We consider the problem of minimizing the sum of a smooth function and a separable convex function. This problem includes as special cases bound-constrained optimization and smooth optimization with l_1-regularization. We propose a (block) coordinate gradient descent method for solving this class of nonsmooth separable problems. We establish global convergence and, under a local Lipschitzian … Read more
This paper focuses on monotone L\”{o}wner operators in Euclidean Jordan algebras and their applications to the symmetric cone complementarity problem (SCCP). We prove necessary and sufficient conditions for locally Lipschitz L\”{o}wner operators to be monotone, strictly monotone and strongly monotone. We also study the relationship between monotonicity and operator-monotonicity of L\”{o}wner operators. As a by-product … Read more
The goal of the current paper is to introduce the notion of certificates which verify the accuracy of solutions of computational problems with convex structure; such problems include minimizing convex functions, variational inequalities corresponding to monotone operators, computing saddle points of convex-concave functions and solving convex Nash equilibrium problems. We demonstrate how the implementation of … Read more
We propose a new iterative approach for solving linear programs over convex cones. Assuming that Slaters condition is satisfied, the conic problem is transformed to the minimization of a convex differentiable function. This “agumented primal-dual function” or “apd-function” is restricted to an affine set in the primal-dual space. The evaluation of the function and its … Read more
We study primal solutions obtained as a by-product of subgradient methods when solving the Lagrangian dual of a primal convex constrained optimization problem (possibly nonsmooth). The existing literature on the use of subgradient methods for generating primal optimal solutions is limited to the methods producing such solutions only asymptotically (i.e., in the limit as the … Read more
In this paper a survey and refinement of its recent results in the discrete optimal control theory are presented. The step control problem depending on a parameter is investigated. No smoothness of the cost function is assumed and new versions of the discrete maximum principle for the step control problem are derived Citationsubmited to the … Read more
The semi-Lagrangian Relaxation (SLR) method has been introduced in Beltran et al. (2006) to solve the p-median problem. In this paper we apply the method to the Uncapacitated Facility Location (UFL) problem. We perform computational experiments on two main collections of UFL problems with unknown optimal values. On one collection, we manage to solve to … Read more
This paper is a contribution to the interval analysis and separability of convex sets. Separation is a familiar principle and is often used not only in optimization theory, but in many economic applications as well. In real problems input data are usually not known exactly. For the purpose of this paper we assume that data … Read more
We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat … Read more
We propose a moment relaxation for two problems, the separation and covering problem with semi-algebraic sets generated by a polynomial of degree d. We show that (a) the optimal value of the relaxation finitely converges to the optimal value of the original problem, when the moment order r increases and (b) there exist probability measures … Read more