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# lecture5 - Main Points of this Lecture Def basic feasible...

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e62: lecture 4 29/09/10 Main Points of this Lecture Def: basic feasible solution = feasible solution at intersection of N linearly independent constraint boundaries Def: vertex of a polytope = an element that is not between any others Thm: vertex = basic feasible solution Thm: there is usually a vertex that is an optimal solution 1 Concepts will be useful for analysis and algorithm development e62: lecture 4 29/09/10 Basic Feasible Solutions Polytope: Def. Linearly independent constraints Def. Basic solution = intersection of N lin. indep. boundaries Combinatorial representation Def. Basic feasible solution = basic solution that is feasible Thm: there are a finite number of BFSs 2 P = { x N | Ax b } 1 2 2 4 - 1 1 5 1 x 1 x 2 b 1 b 2 b 3 b 4

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e62: lecture 4 29/09/10 Convexity Def: w is a convex combination of u and v if Def: a set is convex if it includes all convex combinations of its elements Thm:
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