{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 5

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern