Question: 1. Recall that a function f : R n R is convex if for all x, y R n and [0, 1], f(x) + (1

1. Recall that a function f : R n R is convex if for all x, y R n and [0, 1], f(x) + (1 )f(y) f(x + (1 )y). Using this definition, show that

(a) f(x) = wf1(x) is a convex function for x R n whenever f1 : R n R is a convex function and w 0

(b) f(x) = f1(x) + f2(x) is a convex function for x R n whenever f1 : R n R and f2 : R n R are convex functions

(c) f(x) = f1(Ax + b) is a convex function for x R m whenever f1 : R n R is a convex function and A R nm and b R n . Note: this notation means that A is a matrix with n rows and m columns.

Prof says just use algebra. Thanks!

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

L(y) {x e? : y} (2) Problem 3. Let f : R n + R be convex and Cl (R n ) (it is continuously differentiable). Consider the set: fl(x) {x' e Rn : - x) O} Fix a: Rn and let y f (x). Prove that L(y) C H...

Please help with this question. Thank you. 1. [2pts] Linear Models. Recall that a function f : R - R is linear if both of the following conditions hold. . For all x, y ER , f(xty) = f(x) + f(y). ....

Question 2.2 (1 1 marks) Consider the function f : R - R with f(x) = -2(x - 3)(x2 + 1). a) Determine an antiderivative of f (x). b) Describe the area of the region {(x, y) ER2 : x 2 -1 and 0 Sy 0...

Find the derivatives of the function f for n 1 2 3 and 4 f x x sin x n 1 n 2 n 3 n 4 f x f x f x f x Use the results to write a general rule for f x in terms of n f x

Problem 1 (2 marks). A function f: R" -> R is called weakly increasing if f (x) > f(y) whenever x _ y. It is called strictly increasing if f (x) > f(y) whenever x > y. Finally, f is called strongly...

Section 2-1 : Tangent Lines And Rates Of Change - Practice Problems 1. For the function f (r) = 3(x + 2) and the point P given by x = -3 answer each of the following questions. (a) For the points Q...

URGENT, no explanation for multiple choice answers Question 1 (1 point) The statement "Whether or not lim f(x) exists, depends on how f(a) is defined." is true x -a sometimes always O neverQuestion 2...

(1 point) Consider the function f (x) = x|x|. a.) On the interval (-co, 0), f" (x) = b.) On the interval (0, co), f" (x) = Thus, c.) On the interval (-co, 0), f(x) is Concave down v d.) On the...

please code these questions 3 and 4 on matlab urgent assistance needed with full details this is all info i have Lagrange Polynomial Interpolation Recall, given p + 1 distinct points {Xi}}=0, the...

1. Prove that a function f is convex if and only if for all n 2 0, for all xo, 21, ..., In E dom(f), for all to, t1, . .., tn 2 0 such that ELoti = 1, n n f ( [ timi ) s [tif(Ii). i=0 i=0 2. Let f :...

1. Suppose that you are a member of the board of directors for a large non-profit organization, and the board is required to draw up a constitution to govern it's activities . One of the things that...

Iridium metal, Ir, crystallizes in a face-centered cubic (close-packed) structure. The edge length of the unit cell was found by x-ray diffraction to be 383.9 pm. The density of iridium is 22.42...

Discuss the differences between U . S . GAAP and IFRS in Accounting for Losses and Gains in Defined Benefit Plans and in Reporting Pension Expenses.

On what date was the Tax Reform Act of 1 9 8 6 enacted, and what signi cance does this date have for losses derived from passive activities in 2 0 2 3 ?