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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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QUESTIONS: 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: cTx min x Ax <1 s.t. where A E S", and c +0. d. Consider the following convex optimization problem min fo(x) s.t. x EN where O CR" 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|l. Formulate it into a linear programming problem. g. f: R2 → R with f(x1, x2) = |x1| + |x2| +3x3 +3x3 + (x1 – 18x2)ª + 2el0x1 +2x2-5 is a convex function on R². 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||} holds for all x, y. j. Let f(x) = ||x||. Find out its conjugate function f* and prove your conclusion.
Transcribed Image Text:QUESTIONS: 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: cTx min x Ax <1 s.t. where A E S", and c +0. d. Consider the following convex optimization problem min fo(x) s.t. x EN where O CR" 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|l. Formulate it into a linear programming problem. g. f: R2 → R with f(x1, x2) = |x1| + |x2| +3x3 +3x3 + (x1 – 18x2)ª + 2el0x1 +2x2-5 is a convex function on R². 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||} holds for all x, y. j. Let f(x) = ||x||. Find out its conjugate function f* and prove your conclusion.
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