An Adaptive Penalty Method for Inequality Constrained Minimization Problems

TL;DR

This paper introduces an adaptive penalty method that seamlessly transitions into an active set method for inequality constrained minimization, combining ease of implementation with exact constraint enforcement.

Contribution

It proposes a novel adaptive penalty approach that approximates the active set method, improving efficiency and accuracy in solving constrained optimization problems.

Findings

The method effectively combines penalty and active set advantages.

It adaptively updates the penalty parameter at each iteration.

Numerical experiments demonstrate improved convergence.

Abstract

The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.