Minimax
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Minimax and Applications
Author: Ding-Zhu Du
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-12-01
Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as min max f(x, y) (1) ",EX !lEY where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf(x,y) = maxminf(x,y). (2) "'EX !lEY !lEY "'EX The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way. The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value f(x*, y*) = minmaxf(x, y). (3) "'EX !lEY There are two developments in minimax theory that we would like to mention.
Minimax and Applications
Author: Ding-Zhu Du
language: en
Publisher: Springer Science & Business Media
Release Date: 1995-10-31
Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as min max f(x, y) (1) ",EX !lEY where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf(x,y) = maxminf(x,y). (2) "'EX !lEY !lEY "'EX The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way. The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value f(x*, y*) = minmaxf(x, y). (3) "'EX !lEY There are two developments in minimax theory that we would like to mention.
Minimax Under Transportation Constrains
Author: Vladimir Tsurkov
language: en
Publisher: Springer Science & Business Media
Release Date: 1999-04-30
Transportation problems belong to the domains mathematical program ming and operations research. Transportation models are widely applied in various fields. Numerous concrete problems (for example, assignment and distribution problems, maximum-flow problem, etc. ) are formulated as trans portation problems. Some efficient methods have been developed for solving transportation problems of various types. This monograph is devoted to transportation problems with minimax cri teria. The classical (linear) transportation problem was posed several decades ago. In this problem, supply and demand points are given, and it is required to minimize the transportation cost. This statement paved the way for numerous extensions and generalizations. In contrast to the original statement of the problem, we consider a min imax rather than a minimum criterion. In particular, a matrix with the minimal largest element is sought in the class of nonnegative matrices with given sums of row and column elements. In this case, the idea behind the minimax criterion can be interpreted as follows. Suppose that the shipment time from a supply point to a demand point is proportional to the amount to be shipped. Then, the minimax is the minimal time required to transport the total amount. It is a common situation that the decision maker does not know the tariff coefficients. In other situations, they do not have any meaning at all, and neither do nonlinear tariff objective functions. In such cases, the minimax interpretation leads to an effective solution.