Theorem of Alternative for Extended Homogeneous Linear System and its Application in Conic Optimization

TL;DR

This paper introduces a new framework for infeasible-start primal-dual methods in Conic Optimization based on a Gordan Theorem of Alternative, providing polynomial-time algorithms with quadratic convergence potential.

Contribution

It develops a novel approach leveraging a Gordan Theorem of Alternative to construct primal-dual methods with guaranteed polynomial complexity and hot-start capabilities.

Findings

Proves polynomial-time complexity for Damped Newton and path-following schemes.

Ensures residuals are at machine precision regardless of duality gap target.

Provides a framework for efficient infeasible-start conic optimization algorithms.

Abstract

In this paper, we develop a new framework for constructing infeasible-start primal-dual methods for Conic Optimization. Our approach can be seen as a straightforward consequence of Gordan Theorem of Alternative. Given by the target upper bound $\epsilon > 0$ for the duality gap as the only input parameter, we form an auxiliary convex problem of minimizing barrier function with linear equality constraints. Its solution can be easily transformed to the requested output. This function can be minimized by different schemes of Unconstrained Optimization, with possible quadratic convergence in the end of the process. In our paper, we analyze the Damped Newton Method and a short-step path-following scheme. For both of them, we prove polynomial-time complexity results. Our methods are able to benefit from the hot-start opportunities. We can ensure the residual of the linear equality constraints in the primal and dual problems to be at the level of machine accuracy, independently on the accuracy parameter $\epsilon$.