Utility-based power control in wireless networks has been widely recognized as an effective mechanism to managing co-channel interferences. It is based on the maximization of system utility subject to power constraints, which is referred to as power control optimization problem. Global coupling between the mutual interference of wireless channels increases the difficulty of searching global opt...

We study competitive economy equilibrium computation. We show that, for the first time, the equilibrium sets of the following two markets: 1. A mixed Fisher and ArrowDebreu market with homogeneous and log-concave utility functions; 2. The Fisher and Arrow-Debreu markets with several classes of concave non-homogeneous utility functions; are convex or log-convex. Furthermore, an equilibrium can b...

We consider composite minimax optimization problems where the goal is to find a saddle-point of large sum non-bilinear objective functions augmented by simple regularizers for primal and dual variables. For such problems, under average-smoothness assumption, we propose accelerated stochastic variance-reduced algorithms with optimal up logarithmic factors complexity bounds. In particular, strong...

Concave functions play a fundamental role in the structure of and minimization of badness-of-fit functions used in data analysis when extreme values of either parameters or data need to be penalized. This paper summarizes my findings about this role. I also describe three examples where concave functions are useful: building measures of badness-of-fit, building robust M -estimators, and buildin...

We investigate models for unsupervised learning with concave log-likelihood functions. We begin with the most well-known example, IBM Model 1 for word alignment (Brown et al., 1993) and analyze its properties, discussing why other models for unsupervised learning are so seldom concave. We then present concave models for dependency grammar induction and validate them experimentally. We find our ...

Moreover, if p , q, then the inequality is strict. A proof: The map u 7→ −u log u is strictly concave on [0, 1]; this follows from the fact that its derivative −(1 + log u) is strictly decreasing on [0, 1]. Now, a sum of concave functions is concave, so we conclude that H is concave. Moreover, if p , q, then they differ in some coordinate; strict concavity of the map u 7→ −u log u applied to th...

We examine models for unsupervised learning with concave log-likelihood functions. We begin with the most well-known example, IBM Model 1 for word alignment (Brown et al., 1993), and study its properties, discussing why other models for unsupervised learning are so seldom concave. We then present concave models for dependency grammar induction and validate them experimentally. Despite their sim...

Abstract. A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. It is known that gradient maps of exponentially concave functions are solutions of a MongeKantorovich optimal transport problem and allow for a better gradient approximation than those of ordinary concave functions. The approximation error, called L-diver...

The seminal work by Edmonds [9] and Lovász [39] shows the strong connection between submodular functions and convex functions. Submodular functions have tight modular lower bounds, and a subdifferential structure [16] in a manner akin to convex functions. They also admit polynomial time algorithms for minimization and satisfy the Fenchel duality theorem [18] and the Discrete Seperation Theorem ...

Given a set H of functions defined on X, á function f:X↦R¯ is called abstract H-convex if it the upper envelope its H-minorants, i.e. such minorants which belong to H; and f regularly maximal (with respect pointwise ordering) H-minorants. In paper we first present basic notions (regular) H-convexity for case when an functions. For this general sufficient condition based Zorn's lemma be formulat...