Stochastic Approximation
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S-model
See
▶ Learning Curves
▶ Game Theory ▶ Saddle-Point of a Function
SA Saddle-Point Problem ▶ Sensitivity Analysis ▶ Simulated Annealing ▶ Stochastic Approximation
Saddle-Point of a Function For an arbitrary payoff function F(X, Y), the point (X 0, Y 0) is a saddle point if F(X 0, Y) F(X 0, Y 0) F(X, Y 0).
For the mathematical-programming problem: Minimize f (x), subject to {gi (x) bi}, the saddle-point problem is to find vectors x0 and y0 such that F(x0, y) F(x0, y0) F(x, y0), where F(x, y) is the associated Lagrangian function, y 0.
See ▶ Saddle-Point of a Function
See ▶ Saddle-Point Problem
Safety Igor Ushakov Qualcomm Inc., San Diego, CA, USA
Saddle-Point of a Game For a zero-sum, two-person game, if an element aij of the payoff matrix is the minimum of its row and maximum of its column, it is a saddle point. The value of the game is equal to the value of the saddle point, with the maximizing player’s optimal strategy being the pure strategy i and the minimizing player’s optimal strategy being the pure strategy j.
Safety is a property of a system that permits the system to operate without dangerous consequences for people (including serving personnel) and the environment. For many systems (as aircraft, submarines, chemical plants, nuclear power stations, etc.), some kinds of failures can lead to catastrophic results. In these cases, the safety indices coincide with reliability
S.I. Gass, M.C. Fu (eds.), Encyclopedia of Operations Research and Management Science, DOI 10.1007/978-1-4419-1153-7, # Springer Science+Business Media New York 2013
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Sample Average Approximation
indices after the choice of the appropriate criteria for defining failure. These might be the (complementary) probability of successful operation without accident, the mean time to accident appearance, etc. Sometimes the safety of systems (as for dams of hydro-power stations, constructions in seismic zones, etc.) is considered to be only under the influence of nature. In this case, probabilistic measures may be insufficient and one should instead consider conditional safety under some specified levels of external influence. But many systems may be harmful even under ideal conditions, without accidents. Examples are various chemical and metallurgical technological processes, power stations, and other objects polluting the environment with various toxic substances. To quantify, begin by letting f(t) be the poisonous emission function in time. One useful index of safety of such a system is the condition that, for some specified time interval of width D, Z
tþD
t
f ðtÞdt f
½f ðtÞ fðtÞþ dt f 0
tþD
Ushakov, I. A. (Ed.). (1994). Handbook of reliability engineering. New York: Wiley.
Sample Average Approximation Alexander Shapiro Georgia Institute of Technology, Atlanta, GA, USA
Introduction
Min f0 ðxÞ subject to fi ðxÞ 0; i ¼ 1; . . . ; q: x2w
where [·]+ denotes the positive part of the number in brackets. Many harmful processes (radioactive emission, dioxide pollution, etc.) exponentially calm do
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