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Questions and Answers
Convex functions are always concave.
Convex functions are always concave.
False (B)
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).
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 (A)
The function f(x) = |x|^3 is a convex function on R.
The function f(x) = |x|^3 is a convex function on R.
True (A)
The max function max(x) = max{x1, x2, ..., xn} is a convex function on Rn.
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.
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.
Which of the following functions is strictly convex?
Which of the following functions is strictly convex?
Which of the following functions is concave?
Which of the following functions is concave?
Which of the following is an example of a convex function defined on Rn?
Which of the following is an example of a convex function defined on Rn?
Which of the following is a concave function on R++?
Which of the following is a concave function on R++?
Which of the following functions is an example of a convex function on R?
Which of the following functions is an example of a convex function on R?
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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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