Showing papers on "Concave function published in 1985"
TL;DR: This paper is concerned with the problem of finding the global minimum of a concave function over a closed, convex, possibly unbounded set in R n.
Abstract: In this paper we are concerned with the problem of finding the global minimum of a concave function over a closed, convex, possibly unbounded set in Rn. The intrinsic difficulty of this problem is due to the fact that a local minimum of the objective function may fail to be a global one—which makes the conventional methods of local optimization almost useless.
81 citations
TL;DR: A new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron of the branch-and-bound type finds a globally optimal extreme point solution in a finite number of steps.
Abstract: We present a new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron. The algorithm is of the branch-and-bound type. It finds a globally optimal extreme point solution for this problem in a finite number of steps. One of the major advantages of the algorithm is that the linear programming subproblems solved during the branch-and-bound search each have the same feasible region. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some preliminary computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several bilinear programming problems with the code.
60 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
01 Jan 1985
TL;DR: In this paper, a branch and bound edge search procedure was developed for the problem under the assumption that the concave function g is differentiable, where g is a convex function.
Abstract: Linear programs with an additional reverse convex constraint like (P) have been first studied by Bansal and Jacobsen [3,4], Hillestad [6] and also Hillestad and Jacobsen [7]. In [3,4] the special problem of optimizing a network flow capacity under economies-of-scale was discussed. In [6] a branch and bound edge search procedure was developed for the problem (P) under the assumption that the concave function g is differentiable. In [7] ,
31 citations
TL;DR: In this article, a nonconvex programming problem of the type: Minx∈s Maxi{ϕi(fi(x)/gi(x) − τi), i = 1, …, m} where fi are convex functions, gi are concave functions over the convex subset S of Rn and ϕi are nondecreasing gauge functions.
Abstract: In public policy decision making and in capital planning fractional criterion functions occur. For a given set of desirable target values (goals) τi, this paper develops an algorithm for solving a nonconvex programming problem of the type: Minx∈s Maxi{ϕi(fi(x)/gi(x) − τi), i = 1, …, m} where fi are convex functions, gi are concave functions over the convex subset S of Rn and ϕi are nondecreasing gauge functions. Here ϕi(·) is the penalty incurred whenever the fractional objective fi/gi deviates from the target value τi, the problem is then to choose an x that minimizes the maximum penalty incurred.
18 citations
TL;DR: Given a polytope and an arbitrary subset of its vertices, it is shown how to construct a differentiable concave function that assumes any arbitrary value within a specified e-tolerance at each vertex of the subset, with each vertex in the subset a strong local constrained minimum.
Abstract: Given a polytope and an arbitrary subset of its vertices, we show how to construct a differentiable concave function that assumes any arbitrary value within a specified e-tolerance at each vertex of the subset, with each vertex in the subset a strong local constrained minimum. We also show how this construction method can be used to generate test problems for linearly constrained concave minimization algorithms.
6 citations
01 Jan 1985
TL;DR: In this article, it was shown that the negative part of a concave function has uniformly bounded jumps on all line segments in the set and the function is bounded below on the half-lines emanating from a single element of the set in every extreme direction.
Abstract: Hirsch and Hoffman (1961) studied the problem of minimizing a real-valued concave function on a closed convex set containing no lines. They showed that if the closure of the collection of extreme points is bounded and the function is lower semicontinuous thereon, the function attains its minimum on the set at an extreme point if and only if the function is bounded below on each half-line in the set. It is shown in this paper that this is so if and only if the negative part of the function has uniformly bounded jumps on all line segments in the set and the function is bounded below on the half-lines emanating from a single element of the set in every extreme direction. If also the set is bounded below and the directional derivative of the function at infinity in every extreme direction is linear in the directions, it is shown that the function attains its minimum on the set if and only if a pair of linear functions attains its lexicographic minimum on the set.
2 citations
TL;DR: The Midpoint Rule is better than the Trapezoidal Rule for Concave Functions as discussed by the authors, but it is not the best rule for convex functions, and it does not have a constant number of points.
Abstract: (1985). Behold! The Midpoint Rule is Better Than the Trapezoidal Rule for Concave Functions. The College Mathematics Journal: Vol. 16, No. 1, pp. 56-56.
1 citations