We study the iterate convergence of strong descent algorithms applied to convex functions. We assume that the function satisfies a very simple growth condition around its minimizers, and then show that the trajectory described by the iterates generated by any such method has finite length, which proves that the sequence of iterates converge. CitationFederal University … Read more
The primal-dual algorithm recently proposed by Chambolle \& Pock (abbreviated as CPA) for structured convex optimization is very efficient and popular. It was shown by Chambolle \& Pock in \cite{CP11} and also by Shefi \& Teboulle in \cite{ST14} that CPA and variants are closely related to preconditioned versions of the popular alternating direction method of … Read more
We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. ArticleDownload View PDF
This paper considers computation of Fr\’echet and limiting normal cones to a finite union of polyhedra. To this aim, we introduce a new concept of normally admissible stratification which is convenient for calculations of such cones and provide its basic properties. We further derive formulas for the above mentioned cones and compare our approach to … Read more
Let $g$ be a continuously differentiable function whose derivative is matrix monotone on positive semi-axis. Such a function induces a function $\phi (x)=tr(g(x))$ on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that $\phi (x) -\ln \det(x)$ is a self-concordant function on the interior of the cone. We also show that … Read more
A new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on the graph of the product of a matrix with its transpose. Closed form expressions for the support … Read more
This paper analyses the behavior of the augmented Lagrangian algorithm when it deals with an infeasible convex quadratic optimization problem. It is shown that the algorithm finds a point that, on the one hand, satisfies the constraints shifted by the smallest possible shift that makes them feasible and, on the other hand, minimizes the objective … Read more
The alternating direction method of multipliers (ADMM) has been widely used for solving structured convex optimization problems. In particular, the ADMM can solve convex programs that minimize the sum of $N$ convex functions with $N$-block variables linked by some linear constraints. While the convergence of the ADMM for $N=2$ was well established in the literature, … Read more
We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. … Read more
Primal-dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear compositions, and infimal convolutions of simple functions so that each simple term is processed individually via proximal mappings, gradient mappings, and … Read more