Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Subadditive periodic functions

Abstract: Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function \(f_{0}:[0,1)\rightarrow \mathbb{R}\) is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.

A Convex Optimization Framework for Almost Budget Balanced Allocation of a Divisible Good

Abstract: We address the problem of allocating a single divisible good to a number of agents. The agents have concave valuation functions parameterized by a scalar type. The agents report only the type. The goal is to find allocatively efficient, strategy proof, nearly budget balanced mechanisms within the Groves class. Near budget balance is attained by returning as much of the received payments as rebates to agents. Two performance criteria are of interest: the maximum ratio of budget surplus to efficient surplus, and the expected budget surplus, within the class of linear rebate functions. The goal is to minimize them. Assuming that the valuation functions are known, we show that both problems reduce to convex optimization problems, where the convex constraint sets are characterized by a continuum of half-plane constraints parameterized by the vector of reported types. We then propose a randomized relaxation of these problems by sampling constraints. The relaxed problem is a linear programming problem (LP). We then identify the number of samples needed for “near-feasibility” of the relaxed constraint set. Under some conditions on the valuation function, we show that value of the approximate LP is close to the optimal value. Simulation results show significant improvements of our proposed method over the Vickrey-Clarke-Groves (VCG) mechanism without rebates. In the special case of indivisible goods, the mechanisms in this paper fall back to those proposed by Moulin, by Guo and Conitzer, and by Gujar and Narahari, without any need for randomization. Extension of the proposed mechanisms to situations when the valuation functions are not known to the central planner are also discussed.

Optimization over equilibrium sets

Abstract: We introduce a certain notion of equilibrium points, which constitute a generalization of Pareto efficient points, and we propose a branch-and-bound method to maximize a concave function over the set of all equilibrium points of a given set

On existence and uniqueness of maximum likelihood estimates in quantal and ordinal response models

Abstract: For quantal and ordinal response models, conditions on existence and uniqueness of maximum likelhood estimates are presented. Results are derived from general results on direction sets and spaces associated with a proper concave function. If each summand of the log likelihood is in any direction either strictly concave or affine, necessary and sufficient conditions are obtained. If all cell counts are strictly positive, then it is shown that estimates always exist, and that they are unique if all parameters are identifiable. If estimates exist without being unique, results on uniquely estimable linear functions are given, paralleling corresponding results in linear regression. An extension of the maximum likelihood principle is outlined yielding similar results even if the likelihood does not attain its supremum. The logit model, the linear probability model, cumulative and sequential models and binomial response models are considered in detail.