Polynomial iteration complexity of a path-following smoothing Newton method for symmetric cone programming

TL;DR

This paper introduces a path-following smoothing Newton method for symmetric cone programming, proving it has polynomial iteration complexity and demonstrating competitive numerical performance.

Contribution

It develops a new self-concordant convex-concave barrier function and a path-following method with proven polynomial iteration complexity for symmetric cone programming.

Findings

Achieves an iteration complexity of O(√ν ln(1/ε))

Proves the reduced BAL function is self-concordant convex-concave

Numerical results show competitiveness with existing solvers

Abstract

It has long remained open whether smoothing Newton methods (SNMs) for symmetric cone programming (SCP) admit polynomial iteration complexity. A key difficulty lies in the lack of an analogue of the self-concordant convex framework underlying interior-point methods (IPMs). In this paper, inspired by Nemirovski's self-concordant convex-concave theory, we address this open problem by introducing a reduced barrier augmented Lagrangian (BAL) function. We prove that the reduced BAL function is self-concordant convex-concave and establish that the parameterized smooth system arising in SNMs coincides with the first-order optimality conditions of an associated minimax problem. Motivated by this equivalence, we propose a path-following smoothing Newton method (PFSNM). The reduced BAL function induces a central path and an associated neighborhood, which provide estimates for the Newton decrement needed for the path-following analysis. As a result, the method achieves an iteration complexity of $\mathcal{O}(\sqrt{\nu}\ln(1/\varepsilon))$, matching the best-known short-step complexity for IPMs. Numerical results on standard benchmarks show that PFSNM is competitive with several well-known interior-point solvers, and the observed performance is consistent with the theoretical development.