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Unformatted text preview: Lecture 3 Simplex Method I January 27, 2009 Solving Linear Programs The graphical method is only applicable for simple problems (e.g. problems with two variables). However, it provides some very important observations. • The feasible region has finite many vertices (corner points); • If the problem is bounded, then at least one optimal solution is also a vertex of the feasible region (the problem could have multiple optimal solutions, but at least one of them is a vertex of the feasible region); • If a vertex is not an optimal solution then there exists an adjacent vertex which is better than the current vertex in terms of the objective function value. These observations form the basis of the Simplex Method . 2 The Idea of the Simplex Method • Only consider the vertices (therefore the algorithm ends in finite steps). • Start form a vertex, keep moving to a better adjacent vertex until an optimal solution is reached (iterative method, each iteration corresponding to a vertex). To implement the idea, we need to know • how to represent the vertices algebraically; • how to check whether a vertex is an optimal solution; • how to get a better adjacent vertex if a vertex is not optimal. 3 Linear Programs in the Standard Form A linear program is in the standard form if • All constraints (except those nonnegativity constraints of variables) are equalities; • All variables are nonnegative; • The objective function and the constraints are simplified so that variables appear at most once on the lefthand side and any constant term appears on the righthand side. (noand any constant term appears on the righthand side....
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 Spring '09
 Nedich
 Linear Programming, Optimization, Basic Solutions, X1, Simplex algorithm, Polytope

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