Lecture_March2_MATH4321_12S

# Lecture_March2_MATH4321_12S - Solving 2-Person 0-sum games...

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Solving 2 Solving 2 - - Person 0 Person 0 - - sum games by sum games by linear programming linear programming The basic problem of linear programming, determining the optimal value of a linear function subject to linear constraints, arise in a wide variety of situations. In 1939the Russian mathematician L.V. Kantorovich published a monograph entitled “Mathematical Methods in the Organization and Planning of Production”. Kantorovich went unrecognized in the West. Through the works of Frank Hitchcock and George Stigler and their efforts in World War II, it became clear that a feasible method for solving linear programming problems was needed. Then in 1947 George Dantzig developed the simplex method. John von Neumann recognized the importance of the concept of duality, and the first proof of the Duality Theorem is that of Gale, Kuhn and Tucker. The simplex method gives an algorithm to go from one vertex to another vertex in searching for the optimal value.

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Polynomial time algorithm Polynomial time algorithm In 1979, the Russian mathematician Leonid Khatchian announced a polynomial time algorithm for the resolution of the linear programming problem. The algorithm uses a sequence of ellipsoids to drive to a solution of a linear programming problem. A polynomial time linear programming algorithm using an interior point method was found by Karmarkar (1984). Arguably, interior point methods were known as early as The media hype accompanying Karmarkar's announcement led to these methods receiving a great deal of attention. However, it should be noted that while Karmarkar claimed that his implementation was much more efficient than the simplex method , the potential of interior point method was established only later. By 1994, there were more than 1300 published papers on interior point methods.
Reduction to a Linear Programming Problem. A Linear Program is defined as the problem of choosing real variables to maximize or minimize a linear function of the variables, called the objective function, subject to linear constraints on the variables. The constraints may be equalities or inequalities.

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The Primal Problem of a linear programming problem is in the following form. Choose x 1 ,…,x m To minimize c 1 x 1 +…+ c m x m Subject to the constraints a 11 x 1 + … + a m1 x m b 1 a 1n x 1 + … + a mn x m b n and x i 0 for i=1,…,m
The Dual Problem to the primal problem is to choose y 1 ,…,y n To maximize b 1 y 1 +…+ b n y n Subject to the constraints a 11 y 1 + … +a 1n y n c 1 a m1 y 1 + … +a mn y n c m and y j 0 for j=1,…,n

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## This note was uploaded on 03/13/2012 for the course MATH 4321 taught by Professor Cheng during the Spring '12 term at HKUST.

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Lecture_March2_MATH4321_12S - Solving 2-Person 0-sum games...

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