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 deFned 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. 11 01 10 - 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 Fgure. 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 totheleftofline(2).Asimilarargument 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 12 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 de±nitions. First recall, that Remark 32. Let a , b n .Then a T b = 0 if and only if a , bareorthogonal, a T b < 0 if and only if the angle between a , bislessthan 90 o , a T b > 0 if and only if the angle between a , bislargerthan 90 o . Let a be a non-zero vector with n components and let β ,wede±ne 1. H : = { x n : a T x = } is a hyperplane ,and 2. F : = { x n : a T x } is a half-space . Consider the following inequality, a T x . ( ± ) c ± Department of Combinatorics and Optimization, University of Waterloo Fall 2011
CHAPTER 5. GEOMETRY OF OPTIMIZATION 131 Hence, H is the set of points satisfying constraint ( ± )w i thequa l i tyand F is the set of points satisfying constraint ( ± ). Suppose that ¯ x H and let x be any other point in H .T h e n a T ¯ x = a T x = β .Equ iv a l en t ly , 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 - ¯ xareorthogonal , 2. F is the set of points x for which a and x - ¯ xformanangleofatleast 90 o . We illustrate the previous remark in the following ±gure. Thelineisthehyperplane H and the shaded region is the halfspace F . a H F x - ¯ ¯ x x x x - ¯ x In 2 ahype rp laneisal ine ,i .e . a1 -d imens iona lob jec t . Wha tabout in n ?C o n s i d e r t h e hyperplane H : = { x n : a T x = 0 } .Then H is a vector space and we know how to de±ne 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.

## This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.

### Page1 / 14

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online