. Throughout the paper, we generalize the concept of the central path and the analytic centre in terms of these gauges, introduce the differential barrier concept and establish its relationship with strict quasi-concavity.

Keywords:

• analytic centre
• barrier
• central path
• concave gauge
• differential barrier
• interior point methods
• linear programmes
• penalty function
• separable functions
• strict quasi-concave

AMS Subject Classifications:

• 90C05
• 90C51
• 49M30
• 49N15

Acknowledgements

The anonymous referees and A. Jourani are kindly acknowledged. Their comments greatly helped to improve the paper.

Notes

1 Recall that the idea of a function penalizing inequality constraints of an optimization problem is due to Courant [27] in 1943.