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Understanding Convex Functions Quiz

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Questions and Answers

Which of the following is a convex function?

Logarithm function: $log(x)$, for any $x > 0$

Powers of absolute value: $|x|^p$, for $p \geq 1$

Negative part function: $min\{0, x\}$

Exponential function: $e^x$, for any $x \in R$ (correct)

Which function is a concave function?

Sum of squares function: $\|x\|^2 = x_1^2 + \cdots + x_n^2$

Entropy function: $-x \log x$ for $x > 0$ (correct)

Affine function: $ax + b$, for any $a, b \in R$

Powers function: $x^\alpha$, for $0 \leq \alpha \leq 1$

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

Sum of squares: $\|x\|^2 = x_1^2 + \cdots + x_n^2$

Softmax or log-sum-exp function: $log(exp(x_1) + \cdots + exp(x_n))$

General affine function: $f(X) = tr(A^TX) + b$ (correct)

Max function: $max(x) = max\{x_1, x_2, \ldots, x_n\}$

Which of the following functions is strictly convex?

Exponential function: $e^x$, for any $x \in R$ (D) Signup and view all the answers

Which of the following functions is a concave function?

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

Which of the following best defines a convex function?

A function where the domain is convex and satisfies $f( \theta x + (1 - \theta)y) \leq \theta f(x) + (1 - \theta)f(y)$ (D) Signup and view all the answers

Which of the following is an example of a concave function on $R^{n}$?

Logarithm: $log(x)$ (A) Signup and view all the answers

Which function is a strictly convex function?

Exponential: $e^x$ (D) Signup and view all the answers

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

Sum of squares: $|x|^2 = x_1^2 + x_2^2 + ... + x_n^2$ (B) Signup and view all the answers

Which of the following is NOT an example of a convex function on $R^{n}$?

Powers: $x^\alpha$ for $\alpha geq 1$ or $\alpha leq 0$ (C) Signup and view all the answers