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
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
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
We propose a method to approximate a class of exponential multivariate integrals using moment relaxations. Using this approach, both lower and upper bounds of the integrals are obtained and we show that these bound values asymptotically converge to the real value of the integrals when the moment degree r increases. We further demonstrate the method … Read more
We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets. Our main result is a randomized poly-time algorithm which approximates $V(K_1,…,K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small.\\ Even such rate is impossible to achieve … Read more
It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, … Read more
A convex variational framework is proposed for solving inverse problems in Hilbert spaces with a priori information on the representation of the target solution in a frame. The objective function to be minimized consists of a separable term penalizing each frame coefficient individually and of a smooth term modeling the data formation model as well … Read more
When is the linear image of a closed convex cone closed? We present very simple, and intuitive necessary conditions, which 1) unify, and generalize seemingly disparate, classical sufficient conditions: polyhedrality of the cone, and “Slater” type conditions; 2) are necessary and sufficient, when the dual cone belongs to a class, that we call nice cones. … Read more