MIT6_235S10_hw01 - 6.253: Convex Analysis and Optimization...

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Unformatted text preview: 6.253: Convex Analysis and Optimization Homework 1 Prof. Dimitri P. Bertsekas Spring 2010, M.I.T. Problem 1 (a) Let C be a nonempty subset of Rn , and let λ1 and λ2 be positive scalars. Show that if C is convex, then (λ1 + λ2 )C = λ1 C + λ2 C . Show by example that this need not be true when C is not convex. (b) Show that the intersection ∩i∈I Ci of a collection {Ci | i ∈ I } of cones is a cone. (c) Show that the image and the inverse image of a cone under a linear transformation is a cone. (d) Show that the vector sum C1 + C2 of two cones C1 and C2 is a cone. (e) Show that a subset C is a convex cone if and only if it is closed under addition and positive scalar multiplication, i.e., C + C ⊂ C , and γ C ⊂ C for all γ > 0. Problem 2 Let C be a nonempty convex subset of Rn . Let also f = (f1 , . . . , fm ), where fi : C �→ �, i = 1, . . . , m, are convex functions, and let g : Rm �→ R be a function that is convex and monotonically nondecreasing over a convex set that contains the set {f (x) | x ∈ C }, in the sense that for all u1 , u2 in this set such that u1 ≤ u2 , we have g (u1 ) ≤ g (u2 ). Show that the function h defined by h(x) = g (f (x)) is convex over C . If in addition, m = 1, g is monotonically increasing and f is strictly convex, then h is strictly convex. Problem 3 Show that the following functions from Rn to (−∞, ∞] are convex: (a) f1 (x) = ln(ex1 + · · · + exn ). (b) f2 (x) = �x�p with p ≥ 1. � (c) f3 (x) = eβx Ax , where A is a positive semidefinite symmetric n × n matrix and β is a positive scalar. (d) f4 (x) = f (Ax + b), where f : Rm �→ R is a convex function, A is an m × n matrix, and b is a vector in Rm . Problem 4 Let X be a nonempty bounded subset of Rn . Show that cl(conv (X )) = conv (cl(X )). In particular, if X is compact, then conv(X ) is compact. 1 Problem 5 Construct an example of a point in a nonconvex set X that has the prolongation property, but is not a relative interior point of X . 2 MIT OpenCourseWare http://ocw.mit.edu 6.253 Convex Analysis and Optimization Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. ...
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MIT6_235S10_hw01 - 6.253: Convex Analysis and Optimization...

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