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a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y) is a convex function.

a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y) is a convex function.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.2: Mappings
Problem 18E: 18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
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a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y) is a convex function. b. Consider the following optimization problem min fo(x) s.t. Ax = b where A E R"X" and b e R". Let x* E Xopt. Write out an equivalent condition that characterizes x* and prove your conclusion. c. Solve the following optimization problem by first writing out its equivalent optimization problem: min c'x s.t. xT Ax <1 where A E S+ and c + 0. d. Consider the following convex optimization problem min fo(x) s.t. x E N where N C R" is a convex set. Let x* E Xopt. Write out an equivalent condition that characterizes x* and prove your conclusion. e. Write out the standard form a standard convex optimization problem and show that the feasible set and the solution set of a standard convex optimization problem are convex. f. Consider the unconstrained optimization problem where fo(x) = ||Ax – b||. Formulate it into a linear programming problem. g. f : R? → R with f(x1, x2) = |xı|+|x2| + 3x3 + 3x3 + (x1 – 18x2)4 + 2e10x1+2x2=5 is a convex function on R?. - h. Let f(X) = Amax(X) and dom(f) = S". Prove that f is a convex function. i. A differentiable function f : R" → R is strongly convex with constant m if and only if (Vf(x) – Vf(y))" (x – y) > m||x – y||3 holds for all x, y. j. Let f (x) = ||x||. Find out its conjugate function f* and prove your conclusion.
Transcribed Image Text:a. If f(x,y) is convex with respect to (x,y) and C is a convex set, show that g(x) = infyec f(x,y) is a convex function. b. Consider the following optimization problem min fo(x) s.t. Ax = b where A E R"X" and b e R". Let x* E Xopt. Write out an equivalent condition that characterizes x* and prove your conclusion. c. Solve the following optimization problem by first writing out its equivalent optimization problem: min c'x s.t. xT Ax <1 where A E S+ and c + 0. d. Consider the following convex optimization problem min fo(x) s.t. x E N where N C R" is a convex set. Let x* E Xopt. Write out an equivalent condition that characterizes x* and prove your conclusion. e. Write out the standard form a standard convex optimization problem and show that the feasible set and the solution set of a standard convex optimization problem are convex. f. Consider the unconstrained optimization problem where fo(x) = ||Ax – b||. Formulate it into a linear programming problem. g. f : R? → R with f(x1, x2) = |xı|+|x2| + 3x3 + 3x3 + (x1 – 18x2)4 + 2e10x1+2x2=5 is a convex function on R?. - h. Let f(X) = Amax(X) and dom(f) = S". Prove that f is a convex function. i. A differentiable function f : R" → R is strongly convex with constant m if and only if (Vf(x) – Vf(y))" (x – y) > m||x – y||3 holds for all x, y. j. Let f (x) = ||x||. Find out its conjugate function f* and prove your conclusion.
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