Concave function

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

Linear Contextual Bandits with Global Constraints and Objective

Abstract: We consider the linear contextual bandit problem with global convex constraints and a concave objective function. In each round, the outcome of pulling an arm is a vector, that depends linearly on the context of that arm. The global constraints require the average of these vectors to lie in a certain convex set. The objective is a concave function of this average vector. This problem turns out to be a common generalization of classic linear contextual bandits (linContextual) [Auer 2003], bandits with concave rewards and convex knapsacks (BwCR) [Agrawal, Devanur 2014], and the online stochastic convex programming (OSCP) problem [Agrawal, Devanur 2015]. We present algorithms with near-optimal regret bounds for this problem. Our bounds compare favorably to results on the unstructured version of the problem [Agrawal et al. 2015, Badanidiyuru et al. 2014] where the relation between the contexts and the outcomes could be arbitrary, but the algorithm only competes against a fixed set of policies.

Production Smoothing Under Piecewise Concave Costs, Capacity Constraints and Nondecreasing Requirements

Abstract: The production smoothing problem with known demands that are assumed to increase with time is treated. The production and inventory costs are concave. The cost of increasing production from one period to the next is a concave function of the increase; similarly, the cost of decreasing production is a concave function of the decrease. Backlogging is not permitted. In each period, a fixed production capacity that does not vary with time is available. The problem it similar to that discussed by Zangwill, except for the important difference of capacity constraint. Feasible production plans are partitioned into sets on the basis of production differences from period to period---those with production increases in all N periods, those with a decrease only in the final period, etc. A minimum cost plan is an extreme point of one of these sets. We show that an extreme point plan is such that in between periods with zero inventory there is at most one sequence of periods when production is neither zero nor capacity and within these periods, production does not change. An algorithm is proposed for generating extreme point production plans. It is shown that finding the minimum cost production plan is equivalent to finding the shortest route through an acyclic network of extreme point production plans. The approach put forth enables a complete solution to the problem discussed.

On the optimality of given feedback controls

Abstract: We show how to construct a Lagrange problem which has the property that its extremals are the solutions of a given differential equation and which satisfies certain structural assumptions. These assumptions require that the integrand is either a concave function or that it is additively separable. In the first case, which is relevant in economics, we present a continuous-time analogue of the indeterminacy result of Boldrin and Montrucchio. The second case is illustrated by a minimum principle for the logistic growth function.

How to Pack Your Items When You Have to Buy Your Knapsack

Abstract: In this paper we consider a generalization of the classical knapsack problem. While in the standard setting a fixed capacity may not be exceeded by the weight of the chosen items, we replace this hard constraint by a weight-dependent cost function. The objective is to maximize the total profit of the chosen items minus the cost induced by their total weight. We study two natural classes of cost functions, namely convex and concave functions. For the concave case, we show that the problem can be solved in polynomial time; for the convex case we present an FPTAS and a 2-approximation algorithm with the running time of \(\mathcal{O}(n \log n)\), where n is the number of items. Before, only a 3-approximation algorithm was known.