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K-convex Function
''K''-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ... which is crucial in the proof of the optimality of the (s,S) policy in inventory control theory. The policy is characterized by two numbers and , S \geq s, such that when the inventory level falls below level , an order is issued for a quantity that brings the inventory up to level , and nothing is ordered otherwise. Gallego and Sethi Gallego, G. and Sethi, S. P. (2005). ''K''-convexity in ℜn. ''Journal of Optimization Theory & Applications,'' 127(1):71-88. have generalized the concept of ''K''-convexity to higher dimensional Euclidean spaces. Definition Two equivalent definitions are as follows: Definition 1 (The orig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Herbert Scarf
Herbert Eli "Herb" Scarf (July 25, 1930 – November 15, 2015) was an American mathematical economist and Sterling Professor of Economics at Yale University. Education and career Scarf was born in Philadelphia, the son of Jewish emigrants from Ukraine and Russia, Lene (Elkman) and Louis Scarf. During his undergraduate work he finished in the top 10 of the 1950 William Lowell Putnam Mathematical Competition, the major mathematics competition between universities across the United States and Canada. He received his PhD from Princeton in 1954, supervised by Salomon Bochner. Contributions Among his notable works is a seminal paper in cooperative game in which he showed sufficiency for a core (economics) in general balanced games. Sufficiency and necessity had been previously shown by Lloyd Shapley for games where players were allowed to transfer utility between themselves freely. Necessity is shown to be lost in the generalization. Recognition Scarf received the 1973 Frederi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the quadratic function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function whose graph is shaped like a cup \cup, while a concave function's graph is shaped like a cap \cap. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematical Optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Inventory Theory
Material theory (or more formally the mathematical theory of inventory and production) is the sub-specialty within operations research and operations management that is concerned with the design of production/inventory systems to minimize costs: it studies the decisions faced by firms and the military in connection with manufacturing, warehousing, supply chains, spare part allocation and so on and provides the mathematical foundation for logistics. The inventory control problem is the problem faced by a firm that must decide how much to order in each time period to meet demand for its products. The problem can be modeled using mathematical techniques of optimal control, dynamic programming and network optimization. The study of such models is part of inventory theory. Issues One issue is infrequent large orders vs. frequent small orders. Large orders will increase the amount of inventory on hand, which is costly, but may benefit from volume discounts. Frequent orders are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |