Computing conjugate barrier information for nonsymmetric cones

TL;DR

This paper develops methods to efficiently compute the conjugate barrier function's gradient for seven nonsymmetric cones, aiding interior point algorithms that require such derivative information.

Contribution

It provides explicit procedures to compute conjugate barrier gradients for seven nonsymmetric cones, enhancing the efficiency of interior point methods.

Findings

Derived gradient computation methods for seven nonsymmetric cones

Facilitated closed-form expressions for inverse Hessian operators

Improved efficiency of interior point algorithms for nonsymmetric cones

Abstract

The recent interior point algorithm by Dahl and Andersen [10] for nonsymmetric cones as well as earlier works [16,19] require derivative information from the conjugate of the barrier function of the cones in the problem. Besides a few special cases, there is no indication of when this information is efficient to evaluate. We show how to compute the gradient of the conjugate barrier function for seven useful nonsymmetric cones. In some cases this is helpful for deriving closed-form expressions for the inverse Hessian operator for the primal barrier.