Primal-dual Active Set Method Research Articles

Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem

It is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem. Existence of a Lagrange multiplier and an optimality system are established. This allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations. Here, in particular, the primal-dual active set method, considered as a semismooth Newton method, is analyzed, and superlinear convergence is proved. As a by-product we obtain that the Lagrange multiplier associated with the box constraints acts as an edge detector. Numerical results for image denoising and zooming/resizing show the efficiency of the new approach.

An Infeasible Active Set Method for Quadratic Problems with Simple Bounds

A primal-dual active set method for quadratic problems with bound constraints is presented. Based on a guess on the active set, a primal-dual pair (x,s) is computed that satisfies the first order optimality condition and the complementarity condition. If (x,s) is not feasible, a new active set is determined, and the process is iterated. Sufficient conditions for the iterations to stop in a finite number of steps with an optimal solution are provided. Computational experience indicates that this approach often requires only a few (less than 10) iterations to find the optimal solution.

Optimal Control of Elliptic Variational Inequalities

Optimality systems for optimal control problems governed by elliptic variational inequalities are derived. Existence of appropriately defined Lagrange multipliers is proved. A primal—dual active set method is proposed to solve the optimality systems numerically. Examples with and without lack of strict complementarity are included.