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Unformatted text preview: Lecture 14 The Simplex Method UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Wednesday, February 11th, 2009 Recap: Testing for Optimality Theorem If x * is a solution of an LP, then there is a basic solution of A T a λ a = c with λ * a ≥ . There are a finite number of basic solutions of A T a λ a = c . In particular, there are at most m a n = m a ! n !( m a n )! basic solutions. UCSD Center for Computational Mathematics Slide 2/41, Wednesday, February 11th, 2009 Recap: Testing for Optimality Assume that ¯ x is feasible with activeset matrix A a . if λ a = solve ( A T a , c ) fails then ¯ x is not a minimizer; else if λ a ≥ 0 for some basic solution of A T a λ a = c ; ¯ x is a minimizer; else ¯ x is not a minimizer; end if UCSD Center for Computational Mathematics Slide 3/41, Wednesday, February 11th, 2009 Vertex solutions The optimality conditions make no assumptions about x * being a vertex of FF . Example: minimize x ∈ R n c T x subject to Ax ≥ b , with c = 2 2 ! , A = 1 1 1 1 , b = 1 ¯ x = 1 2 1 2 ! is feasible with active set A (¯ x ) = { 1 } and A a = ( 1 1 ) . ⇒ ¯ x is not a vertex. UCSD Center for Computational Mathematics Slide 4/41, Wednesday, February 11th, 2009 Solving A T a λ a = c for the Lagrange multipliers gives 1 1 ! λ a = 2 2 ! ⇒ λ a = 2 > ⇒ ¯ x is optimal . UCSD Center for Computational Mathematics Slide 5/41, Wednesday, February 11th, 2009 x 1 x 2 ¯ x = 1 2 1 2 c T x = constant x 1 x 2 vertex solutions ¯ x = 1 2 1 2 c T x = constant Theorem (Existence of a vertex minimizer) Consider the linear programming problem of minimizing c T x subject to Ax ≥ b. If rank( A ) = n and the optimal objective value ‘ * is finite, then a vertex minimizer must exist. Proof: The assumption of a finite optimal c T x implies existence of a feasible x such that c = A T λ , λ ≥ where A is the activeconstraint matrix at x . UCSD Center for Computational Mathematics Slide 8/41, Wednesday, February 11th, 2009 If x is not a vertex, execute the vertexfinding algorithm, starting at x . This algorithm generates a sequence of minimizers { x k } with active sets of strictly increasing size . If x k is not a vertex, we move along a direction p k chosen to “hit” a new constraint and satisfy A k p k = 0. Because c = A T k λ k and A k p k = 0, we have: c T p k = ( A T k λ k ) T p k = λ T k ( A k p k ) = 0 ⇒ ‘ is unchanged when moving from x k along p k ....
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.
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