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Lecture7

# Lecture7 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 7 Dr. Ted Ralphs

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IE417 Lecture 7 1 Reading for This Lecture Chapter 4, Section 2
IE417 Lecture 7 2 Optimality Conditions Inequality Constrained Problems (continued)

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IE417 Lecture 7 3 Fritz-John Necessary Conditions Theorem 1. Consider the feasible region S = { x X : g i ( x ) 0 , i [1 , m ] } where X is a nonempty open set in R n and g i : R n R , i [1 , m ] . Given a feasible x * S , set I = { i : g i ( x * ) = 0 } . Assume that f and g i are differentiable at x * for i I and g i is continuous at x * for i / I . If x * is a local minimum, then there exists μ R m such that μ 0 5 f ( x * ) + X μ i 5 g i ( x * ) = 0 μ i g i ( x * ) = 0 i [1 , m ] μ 0 μ 6 = 0
IE417 Lecture 7 4 Terminology The μ i ’s are called Lagrange multipliers or dual multipliers . The requirement that x * S is called the primal feasibility (PF) condition. The requirement that μ 0 5 f ( x * ) + μ i 5 ( x * ) = 0 is called the dual feasibility (DF) condition.

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Lecture7 - Advanced Mathematical Programming IE417 Lecture...

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