Tree convex sets refer to a collection of sets such that each set in the collection is a subtree of a tree whose nodes are the elements of these sets. They extend the concept of row convex sets each of which is an interval over a total ordering of the elements of those sets. They have been applied to identify tractable Constraint Satisfaction Problems and Combinatorial Auction Problems. Recentl...

In this paper, optimal power allocation and relay selection strategies in energy harvesting cooperative wireless networks are studied. In particular, signal-to-noise ratio (SNR)-maximizing based power allocation and relay selection without and with energy cooperation—via wireless energy transfer—are considered. Moreover, total relay power minimization subject to target end-to-end SNR is investi...

Let X be a Hilbert space and let Cn, n = 1, . . . ,N be convex closed subsets of X . The convex feasibility problem is to find some point x ∈ N ⋂ n=1 Cn, when this intersection is non-empty. In this talk we discuss projection algorithms for finding such a feasibility point. These algorithms have wide ranging applications including: solutions to convex inequalities, minimization of convex nonsmo...

We give in this paper a convergence result concerning parallel asynchronous algorithm with bounded delays to solve a nonlinear fixed point problems. This result is applied to calculate the solution of a strongly monotone operator. Special cases of these operators are used to solve some problems related to convex analysis like minimization of functionals, calculus of saddle point and variational...

Consider the minimization problem with a convex separable objective function over a feasible region defined by linear equality constraint(s)/linear inequality constraint of the form “greater than or equal to” and bounds on the variables. A necessary and sufficient condition and a sufficient condition are proved for a feasible solution to be an optimal solution to these two problems, respectivel...

A number of discrete and continuous optimization problems in machine learning are related to convex minimization problems under submodular constraints. In this paper, we deal with a submodular function with a directed graph structure, and we show that a wide range of convex optimization problems under submodular constraints can be solved much more efficiently than general submodular optimizatio...

We derive lower bounds on the black-box oracle complexity of large-scale smooth convex minimization problems, with emphasis on minimizing smooth (with Hölder continuous, with a given exponent and constant, gradient) convex functions over high-dimensional ‖ · ‖p-balls, 1 ≤ p ≤ ∞. Our bounds turn out to be tight (up to logarithmic in the design dimension factors), and can be viewed as a substanti...