Multiconic Optimization for Symmetric Cones and Hyperbolic Coupling

TL;DR

This paper introduces a novel interior-point algorithm for multiconic optimization problems leveraging hyperbolic coupling, improving control over primal-dual variables and achieving complexity comparable to linear programming.

Contribution

It presents a new framework using hyperbolic coupling for multiconic optimization, enhancing primal-dual variable management and complexity analysis.

Findings

Algorithm's complexity is comparable to the best linear programming methods.

The hyperbolic coupling concept improves control over primal-dual variables.

The approach is effective for large-scale nonlinear multiconic problems.

Abstract

We develop a new interior-point algorithm for solving multiconic optimization problems using the parabolic target space approach. The feasible cone in these problems is composed as a direct product of many small-dimensional cones. Our approach is based on a new concept, called the hyperbolic coupling. This provides a new framework that has an advantage of interdependent pairs of primal-dual variables. In this way, their behaviour is much more controllable. We justify all main steps in the complexity analysis of the algorithm and prove that the overall complexity of solving this type of large-scale nonlinear problems by our algorithm is comparable with the best known complexity for solving linear programming problems of the same dimension.