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An efficient primal dual semismooth Newton method for semidefinite programming

Zhanwang Deng, Jiang Hu, Kangkang Deng, Zaiwen Wen

TL;DR

This paper addresses large-scale convex composite semidefinite programs by developing SSNCP, a primal–dual semismooth Newton method grounded in augmented Lagrangian duality. A correction step is added to identify a manifold where the non-smooth mapping becomes smooth, enabling global convergence under inexact criteria and achieving local superlinear convergence without relying on nonsingularity or strict complementarity. The authors prove an iteration complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-3/2})$ to reach $\varepsilon$-stationarity and demonstrate superior performance on diverse SDP datasets, including the Mittelmann benchmark, often rivaling or surpassing MOSEK and SDPNAL+. The approach integrates an efficient Newton-step implementation that exploits problem structure and uses a nonmonotone line search to ensure robust globalization, making it suitable for challenging large-scale SDP problems with high accuracy requirements.

Abstract

In this paper, we present an efficient semismooth Newton method, named SSNCP, for solving a class of semidefinite programming problems. Our approach is rooted in an equivalent semismooth system derived from the saddle point problem induced by the augmented Lagrangian duality. An additional correction step is incorporated after the semismooth Newton step to ensure that the iterates eventually reside on a manifold where the semismooth system is locally smooth. Global convergence is achieved by carefully designing inexact criteria and leveraging the $α$-averaged property to analyze the error. The correction steps address challenges related to the lack of smoothness in local convergence analysis. Leveraging the smoothness established by the correction steps and assuming a local error bound condition, we establish the local superlinear convergence rate without requiring the stringent assumptions of nonsingularity or strict complementarity. Furthermore, we prove that SSNCP converges to an $\varepsilon$-stationary point with an iteration complexity of $\widetilde{\mathcal{O}}(\varepsilon^{-3/2})$. Numerical experiments on various datasets, especially the Mittelmann benchmark, demonstrate the high efficiency and robustness of SSNCP compared to state-of-the-art solvers.

An efficient primal dual semismooth Newton method for semidefinite programming

TL;DR

This paper addresses large-scale convex composite semidefinite programs by developing SSNCP, a primal–dual semismooth Newton method grounded in augmented Lagrangian duality. A correction step is added to identify a manifold where the non-smooth mapping becomes smooth, enabling global convergence under inexact criteria and achieving local superlinear convergence without relying on nonsingularity or strict complementarity. The authors prove an iteration complexity of to reach -stationarity and demonstrate superior performance on diverse SDP datasets, including the Mittelmann benchmark, often rivaling or surpassing MOSEK and SDPNAL+. The approach integrates an efficient Newton-step implementation that exploits problem structure and uses a nonmonotone line search to ensure robust globalization, making it suitable for challenging large-scale SDP problems with high accuracy requirements.

Abstract

In this paper, we present an efficient semismooth Newton method, named SSNCP, for solving a class of semidefinite programming problems. Our approach is rooted in an equivalent semismooth system derived from the saddle point problem induced by the augmented Lagrangian duality. An additional correction step is incorporated after the semismooth Newton step to ensure that the iterates eventually reside on a manifold where the semismooth system is locally smooth. Global convergence is achieved by carefully designing inexact criteria and leveraging the -averaged property to analyze the error. The correction steps address challenges related to the lack of smoothness in local convergence analysis. Leveraging the smoothness established by the correction steps and assuming a local error bound condition, we establish the local superlinear convergence rate without requiring the stringent assumptions of nonsingularity or strict complementarity. Furthermore, we prove that SSNCP converges to an -stationary point with an iteration complexity of . Numerical experiments on various datasets, especially the Mittelmann benchmark, demonstrate the high efficiency and robustness of SSNCP compared to state-of-the-art solvers.

Paper Structure

This paper contains 28 sections, 8 theorems, 82 equations, 8 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review
  3. Contributions
  4. Notation
  5. Organization
  6. Preliminaries
  7. An equivalent saddle point problem
  8. A monotone nonlinear system
  9. An efficient semismooth Newton method
  10. Generalized Jacobian of $F$
  11. A correction step
  12. A globalized semismooth Newton method
  13. An efficient implementation of Newton step
  14. Convergence analysis
  15. Global convergence
  16. ...and 13 more sections

Key Result

Lemma 1

Suppose that Assumption assum holds. Given $\sigma > 0$, the strong duality holds for prob:saddle, i.e., where both sides of lemma:strong are equivalent to problem prob:dual0.

Figures (8)

  • Figure 1: The first three columns: the performance of SSNCP for SDP (top row) and SDP+ (bottom row) problems with or without the correction step. The last column: $\|F(\bm{w}^k)\|$ and $\|\bm{w}^k - \bm{w}_*\|$ of SSNCP with correction step.
  • Figure 2: The performance profiles of tested algorithms for Theta problems
  • Figure 3: The performance profiles of tested algorithms for Theta+ problems
  • Figure 4: The performance profiles of tested algorithms for R1TA problems
  • Figure 5: The performance profiles of tested algorithms for RDM problems
  • ...and 3 more figures

Theorems & Definitions (13)

  • Lemma 1
  • Definition 2
  • Lemma 3
  • Theorem 4
  • Definition 5: $C^p$-partial smoothness
  • Definition 6
  • Lemma 7
  • Remark 8
  • Lemma 9
  • Theorem 10
  • ...and 3 more