Convex Functions: Theory and Operations

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Which of the following statements about convex functions is true?

A convex function is defined as a function where f(𝜃x + (1 − 𝜃)y) ≤ 𝜃 f(x) + (1 − 𝜃)f(y) for all x, y in the domain of f and 0 ≤ 𝜃 ≤ 1. (correct)
A convex function is defined as a function where f(𝜃x + (1 − 𝜃)y) > 𝜃 f(x) + (1 − 𝜃)f(y) for all x, y in the domain of f and 0 ≤ 𝜃 ≤ 1.
A convex function is defined as a function where f(𝜃x + (1 − 𝜃)y) < 𝜃 f(x) + (1 − 𝜃)f(y) for all x, y in the domain of f and 0 ≤ 𝜃 ≤ 1.
A convex function is defined as a function where f(𝜃x + (1 − 𝜃)y) ≥ 𝜃 f(x) + (1 − 𝜃)f(y) for all x, y in the domain of f and 0 ≤ 𝜃 ≤ 1.

Which of the following functions is an example of a convex function on R?

negative part: min{0, x}
logarithm: log(x)
affine: ax + b
exponential: e^x (correct)

What is the defining property of a strictly convex function?

The function is defined as strictly convex if for x, y in the domain of f, x ≠ y, 0 < 𝜃 < 1, f(𝜃x + (1 − 𝜃)y) < 𝜃f(x) + (1 − 𝜃)f(y). (correct)
The function is defined as strictly convex if for x, y in the domain of f, x = y, 0 < 𝜃 < 1, f(𝜃x + (1 − 𝜃)y) < 𝜃f(x) + (1 − 𝜃)f(y).
The function is defined as strictly convex if for x, y in the domain of f, x = y, 0 ≤ 𝜃 ≤ 1, f(𝜃x + (1 − 𝜃)y) < 𝜃f(x) + (1 − 𝜃)f(y).
The function is defined as strictly convex if for x, y in the domain of f, x ≠ y, 0 ≤ 𝜃 ≤ 1, f(𝜃x + (1 − 𝜃)y) < 𝜃f(x) + (1 − 𝜃)f(y).

Which of the following functions is an example of a concave function on R?

entropy: −x log x (D) Signup and view all the answers

Which type of functions are examples of convex functions on Rn?

Affine functions (B) Signup and view all the answers

Which function is an example of a convex function on R?

Powers: $x^\alpha$ on $R^{++}$, for $\alpha eq 0$ (C) Signup and view all the answers

Which type of function is an example of a concave function on $R$?

Affine: $ax + b$ on $R$, for any $a, b in R$ (A) Signup and view all the answers

What is the defining property of a strictly convex function?

The function must satisfy the inequality $f(\theta x + (1 - \theta)y) < \theta f(x) + (1 - \theta)f(y)$ (D) Signup and view all the answers

Which type of function is an example of a convex function on $R^n$?

Sum of squares: $|x|^2 = x_1^2 + ... + x_n^2$ (C) Signup and view all the answers

Which type of function is an example of a convex function on $R^{m \times n}$?

General affine function: $f(X) = tr(A^T X) + b$ (B) Signup and view all the answers

Which of the following functions is an example of a concave function on $R$?

exponential: $e^{-x}$ (A) Signup and view all the answers

Which of the following operations preserves convexity?

Composition with an affine function (C) Signup and view all the answers

What is the defining property of a strictly convex function?

$f(\theta x + (1 - \theta)y) < \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in \text{dom} f$, $x \neq y$, and $0 < \theta < 1$ (D) Signup and view all the answers