A Unified Convergence Analysis of a Second-Order Method of Multipliers for Nonlinear Conic Programming

TL;DR

This paper provides a unified convergence analysis of a second-order augmented Lagrangian method for nonlinear conic programming, showing local superlinear convergence under broad conditions without strict complementarity.

Contribution

It introduces a unified framework demonstrating superlinear convergence of a second-order method of multipliers for various nonlinear conic problems under minimal assumptions.

Findings

Method converges locally with superlinear rate.

Convergence holds even with fixed penalty parameters.

Applicable to nonlinear programming, cone programming, and semidefinite programming.

Abstract

In this paper, we accomplish a unified convergence analysis of a second-order method of multipliers (i.e., a second-order augmented Lagrangian method) for solving the conventional nonlinear conic optimization problems.Specifically, the algorithm that we investigated incorporates a specially designed nonsmooth (generalized) Newton step to furnish a second-order update rule for the multipliers.We first show in a unified fashion that under a few abstract assumptions, the proposed method is locally convergent and possesses a (nonasymptotic) superlinear convergence rate, even though the penalty parameter is fixed and/or the strict complementarity fails.Subsequently, we demonstrate that, for the three typical scenarios, i.e., the classic nonlinear programming, the nonlinear second-order cone programming, and the nonlinear semidefinite programming, these abstract assumptions are nothing but exactly the implications of the iconic sufficient conditions that were assumed for establishing the Q-linear convergence rates of the method of multipliers without assuming the strict complementarity.