Convex Functions in Optimization Theory
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Questions and Answers

Convex functions are always concave.

False

A function f(x) is strictly convex if for all x, y in dom f and 0 < 𝜃 < 1, f ( 𝜃x + (1 − 𝜃)y) < 𝜃 f (x) + (1 − 𝜃)f (y).

True

The function f(x) = |x|^3 is a convex function on R.

True

The max function max(x) = max{x1, x2, ..., xn} is a convex function on Rn.

<p>False</p> Signup and view all the answers

The function f(X) = tr(A^T X) + b is a convex function when X ∈ R^{m×n} (m × n matrices) and A, b are constants.

<p>True</p> Signup and view all the answers

Which of the following functions is strictly convex?

<p>Exponential: e^ax, for any a ∈ R</p> Signup and view all the answers

Which of the following functions is concave?

<p>Logarithm: log x on R++</p> Signup and view all the answers

Which of the following is an example of a convex function defined on Rn?

<p>Softmax or log-sum-exp function: log(exp x1 + ... + exp xn)</p> Signup and view all the answers

Which of the following is a concave function on R++?

<p>Logarithm: log x on R++</p> Signup and view all the answers

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

<p>Exponential: e^ax, for any a ∈ R</p> Signup and view all the answers

Study Notes

Convex Functions

  • A function is strictly convex if it satisfies the condition: f (𝜃x + (1 − 𝜃)y) < 𝜃 f (x) + (1 − 𝜃)f (y) for all x, y in dom f and 0 < 𝜃 < 1.

Examples of Convex Functions

  • The function f(x) = |x|^3 is a convex function on R.
  • The max function max(x) = max{x1, x2, ..., xn} is a convex function on Rn.
  • The function f(X) = tr(A^T X) + b is a convex function when X ∈ R^{m×n} (m × n matrices) and A, b are constants.

Important Notes

  • Convex functions are not concave functions.
  • The definition of a strictly convex function is different from a convex function.

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Description

Learn about convex functions in optimization theory, including operations that preserve convexity, constructive convex analysis, perspective, conjugate, and quasiconvexity. Understand the definitions of convex, concave, and strictly convex functions.

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