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non_convex_prob

# non_convex_prob - Issues in Non-Convex Optimization Robert...

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Issues in Non-Convex Optimization Robert M. Freund with assistance from Brian W. Anthony April 22, 2004 c 2004 Massachusetts Institute of Technology. 1

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1 Outline General Nonlinear Optimization Problem Optimality Conditions for NLP Sequential Quadratic Programming (SQP) Method LOQO: Combining Interior-Point Methods and SQP Practical Issues in Solving NLP Problems 2 General Nonlinear Optimization Problem NLP: minimize x f ( x ) s.t. g i ( x ) = 0 , i ∈ E g i ( x ) 0 , i ∈ I n x , n n where f ( x ) : , g i ( x ) : , i ∈ E ∪I , E denotes the indices of the equality constraints, and I denotes the indices of the inequality constraints. 2.1 General Comments Non-convex optimization problems arise in just about every economic and scientific domain: radiation therapy engineering product design economics: Nash equilibria finance: options pricing 2
industrial engineering: traﬃc equilibria, supply chain management many other domains as well Non-convex optimization is hard. Since x x 2 = 0 if and only if x ∈ { 0 , 1 } , we can formulate binary integer optimization as the following nonlinear optimization instance: T BIP: minimize x c x s.t. Ax b 2 = 0 , j = 1 , . . . , n x j x j n x 2.2 Useful Definitions The feasible region F of NLP is the set F = { x | g i ( x ) = 0 for i ∈ E , g i ( x ) 0 for i ∈ I} We have the following definitions of local/global, strict/non-strict min- ima/maxima. Definition 2.1 x ∈ F is a local minimum of NLP if there exists > 0 such ¯ x ) f ( x ) for all x B that f x, ) ∩ F . Definition 2.2 x ∈ F is a global minimum of NLP if f ¯ x ) f ( x ) for all x ∈ F . Definition 2.3 x ∈ F is a strict local minimum of NLP if there exists > 0 ¯ x ) < f ( x ) for all x B = ¯ such that f x, ) ∩ F , x x . Definition 2.4 x ∈ F is a strict global minimum of NLP if f ¯ x ) < f ( x ) for all x ∈ F , x x . = ¯ 3

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Definition 2.5 x ∈ F is a local maximum of NLP if there exists > 0 ¯ x ) f ( x ) for all x B such that f x, ) ∩ F . Definition 2.6 x ∈ F is a global maximum of NLP if f ¯ x ) f ( x ) for all x ∈ F . Definition 2.7 x ∈ F is a strict local maximum of NLP if there exists ¯ x ) > f ( x ) for all x B = ¯ > 0 such that f x, ) ∩ F , x x . Definition 2.8 x ∈ F is a strict global maximum of NLP if f ¯ x ) > f ( x ) for all x ∈ F , x x . = ¯ If x is feasible for NLP, we let I ( x ) denote the indices of the active inequality constraints, namely: I ( x ) := { i ∈ I | g i ( x ) = 0 } . 3 Optimality Conditions for NLP Theorem: Karush-Kuhn-Tucker Necessary Conditions. Suppose that f ( x ) and g i ( x ) , i ∈ E ∪ I , are all differentiable functions. Under mild additional conditions, if ¯ x is a local minimum of NLP, then there exists y ¯ for which x ) + y ¯ i g i x ) = 0 ( i ) f x ) + y ¯ i g i i ∈E i ∈I ( ii ) g i x ) = 0 , i ∈ E ( iii ) g i x ) 0 , i ∈ I ( iv ) ¯ y i 0 , i ∈ I ( v ) y ¯ i · g i x ) = 0 , i ∈ I .
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