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

TL;DR

This paper introduces a novel inexact MBA method for constrained nonconvex, nonsmooth optimization problems with upper-C^2 functions, providing convergence analysis and rate under verifiable conditions.

Contribution

It develops the first implementable inexact MBA method with full convergence guarantees for constrained nonconvex nonsmooth optimization.

Findings

Established full convergence of the inexact MBA method.

Derived local convergence rates under KL property.

Provided verifiable conditions for KL exponent at critical points.

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-{\L}ojasiewicz (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.