An hyperplane h supports c at a point x if c is a

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Unformatted text preview: at x is a supporting hyperplane. An hyperplane H supports the better than set at x if % (x ) is a strict subset of H + = fx 2 Rn : (x y) 0g : The tangent to the better than set at x is also a supporting hyperplane. Fact An optimum is a point where the same hyperlane supports both C and % (x ). Geometry Famous theorem: a non-empty convex set has at least one supporting hyperplane. How do we …nd it? De…nition Let X Rn be a convex set. If x 2 X , the normal cone to X at x is NX (x ) = f 2 Rn : (x y) 0 for all y 2 X g : Elements of NX (x ) are orthogonal to the hyperplane that supports X at x . NX (x ) is the set of all vectors that generate hyperplanes which support X at x . Famous theorem says that the normal cone contains at least one half line. An optimum is a point where an hyperplane supports % (x ) and belongs to NC (x ). This gives some intuition for the following theorem. First Order Conditions for Convex Optimization First Order Conditions for Convex Optimization (KKT) Let C Rn be a convex set and let f : Rn ! R be a di¤erentiable, monotonic, and + quasi-concave function, such that rf (x ) 6= 0 for all x 2 C . Then: x is a solution to max f (x ) x 2C if and only if rf (x ) 2 NC (x ) Why rf (x )? Because the gradient is orthogonal to the level set. Suppose C = fx 2 Rn : gi (x ) 0 with i = 1; :::; mg where each gi (x ) is convex and di¤erentiable. If there exists a strictly feasible point in C ,a the normal cone of C is ( ) m X n NC (x ) = z 2 R : z = 0 and i gi (x ) = 0 for i = 1; ::; m i rgi (x ) with i k =i In this case, the …rst order condition become rf (x ) aA m X i =1 i rgi (x ) = 0 with i point x is strictly feasible if g i (x ) < 0 for i = 1 ; :::; N 0 and i gi (x ) = 0 for each i Details To Remember If the better than set or the constraint sets are not convex: big trouble. If functions are not di¤erentiable: small trouble. If the geometry still works we can …nd a more general theorem. When does the recipe fail? If the constraint quali…cation condition fails. If the object...
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This note was uploaded on 05/13/2013 for the course ECON 2100 taught by Professor Board,o during the Fall '08 term at Pittsburgh.

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