Linear programming and network flows
Addresses the problem of minimizing or maximizing a linear function in the presence of linear equality or inequality constraints. Provided are methods for modeling complex problems via effective algorithms on modern computers. The general theory and characteristics of optimization problems are presented, along with effective solution algorithms. Explores linear programming and network flows, employing polynomial-time algorithms and various specializations of the simplex method. Includes many numerical examples to illustrate theory and techniques.
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LINEAR ALGEBRA CONVEX ANALYSIS
THE SIMPLEX METHOD
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artificial variables assignment problem basic feasible solution basic variables coefficients column compute Consider the following constraints convex convex combination convex set corresponding cost cycle decomposition degenerate dual feasible dual problem dual variables enter the basis Equation example Exercise extreme directions extreme point feasible region Figure foregoing given graph Hence hyperplanes integer iteration Karmarkar's algorithm leaves the basis linear programming problem linearly independent matrix maximal flow Minimize cx Subject Minimum network flow problem nonbasic variables nonnegative Note objective function objective value obtain optimal objective optimal solution out-of-kilter algorithm phase pivot polyhedral set primal and dual revised simplex right-hand-side shortest path shortest path problem Show simplex algorithm simplex method slack variables Solve the following step Subject to Ax subproblem Suppose transportation problem tree unbounded updated upper bound vector xv x2 zero