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FinalReview - Introduction to Mathematical Programming...

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Introduction to Mathematical Programming IE496 Final Review Dr. Ted Ralphs
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IE496 Final Review 1 Course Wrap-up: Chapter 2 In the introduction, we discussed the general framework of mathematical modeling and defined the concept of a linear program . In Chapter 2 , we discussed the geometry of linear programming. We showed how to solve linear programs graphically and showed that the feasible region for an LP is a polyhedron . A (bounded) polyhedron is described by its vertices , or extreme points . Every linear program has an optimal solution that is an extreme point of its feasible region. In order to solve a linear program, we derived an algebraic analogue of an extreme point, called a basic feasible solution . We showed how to construct basic feasible solutions by choosing a basis matrix and solving an associated system of equations. We showed that the set of extreme points and the set of basic feasible solution are the same and that therefore, we need only consider basic feasible solutions when optimizing.
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IE496 Final Review 2 Course Wrap-up: Chapter 3 In Chapter 3 , we first discussed optimality conditions. From a given starting point, we showed how to construct a feasible, improving direction . We derived that the reduced cost of variable j is the cost reduction that occurs from moving in the jth basic direction . We therefore derived that if no variable has negative reduced cost, then the current solution is optimal . We then derived the simplex algorithm .
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