# Z94.1 - Analytical Techniques & Operations Research Terminology

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GAMBLER'S RUIN. The name given to one of the classical topics in probability theory. A game of chance can be related to a series of Bernoulli trials at which a gambler wins a certain predetermined sum of money for every success and loses a second sum of money for every failure. The play may proceed until his initial capital is exhausted and he is ruined. The statistical problems involved are concerned with the probability of the ruin of a player, given the stakes, initial capital and chances of success, and with such matters as the distribution of the length of play. There are many variations to this classical problem, which is closely associated with problems of the random walk and in the limiting case, Brownian motion, in particular, of sequential sampling. [22:117, 5.25, 13:334, 25:144]

GAME THEORY. The study of the following problem: if n players Pl, P2...PN play a given game G, how must the ith player, Pi, play to achieve the most favorable result? Special two-person games can be solved by linear programming methods. More generally, the mathematical study of cooperative/competitive situations. 

GAUSSIAN ELIMINATION. A reduction method for systems of simultaneous linear equations, in which one of the equations is solved for one of the variables in terms of the remaining variables. When these expressions for the solved variable are substituted into the remaining equations, the result is an equivalent system with one less equation and one less variable. A Gaussian elimination step is exactly equivalent to a pivot step. It is a single change of basis and can be expressed functionally as premultiplying by the inverse of an appropriate elementary column matrix. Sufficient repetition of this procedure can yield the numerical solution in case of a nonsingular square system, and a solution of parametric form, (a linear function of a subset of the variables) if the number of variables exceeds the number of equations. 

GEOMETRIC SOLUTION. A graphic method of solving a linear programming problem,   by plotting the halfplanes determined by the constraints and the lines of constant value for the objective function. 

GLOBAL OPTIMUM. A feasible solution (q.v.) which gives a value to the objective   function at least as great (small) as any other in the feasible region. It is contrasted with a local optimum, which yields the best objective function value of all points in some subset of the feasible region. In convex programming problems a local optimum is a global optimum. (See ABSOLUTE MAXIMUM.) 

GOAL PROGRAMMING. A model and associated algorithm to minimize the absolute value of deviations from a set of values called goals subject to technological constraints. For example, one may have goals for profit, market share, and pollution limits.

GRADIENT OF A FUNCTION. A vector at a point, whose direction is the direction of   most rapid change of some function f, and whose magnitude is the rate of change of f in that direction. 

GRADIENT METHODS. A term applied to algorithms which at each iteration in their use seek to improve the current function value by moving in the direction of the gradient of the function f(x) to be optimized. The application of this method in the presence of constraints gives rise to numerous forms depending upon the type of problem and the manner in which the method is modified to handle constraints. 

GRAPH (LINEAR). Compare network. A linear graph consists of a number of nodes or junction points, each joined to some or all of the others by arcs or lines. 

GRAPH, SIGNAL FLOW. (See SIGNAL FLOW GRAPH.)

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