Exterior Distance Function

TL;DR

This paper introduces the exterior distance function (EDF) and exterior point method (EPM) for convex optimization, demonstrating their convergence properties and potential for efficient primal-dual solution computation.

Contribution

The paper presents a novel exterior distance function and an associated exterior point method with proven convergence and linear rate under certain optimality conditions.

Findings

EPM generates primal-dual sequences converging to optimal solutions.

Convergence occurs under minimal assumptions on input data.

Linear convergence rate is achieved with second-order optimality.

Abstract

We introduce and study exterior distance function (EDF) and correspondent exterior point method (EPM) for convex optimization. The EDF is a classical Lagrangian for an equivalent problem obtained from the initial one by monotone transformation of both the objective function and the constraints. The constraints transformation is scaled by a positive scaling parameter. Thus, the EDF is a particular realization of the Nonlinear Rescaling (NR) principle. Along with the "center", the EDF has two extra tools: the barrier (scaling) parameter and the vector of Lagrange multipliers. We show that EPM generates primal - dual sequence, which converges to the primal - dual solution in value under minimum assumption on the input data. Moreover, the convergence is taking place under any fixed interior point as a "center" and any fixed positive scaling parameter, just due to the Lagrange multipliers update. If the second order sufficient optimality condition is satisfied, then the EPM converges with Q-linear rate under any fixed interior point as a "center" and any fixed, but large enough positive scaling parameter.