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

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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....
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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.

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

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