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Electrical Engineering
Oct 3, 2023
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HW 1 EE 588: Optimization for the information and data sciences University of Southern California Assigned on: August 23, 2023 Due date: beginning of class on September 11, 2023 2.5 What is the distance between two parallel hyperplanes {z R" | a2 = b;} and {z R" | a"z = by}? 2.7 Voronoi description of halfspace. Let a and b be distinct points in R™. Show that the set of all points that are closer (in Euclidean norm) to a than b, i.e., {z | ||z —al|2 < ||z—b||2}, is a halfspace. Describe it explicitly as an inequality of the form ¢'z < d. Draw a picture. 2.12 Which of the following sets are convex? (a) A slab, i.e., a set of the form {z R" | a < aTz < B}. (b) A rectangle, i.e., a set of the form {z R" | a; <z < Bi, i =1,...,n}. A rectangle is sometimes called a hyperrectangle when n > 2. (c) A wedge, i.e., {x R" | alz < by, alz < by}. (d) The set of points closer to a given point than a given set, i.e., {o | llz - zoll2 < llo ylla for all y 5} where S C R". (e) The set of points closer to one set than another, i.e., {z | dist(z, S) < dist(z,T)}, where S, T C R", and dist(z,S) = inf{||lz z||2 | z S}. (f) [HUL93, volume 1, page 93] The set {z | z + Sz C S1}, where S1,S2 C R" with S; convex. (g) The set of points whose distance to a does not exceed a fixed fraction 6 of the distance to b, i.e., the set {z | ||z al|]2 < 6||z b||2}. You can assume a # b and 0<6<1. 2.28 Positive semidefinite cone for n = 1, 2, 3. Give an explicit description of the positive semidefinite cone S, in terms of the matrix coefficients and ordinary inequalities, for n =1, 2, 3. To describe a general element of S™, for n = 1, 2, 3, use the notation 1 xr2 rs3 1 xro I, N r2 T4 s . xr2 xrs3 r3 Ts Te
3.2 Level sets of convex, concave, quasiconvex, and quasiconcave functions. Some level sets of a function f are shown below. The curve labeled 1 shows {z | f(z) = 1}, etc. 3 2 Could f be convex (concave, quasiconvex, quasiconcave)? Explain your answer. Repeat for the level curves shown below. 12 3 4 5 6 3.3 Inverse of an increasing convex function. Suppose f : R R is increasing and convex on its domain (a,b). Let g denote its inverse, i.e., the function with domain (f(a), f(b)) and g(f(z)) = z for a < x < b. What can you say about convexity or concavity of g? Additional Exercise Dual cones in R2. Describe the dual cone for each of the following cones. 1. K = {0}, 2. K =R2% 3. K = {(.'Bl,xg) | |:B1| < :BQ}. 4. K = {(z1,22) | 21 + xz2 = 0}.
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