A matrix-free interior point continuous trajectory for linearly constrained convex programming

TL;DR

This paper introduces a matrix-free interior point method using an augmented Lagrangian continuous trajectory for linearly constrained convex programming, avoiding ill-conditioning issues and ensuring convergence to optimal solutions.

Contribution

It develops a novel matrix-free ODE-based interior point approach that guarantees convergence without requiring a variable projection matrix.

Findings

Converges to an optimal solution from any interior feasible point.

Lagrange multipliers converge to dual optimal solutions under strict complementarity.

Proposes new search directions for discrete algorithms based on the ODE system.

Abstract

Interior point methods for solving linearly constrained convex programming involve a variable projection matrix at each iteration to deal with the linear constraints. This matrix often becomes ill-conditioned near the boundary of the feasible region that results in wrong search directions and extra computational cost. A matrix-free interior point augmented Lagrangian continuous trajectory is therefore proposed and studied for linearly constrained convex programming. A closely related ordinary differential equation (ODE) system is formulated. In this ODE system, the variable projection matrix is no longer needed. By only assuming the existence of an optimal solution, we show that, starting from any interior feasible point, (i) the interior point augmented Lagrangian continuous trajectory is convergent; and (ii) the limit point is indeed an optimal solution of the original optimization problem. Moreover, with the addition of the strictly complementarity condition, we show that the associated Lagrange multiplier converges to an optimal solution of the Lagrangian dual problem. Based on the studied ODE system, several possible search directions for discrete algorithms are proposed and discussed.