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MIT6_235S10_hw02 - 6.253 Convex Analysis and Optimization...

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6.253: Convex Analysis and Optimization Homework 2 Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 (a) Let C be a nonempty convex cone. Show that cl ( C ) and ri ( C ) is also a convex cone. (b) Let C = cone ( { x 1 , . . . , x m } ). Show that m ri ( C ) = { a i x i | a i > 0 , i = 1 , . . . , m } . i =1 Problem 2 Let C 1 and C 2 be convex sets. Show that C 1 ri ( C 2 ) = if and only if ri ( C 1 aff ( C 2 )) ri ( C 2 ) = . Problem 3 (a) Consider a vector x such that a given function f : R n �→ R is convex over a sphere centered at x . Show that x is a local minimum of f if and only if it is a local minimum of f along every line passing through x [i.e., for all d R n , the function g : R �→ R , defined by g ( α ) = f ( x + αd ), has α = 0 as its local minimum]. (b) Consider the nonconvex function f : R 2 �→ R given by f ( x 1 , x 2 ) = ( x 2 px 1 2 )( x 2 qx 1 2 ) , where p and q are scalars with 0 < p < q , and x = (0 , 0). Show that f ( y, my 2 ) < 0 for y = 0 and m satisfying p < m < q , so x is not a local minimum of f even though it is a local minimum along every
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