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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 nonempty 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
+
quasiconcave 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.
 Fall '08
 Board,O

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