Convex Functions: Theory and Operations
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Questions and Answers

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?

    <p>entropy: −x log x</p> Signup and view all the answers

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

    <p>Affine functions</p> Signup and view all the answers

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

    <p>Powers: $x^\alpha$ on $R^{++}$, for $\alpha <br /> eq 0$</p> Signup and view all the answers

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

    <p>Affine: $ax + b$ on $R$, for any $a, b <br /> in R$</p> Signup and view all the answers

    What is the defining property of a strictly convex function?

    <p>The function must satisfy the inequality $f(\theta x + (1 - \theta)y) &lt; \theta f(x) + (1 - \theta)f(y)$</p> Signup and view all the answers

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

    <p>Sum of squares: $|x|^2 = x_1^2 + ... + x_n^2$</p> Signup and view all the answers

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

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

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

    <p>exponential: $e^{-x}$</p> Signup and view all the answers

    Which of the following operations preserves convexity?

    <p>Composition with an affine function</p> Signup and view all the answers

    What is the defining property of a strictly convex function?

    <p>$f(\theta x + (1 - \theta)y) &lt; \theta f(x) + (1 - \theta)f(y)$ for all $x, y \in \text{dom} f$, $x \neq y$, and $0 &lt; \theta &lt; 1$</p> Signup and view all the answers

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

    <p>Sum of squares: $|x|^2 = x_1^2 + x_2^2 + ... + x_n^2$</p> Signup and view all the answers

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

    <p>General affine function: $f(X) = \text{tr}(A^T X) + b$</p> Signup and view all the answers

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