# Simplex - ESI 6417 Simplex Method 1 Ref. Chapters 3 and 4,...

This preview shows pages 1–5. Sign up to view the full content.

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

View Full Document

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.

Unformatted text preview: ESI 6417 Simplex Method 1 Ref. Chapters 3 and 4, BJS Consider linear programs of the form: min c T x s.t. Ax < b , x > 0. It is also convenient to write the above as min{ c T x : Ax < b , x > 0}. • Theorem 3.3 : If an optimal solution to a linear program exists, then there must be an optimal solution which is also an extreme point. Proof : Let x * be an optimal solution to the above LP. Assume that x * is not an extreme point. Algebraic Characterization of an extreme point • To simplify, consider linear programs of the following forms instead: min c T x s.t. Ax = b , x > 0 or min{ c T x : Ax = b , x > 0}. - Linear programs with inequality constraints can be converted into ones with only equality constraints (nonnegativity constraints excluded) be adding “slacks” and “surplus” variables. For example, min 2 x 1 + 3 x 2 + x 3 s.t. x 1 + x 2 + x 3 < 10 2 x 1 – x 2 – 2 x 3 > 5 x 1 , x 2 , x 3 > 0 - The result (Theorem 2.1) concerning extreme points and directions still applies. Why? ESI 6417 Simplex Method 2 Ref. Chapters 3 and 4, BJS • Basic feasible solutions to linear programs - There are two notions in the above heading: “basic” and “feasible.” - Consider the feasible region of the (equality constrained) LP, i.e., Ax = b , x > 0. For example, x 1 +2 x 2 + 1 x 4 + 2 x 5 = 2 3 x 3 + 2 x 4 + x 5 = 3 x 1 , x 2 , x 3 , x 4 , x 5 > 0 - A graphical interpretation of a feasible solution. - How many solutions are there in the feasible region? - Ignoring the nonnegativity constraints, what is a mathematical expression for a solution to the above linear system? ESI 6417 Simplex Method 3 Ref. Chapters 3 and 4, BJS - Assumptions : ¡ A is m × n and b ∈ R m (i.e., b is m ×1). (Thus, x , the vectors of decision variables, must be n ×1 or x ∈ R n .) ¡ Rank( A , b ) = Rank( A ) = m . (Why necessary?) - Basic solutions : We obtain a basic solution by rearranging columns of A so that A can be decomposed into two submatrices as follows: A = [ B : N ], where B is an m × m invertible matrix and N is an m ×( n – m ). Similarly, decompose x into . Then, satisfies Ax = b and is called a “basic” solution. ¡ When B-1 b > 0, then the solution satisfies the nonnegativity constraints and is, thus, feasible the linear program. In this case, we call a “basic (and) feasible” solution ¡ Example: x 1 +2 x 2 + 1 x 4 + 2 x 5 = 2 3 x 3 + 2 x 4 + x 5 = 3 x 1 , x 2 , x 3 , x 4 , x 5 > 0 - With respect to the feasible region X ={ x : Ax = b , x > 0}, does a basic feasible solution satisfy the definition of an extreme point? Why? (Assume that x B > 0.) ESI 6417 Simplex Method 4 Ref. Chapters 3 and 4, BJS - Terminology: ¡ B = a basic matrix or basis ¡ N = a non-basic matrix ¡ x B = a vector of basic variables or variables in the basis ¡ x N = a vector of non-basic variables....
View Full Document

## This note was uploaded on 09/19/2008 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

### Page1 / 11

Simplex - ESI 6417 Simplex Method 1 Ref. Chapters 3 and 4,...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online