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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.
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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.
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Which of the following functions is strictly convex?
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Which of the following functions is concave?
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Which of the following is an example of a convex function defined on Rn?
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Which of the following is a concave function on R++?
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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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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.
