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Chapter 5 - Chapter 5 Geometry of optimization In this...

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Chapter 5 Geometry of optimization In this chapter we introduce a number of geometric concepts and will interpret much of the material defined in the previous chapters through the lens of geometry. Questions that we will address include: what can we say about the shape of the set of solutions to a linear program, how are basic feasible solutions distinguished from the set of all feasible solutions, what does that say about the simplex algorithm, is there a geometric interpretation of complementary slackness? 5.1 Feasible solutions to linear programs and polyhedra Consider the following linear program, max ( c 1 , c 2 ) x s.t. 1 1 0 1 1 0 - 1 0 0 - 1 x 3 2 2 0 0 ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) (5.1) We represented the set of all feasible solutions to (5.1) in the following figure. The set of all points ( x 1 , x 2 ) T satisfying constraint (2) with equality correspond to line (2). The set of all points satisfying constraint (2) correspond to all points to the left of line (2). A similar argument 129
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5.1. FEASIBLE SOLUTIONS TO LINEAR PROGRAMS AND POLYHEDRA 130 1 2 x 1 x 2 (3) (1) (2) 3 1 2 3 0 (4) (5) Figure 5.1: Feasible region holds for constraints (1),(3),(4) and (5). Hence, the set of all feasible solutions of (5.1) is the shaded region (called the feasible region ). Looking at examples in 2 as above can be somewhat misleading however. In order to get the right geometric intuition we need to introduce a number of definitions. First recall, that Remark 32. Let a , b n . Then a T b = 0 if and only if a , b are orthogonal, a T b < 0 if and only if the angle between a , b is less than 90 o , a T b > 0 if and only if the angle between a , b is larger than 90 o . Let a be a non-zero vector with n components and let β , we define 1. H : = { x n : a T x = β } is a hyperplane , and 2. F : = { x n : a T x β } is a half-space . Consider the following inequality, a T x β . ( ) cDepartment of Combinatorics and Optimization, University of Waterloo Fall 2011
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CHAPTER 5. GEOMETRY OF OPTIMIZATION 131 Hence, H is the set of points satisfying constraint ( ) with equality and F is the set of points satisfying constraint ( ). Suppose that ¯ x H and let x be any other point in H . Then a T ¯ x = a T x = β . Equivalently, a T ( x - ¯ x ) = 0, i.e. a and x - ¯ x are orthogonal. This implies (1) in the following remark, we leave (2) as an exercise, Remark 33. Let ¯ x H. 1. H is the set of points x for which a and x - ¯ x are orthogonal, 2. F is the set of points x for which a and x - ¯ x form an angle of at least 90 o . We illustrate the previous remark in the following figure. The line is the hyperplane H and the shaded region is the halfspace F . a H F x - ¯ x ¯ x x x x - ¯ x In 2 a hyperplane is a line, i.e. a 1-dimensional object. What about in n ? Consider the hyperplane H : = { x n : a T x = 0 } . Then H is a vector space and we know how to define its dimension. Recall, that for any m × n matrix A we have the relation, dim { x : Ax = 0 } + rank ( A ) = n .
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