Topic
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: Sharp bounds on the value of perfect information for static and dynamic simple recourse stochastic programming problems are presented and some recent extensions of Jensen's upper bound and the Edmundson-Madansky lower bound are used.
Abstract: We present sharp bounds on the value of perfect information for static and dynamic simple recourse stochastic programming problems. The bounds are sharper than the available bounds based on Jensen's inequality. The new bounds use some recent extensions of Jensen's upper bound and the Edmundson-Madansky lower bound on the expectation of a concave function of several random variables. Bounds are obtained for nonlinear return functions and linear and strictly increasing concave utility functions for static and dynamic problems. When the random variables are jointly dependent, the Edmundson-Madansky type bound must be replaced by a less sharp "feasible point" bound. Bounds that use constructs from mean-variance analysis are also presented. With independent random variables the calculation of the bounds generally involves several simple univariate numerical integrations and the solution of several similar nonlinear programs. These bounds may be made as sharp as desired with increasing computational effort. The bounds are illustrated on a well-known problem in the literature and on a portfolio selection problem.
49 citations
26 Jun 1961
TL;DR: In this article, it was shown that the problem of maximizing a concave function subject to linear constraints does not have a dual, as is the case in linear programming, in which primal optimizing variables do not appear.
Abstract: : The problem of maximizing a concave function subject to linear constraints does not have a dual, as is the case in linear programming, in which primal optimizing variables do not appear. As a special case of the principal result it follows that such a dual does indeed exist whenever the objective function is also homogeneous.
49 citations
Journal Article•
TL;DR: In this article, a new technique is presented to detect singularities that propagate along Lipschitz arcs and, more generally, along sets of higher dimensions, which is then used to analyze the singular set of the distance function from a closed subset of R.
Abstract: A semiconcave function on an open domain of R" is a function that can
be locally represented as the sum of a concave function plus a smooth one. The
local structure of the singular set (non-differentiability points) of such a function
is studied in this paper. A new technique is presented to detect singularities that
propagate along Lipschitz arcs and, more generally, along sets of higher dimension.
This approach is then used to analyze the singular set of the distance function from
a closed subset of R^n
49 citations
TL;DR: In this article, it was shown that the convex envelope of a differentiable, or C 1, α -function f is C 1, or c 1,α respectively, provided that the function satisfies the very mild growth condition that f (x ) tends to ∞ if x | does so.
Abstract: We prove that the convex envelope of a differentiable, or C 1, α -function f is C 1 , or C 1, α respectively, provided only that the function satisfies the very mild growth condition that f ( x ) tends to +∞ if | x | does so.
49 citations
TL;DR: In this article, the authors considered a generalized version of the newsboy problem and assumed that the product (e.g., donuts, vegetables, fashionwear) perishes in two periods, and one-period old items and fresh (new) items may be substituted if one of them is out of stock.
Abstract: In this paper we consider a generalized version of the newsboy problem and assume that the product (e,g., donuts, vegetables, fashionwear) perishes in two periods, and one-period-old items and fresh (new) items may be substituted if one of them is out of stock. We show that the one-period expected profit expression is a concave function of the order quantity of new items. We model this problem in the stochastic control theory framework and find optimal stationary policies after transforming the control problem to a Markov decision problem. We provide several numerical examples and discuss the sensitivity of the optimal policy to changes in the problem parameters.
48 citations