Abstract
Banjac et al. (J Optim Theory Appl 183(2):490–519, 2019) recently showed that the Douglas–Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite-dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to real Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.
Highlights
Banjac et al (J Optim Theory Appl 183(2):490–519, 2019) recently showed that the Douglas–Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. They showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility
We extend the result to real Hilbert spaces and a general nonempty closed convex set
We show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates
Summary
Due to its very good practical performance and ability to handle nonsmooth functions, the Douglas–Rachford algorithm has attracted a lot of interest for solving convex optimization problems. Results on the asymptotic behavior of the Douglas–Rachford algorithm for infeasible problems are very scarce, and most of them study some specific cases such as. There have been some recent results studying a more general setting [6,7], they impose some additional assumptions on feasibility of either the primal or the dual problem. The authors in [8] consider a problem of minimizing a convex quadratic function over a particular constraint set, and show that the iterates of the Douglas–Rachford algorithm generate an infeasibility certificate when the problem is primal and/or dual strongly infeasible. We show that a similar analysis can be used to prove that the proximal-point algorithm for solving the same class of problems generates similar infeasibility certificates. 4 and 5 analyze the asymptotic behavior of the Douglas–Rachford and proximal-point algorithms, respectively, and show that they provide infeasibility certificates for the considered problem
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