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Problem 4 (Geometric Programming): In this exercise, we discuss a class of nonconvex geometric programs that can be reformulated as convex optimization problems. a) Let a € R" be given with a; = 1. Show that the matrix A := diag(a) – aaT is positive semidefinite. (Here, diag(a) is a n x n diagonal matrix with a on its diagonal).

Problem 4 (Geometric Programming): In this exercise, we discuss a class of nonconvex geometric programs that can be reformulated as convex optimization problems. a) Let a € R" be given with a; = 1. Show that the matrix A := diag(a) – aaT is positive semidefinite. (Here, diag(a) is a n x n diagonal matrix with a on its diagonal).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 4 (Geometric Programming): In this exercise, we discuss a class of nonconvex geometric programs that can be reformulated as convex optimization problems. a) Let a € R" be given with C1 aị = 1. Show that the matrix A := diag(a) – aaT is positive semidefinite. (Here, diag(a) is a n x n diagonal matrix with a on its diagonal). Hint: The Cauchy-Schwarz inequality rTy < ||r||||| x, y E R", can be helpful. b) We define f : R" → R, f(x) := log(E1 exp(x;)). Show that f is a convex function. c) Convert the following optimization problem into a convex problem: 13 min-eR³ max subject to rỉ + 2 < 2r2 (1) x1, 12, 13 > 0. Hint: Substitute the variables r; in an appropriate way and apply the result of part b). d) Use CVX (in MATLAB or Python) to solve problem (1).
Transcribed Image Text:Problem 4 (Geometric Programming): In this exercise, we discuss a class of nonconvex geometric programs that can be reformulated as convex optimization problems. a) Let a € R" be given with C1 aị = 1. Show that the matrix A := diag(a) – aaT is positive semidefinite. (Here, diag(a) is a n x n diagonal matrix with a on its diagonal). Hint: The Cauchy-Schwarz inequality rTy < ||r||||| x, y E R", can be helpful. b) We define f : R" → R, f(x) := log(E1 exp(x;)). Show that f is a convex function. c) Convert the following optimization problem into a convex problem: 13 min-eR³ max subject to rỉ + 2 < 2r2 (1) x1, 12, 13 > 0. Hint: Substitute the variables r; in an appropriate way and apply the result of part b). d) Use CVX (in MATLAB or Python) to solve problem (1).
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