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Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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TL;DR: In this paper, the problem of a seller dynamically pricing $d$ distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution, was studied.
Abstract: We study the problem of a seller dynamically pricing $d$ distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution. The goods are not in limited supply, but can only be produced at a limited rate and are costly to produce. The seller observes only the bundle of goods purchased at each day, but nothing else about the buyer's valuation function. Our main result is a dynamic pricing algorithm for optimizing welfare (including the seller's cost of production) that runs in time and a number of rounds that are polynomial in $d$ and the approximation parameter. We are able to do this despite the fact that (i) the price-response function is not continuous, and even its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not observable to the seller.
We derive this result as an application of a general technique for optimizing welfare over \emph{divisible} goods, which is of independent interest. When buyers have strongly concave, Holder continuous valuation functions over $d$ divisible goods, we give a general polynomial time dynamic pricing technique. We are able to apply this technique to the setting of unit demand buyers despite the fact that in that setting the goods are not divisible, and the natural fractional relaxation of a unit demand valuation is not strongly concave. In order to apply our general technique, we introduce a novel price randomization procedure which has the effect of implicitly inducing buyers to "regularize" their valuations with a strongly concave function. Finally, we also extend our results to a limited-supply setting in which the number of copies of each good cannot be replenished.
12 citations
01 Jan 1995
TL;DR: In this paper, the authors study a characterization of a convex set where the minimax value of the convex function is obtained by minimizing a set of continuous concave functions.
Abstract: Consider the problem min x∈X max i∈I f i (x) where X is a convex set, I is a finite set of indices and the f i (x)’s are continuous concave functions of x. In this article, we study a characterization of x ∈ X at which the minimax value is achieved. We also study some applications of the characterization.
12 citations
01 Jul 2002
12 citations
TL;DR: In this article, the authors studied the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients and proved that the average solution of the average SDE with jumps converges to that of the standard one in the sense of mean square and also in probability.
Abstract: In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients. By the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.
12 citations
TL;DR: In this paper, it is shown that every reasonable concave preference ordering possesses a concave utility function assuming values in a suitable non-standard extension of the reals, unless a certain finiteness (or piecewise linearity) condition holds.
12 citations