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

Yu-Hong Dai, Ruoyu Diao, Xin-Wei Liu, Rui-Jin Zhang

Abstract

Whether polynomial iteration complexity can be established for smoothing Newton methods (SNMs) in symmetric cone programming (SCP) remains a long-standing open problem. A key difficulty lies in the lack of an analogue of the self-concordant convex framework in interior-point methods (IPMs). In this paper, we answer this question affirmatively. We introduce a reduced smoothing barrier augmented Lagrangian (SBAL) function and prove that it is self-concordant convex-concave, which extends the classical self-concordant theory beyond the convex setting. Furthermore, we show that the parameterized smooth equations associated with SNMs are equivalent to the first-order optimality conditions of a minimax problem whose objective is the reduced SBAL function. Motivated by this equivalence, we propose a path-following smoothing Newton method (PFSNM). The reduced SBAL function induces a central path and an associated neighborhood, which provide estimates of the Newton decrement needed for the path-following analysis. As a result, the method is proven to achieve an iteration complexity of $\mathcal{O}( \sqrtν \ln(1/\varepsilon) )$, matching the best-known short-step bound for IPMs. Numerical results on standard benchmarks show that PFSNM is competitive with several well-known interior-point solvers, providing computational support for the polynomial iteration complexity.

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

Abstract

Whether polynomial iteration complexity can be established for smoothing Newton methods (SNMs) in symmetric cone programming (SCP) remains a long-standing open problem. A key difficulty lies in the lack of an analogue of the self-concordant convex framework in interior-point methods (IPMs). In this paper, we answer this question affirmatively. We introduce a reduced smoothing barrier augmented Lagrangian (SBAL) function and prove that it is self-concordant convex-concave, which extends the classical self-concordant theory beyond the convex setting. Furthermore, we show that the parameterized smooth equations associated with SNMs are equivalent to the first-order optimality conditions of a minimax problem whose objective is the reduced SBAL function. Motivated by this equivalence, we propose a path-following smoothing Newton method (PFSNM). The reduced SBAL function induces a central path and an associated neighborhood, which provide estimates of the Newton decrement needed for the path-following analysis. As a result, the method is proven to achieve an iteration complexity of , matching the best-known short-step bound for IPMs. Numerical results on standard benchmarks show that PFSNM is competitive with several well-known interior-point solvers, providing computational support for the polynomial iteration complexity.

Paper Structure

This paper contains 23 sections, 20 theorems, 172 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Organization
  3. Notation
  4. Preliminaries
  5. Euclidean Jordan algebras and symmetric cones
  6. Self-concordant convex-concave functions
  7. A minimax reformulation of the smoothing Newton method
  8. From smoothing CHKS function to the reduced SBAL function
  9. Properties of the reduced SBAL function
  10. Equivalent characterization of the parameterized smooth equations
  11. A path-following smoothing Newton method
  12. Neighborhood of the central path
  13. Two-phase path-following framework
  14. Linear system in the algorithm
  15. Complexity analysis
  16. ...and 8 more sections

Key Result

theorem 1

Let $\mathbb{E}$ be a Euclidean Jordan algebra of rank $\nu$. For any element $z \in \mathbb{E}$, there exist pairwise orthogonal primitive idempotents $\{v_{1},\ldots,v_{\nu}\}$ and unique real eigenvalues $\lambda_{1}(z),\ldots,\lambda_{\nu}(z)$ such that where the idempotents satisfy $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Summary of equivalence relationships
  • Figure 2: Structure of the SOCP Schur complement in PFSNM
  • Figure 3: Performance profiles of PFSNM, SDPT3, SeDuMi, ECOS, and Clarabel on the NETLIB collection.
  • Figure 4: Performance profiles of PFSNM, SDPT3, SeDuMi, ECOS, and Clarabel on the Maros--Mészáros collection.
  • Figure 5: Performance profiles of PFSNM, SDPT3, SeDuMi, ECOS, and Clarabel on SOCP problems constructed from SuiteSparse matrices.

Theorems & Definitions (48)

  • definition 1
  • theorem 1
  • definition 2: nemirovski1999self
  • theorem 2: nesterov1994interior
  • definition 3
  • remark 1
  • proposition 1
  • proof
  • theorem 3
  • proof
  • ...and 38 more