Convergence rate estimates for penalty methods revisited

A. F. Izmailov () and M. V. Solodov ()
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A. F. Izmailov: Lomonosov Moscow State University, MSU
M. V. Solodov: IMPA – Instituto de Matemática Pura e Aplicada

Computational Optimization and Applications, 2023, vol. 85, issue 3, No 10, 973-992

Abstract: Abstract For the classical quadratic penalty, it is known that the distance from the solution of the penalty subproblem to the solution of the original problem is at worst inversely proportional to the value of the penalty parameter under the linear independence constraint qualification, strict complementarity, and the second-order sufficient optimality conditions. Moreover, using solutions of the penalty subproblem, one can obtain certain useful Lagrange multipliers estimates whose distance to the optimal ones is also at least inversely proportional to the value of the parameter. We show that the same properties hold more generally, namely, under the (weaker) strict Mangasarian–Fromovitz constraint qualification and second-order sufficiency (and without strict complementarity). Moreover, under the linear independence constraint qualification and strong second-order sufficiency (also without strict complementarity), we demonstrate local uniqueness and Lipschitz continuity of stationary points of penalty subproblems. In addition, those results follow from the analysis of general power penalty functions, of which quadratic penalty is a special case.

Keywords: Penalty function; Quadratic penalty; Convergence rate; Strong regularity; Linear independence constraint qualification; Strict Mangasarian–Fromovitz constraint qualification; Second-order sufficient optimality conditions; 90C30; 65K05 (search for similar items in EconPapers)
Date: 2023
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10589-023-00476-1

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