A Dual and Interior Point Approach to Solve Convex Min-Max Problems

Abstract In this paper we propose an interior point method for solving the dual form of minmax type problems The dual variables are updated by means of a scaling supergradient method The boundary of the dual feasible region is avoided by the use of a logarithmic barrier function A major dierence with other interior point methods is the nonsmoothness of the objective functio

New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

Social welfare and profit maximization from revealed preferences

Consider the seller's problem of finding optimal prices for her $n$ (divisible) goods when faced with a set of $m$ consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity $O(m^2/\epsilon^2)$, where $\epsilon$ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of $O(\sqrt{mn})$ from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently)

Approximation of fractional Brownian motion by martingales

We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exist a unique martingale closest to fractional Brownian motion in a specific sense. It shown that this martingale has a specific form. Numerical results concerning the approximation problem are given

On a Convex Set with Nondifferentiable Metric Projection

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property

Immobile indices and CQ-free optimality criteria for linear copositive programming problems

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality.publishe