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Convergence analysis of inexact MBA method for constrained upper-$\mathcal{C}^2$ optimization problems

Ruyu Liu, Shaohua Pan

TL;DR

This work develops an inexact moving balls approximation (MBA) method for constrained optimization with upper-$\mathcal{C}^2$ objective and constraint functions, addressing nonconvex and nonsmooth settings. It introduces a local majorization model with an implementable inexactness criterion and builds a KL-based potential function $\Phi_{\widetilde{c}}$ to bridge subproblem solutions and the original problem, proving full convergence under partial BMP and MSCQ and establishing convergence rates when the KL exponent $q\in[1/2,1)$. A verifiable KL-exponent criterion for $\Phi$ is provided, enabling linear to sublinear rates depending on $q$. The method is implemented via iMBA-pgls, with numerical experiments on QDCC problems showing superior performance and efficiency compared to MOSEK-based approaches, especially for large-scale instances, thus offering a practical, convergence-guaranteed tool for constrained nonconvex nonsmooth optimization.

Abstract

This paper concerns a class of constrained optimization problems in which, the objective and constraint functions are both upper-$\mathcal{C}^2$. For such nonconvex and nonsmooth optimization problems, we develop an inexact moving balls approximation (MBA) method by a workable inexactness criterion for the solving of subproblems. By leveraging a global error bound for the strongly convex program associated with parametric optimization problems, we establish the full convergence of the iterate sequence under the partial bounded multiplier property (BMP) and the Kurdyka-Łojasiewicz (KL) property of the constructed potential function, and achieve the local convergence rate of the iterate and objective value sequences if the potential function satisfies the KL property of exponent $q\in[1/2,1)$. A verifiable condition is also provided to check whether the potential function satisfies the KL property of exponent $q\in[1/2,1)$ at the given critical point. To the best of our knowledge, this is the first implementable inexact MBA method with a full convergence certificate for the constrained nonconvex and nonsmooth optimization problem.

Convergence analysis of inexact MBA method for constrained upper-$\mathcal{C}^2$ optimization problems

TL;DR

This work develops an inexact moving balls approximation (MBA) method for constrained optimization with upper- objective and constraint functions, addressing nonconvex and nonsmooth settings. It introduces a local majorization model with an implementable inexactness criterion and builds a KL-based potential function to bridge subproblem solutions and the original problem, proving full convergence under partial BMP and MSCQ and establishing convergence rates when the KL exponent . A verifiable KL-exponent criterion for is provided, enabling linear to sublinear rates depending on . The method is implemented via iMBA-pgls, with numerical experiments on QDCC problems showing superior performance and efficiency compared to MOSEK-based approaches, especially for large-scale instances, thus offering a practical, convergence-guaranteed tool for constrained nonconvex nonsmooth optimization.

Abstract

This paper concerns a class of constrained optimization problems in which, the objective and constraint functions are both upper-. For such nonconvex and nonsmooth optimization problems, we develop an inexact moving balls approximation (MBA) method by a workable inexactness criterion for the solving of subproblems. By leveraging a global error bound for the strongly convex program associated with parametric optimization problems, we establish the full convergence of the iterate sequence under the partial bounded multiplier property (BMP) and the Kurdyka-Łojasiewicz (KL) property of the constructed potential function, and achieve the local convergence rate of the iterate and objective value sequences if the potential function satisfies the KL property of exponent . A verifiable condition is also provided to check whether the potential function satisfies the KL property of exponent at the given critical point. To the best of our knowledge, this is the first implementable inexact MBA method with a full convergence certificate for the constrained nonconvex and nonsmooth optimization problem.

Paper Structure

This paper contains 23 sections, 15 theorems, 115 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Notation
  4. Preliminaries
  5. The family of upper-$\mathcal{C}^2$ functions
  6. Constraint qualifications
  7. Partial BMP of parametric system
  8. Kurdyka-Łojasiewicz property
  9. An inexact MBA method
  10. Convergence analysis
  11. Subsequential convergence
  12. Full convergence and convergence rate
  13. KL property of $\Phi$ with exponent $q\in[1/2,1)$
  14. Numerical experiments
  15. Implementation of Algorithm \ref{['iMBA']}
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $h:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be locally Lipschitz continuous and upper-$\mathcal{C}^2$ at every point of an open convex set $\mathcal{O}'\supset{\rm dom}\,h$. Then, the following two assertions hold.

Figures (2)

  • Figure 1: Performance of the inner loop of Algorithm \ref{['iMBA']} for $f_0$ from \ref{['f0-type1']} with $\omega_0=10^3$
  • Figure 2: Convergence behavior of the iterate sequence for $f_0$ from \ref{['f0-type1']} with $\omega^0=10^4$

Theorems & Definitions (24)

  • Definition 2.1
  • Remark 2.1
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.1
  • Definition 2.5
  • Remark 3.1
  • Lemma 3.1
  • ...and 14 more