L6_fun_geom - Lecture 6 Closed Functions February 5 2007 Lecture 6 Outline I Log-Transformation I Jensen’s Inequality I Level Sets I Closed

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Unformatted text preview: Lecture 6: Closed Functions February 5, 2007 Lecture 6 Outline I Log-Transformation I Jensen’s Inequality I Level Sets I Closed Functions I Convexity and Continuity I Recession Cone and Constancy Space I Optimization Terminology Convex Optimization 1 Lecture 6 Log-Transformation of Variables Useful for transforming a nonconvex function to a convex one A posynomial is a function of y 1 > ,...,y n > of the form g ( y 1 ,...,y n ) = by a 1 1 ··· y a n n with scalars b > and a i > for all i . I A posynomial need not be convex I Log-transformation of variables x i = ln y i for all i I We have convex function f ( x ) = be a 1 x 1 ··· e a n x n = be a x Convex Optimization 2 Lecture 6 Log-Transformation of Functions Replacing f with ln f [when f ( x ) > over dom f ] Useful for: I Transforming non-separable functions to separable ones Example : (Geometric Mean) f ( x ) = (Π n i =1 x i ) 1 /n for x with x i > for all i is non-separable. Using F ( x ) = ln f ( x ) , we obtain a separable F , F ( x ) = 1 n n X i =1 ln x i I Separable structure of objective function is advantageous in distributed optimization Convex Optimization 3 Lecture 6 General Convex Inequality Basic convex inequality : For a convex f , we have for x,y ∈ dom f and α ∈ (0 , 1) , f ( αx + (1- α ) y ) ≤ αf ( x ) + (1- α ) f ( y ) General convex inequality : For a convex f and any convex combination of the points in dom f , we have f m X i =1 α i x i ! ≤ m X i =1 α i f ( x i ) ( x i ∈ dom f and α i > for all i , ∑ m i =1 α i = 1 , m > integer) A convex combination ∑ m i =1 α i x i can be viewed as the expectation of a random vector z having outcomes z = x i with probability α i Convex Optimization 4 Lecture 6 Jensen’s Inequality General convex inequality can be interpreted as: I For a convex f and a (finite) discrete random variable z with outcomes z i ∈ dom f , we have f ( E z ) ≤ E [ f ( z )] General Jensen’s inequality : The above relation holds for a convex f and,...
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This note was uploaded on 08/22/2008 for the course GE 498 AN taught by Professor Angelianedich during the Spring '07 term at University of Illinois at Urbana–Champaign.

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L6_fun_geom - Lecture 6 Closed Functions February 5 2007 Lecture 6 Outline I Log-Transformation I Jensen’s Inequality I Level Sets I Closed

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