Convex Functions and Preservation of Convexity

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Questions and Answers

Which operation does NOT preserve convexity when applied to a simple convex function?

Nonnegative weighted sum

Taking the minimum

Pointwise maximum

Subtracting a linear function (correct)

What is a key property of the log-sum-exp function in relation to convex functions?

It is a composition with a non-affine function.

It is neither convex nor concave.

It can be represented as a nonnegative weighted sum of convex functions. (correct)

It is the pointwise maximum of convex functions.

In the context of Jensen's Inequality, which operation is essential for applying the theorem?

Using a pointwise minimum

Taking the supremum of a function

Applying nonnegative weights to the variables (correct)

Adding random variables

How can perspective operations affect the convexity of a function?

They preserve convexity when applied correctly. Signup and view all the answers

Which statement about epigraphs and sublevel sets is true regarding convex functions?

Epigraphs of convex functions are convex sets. Signup and view all the answers

What is the form of a convex function that is represented by the maximum of several functions?

f(x) = max{f1(x),..., fm(x)} Signup and view all the answers

Which of the following represents a piecewise-linear convex function?

f(x) = max{aTi x + bi | i=1,...,m} Signup and view all the answers

What does the sum of the r largest components of a vector x in Rn produce?

A convex function Signup and view all the answers

Which operation is guaranteed to preserve the convexity of a function?

Taking the maximum of convex functions Signup and view all the answers

Which concept is closely related to the property of a function being convex?

Epigraph and sublevel sets Signup and view all the answers

What is the requirement for a function to be classified as log-concave?

The function must be positive and its logarithm must be concave. Signup and view all the answers

Which of the following functions is log-convex?

The function $f(x) = e^{x^2}$ for $x ext{ in } R$. Signup and view all the answers

Which property does Jensen's Inequality illustrate in relation to convex functions?

The average of the function applied to arguments is less than or equal to the function of the average for convex functions. Signup and view all the answers

For which range of the exponent $a$ is the function $f(x) = x^a$ log-concave?

For $a ≥ 0$. Signup and view all the answers

Which of the following statements is correct regarding operations that preserve convexity?

The composition of a convex function with a linear function preserves convexity. Signup and view all the answers

Which of the following characteristics is true for the cumulative Gaussian distribution function Φ?

It is log-concave. Signup and view all the answers

Study Notes

Convex Functions and Operations that Preserve Convexity

Convex functions can be constructed from simple convex functions through specific operations that maintain their convexity.
Operations that preserve convexity include:
- Nonnegative weighted sums
- Composition with affine functions
- Pointwise maximum and supremum
- General composition
- Minimization
- Perspective transformation

Convexity of Max Functions

If f1, f2, ..., fm are convex functions, the function defined as f(x) = max{f1(x), ..., fm(x)} is also convex.

Examples of Convex Functions

A piecewise-linear function defined as f(x) = max{ai^T x + bi} for i = 1 to m is convex.
A sum of the r largest components of a vector x ∈ Rn, expressed as f(x) = x[i1] + x[i2] + ... + x[ir], where x[i] is the ith largest component, is convex.
Proof is established using the maximum of sums of the largest components.

Log-Concave and Log-Convex Functions

A positive function f is considered log-concave if the logarithm of the function, log f, exhibits concavity:
- The inequality f(θx + (1 - θ)y) ≥ f(x)^θ * f(y)^(1-θ) holds for 0 ≤ θ ≤ 1.
A function f is log-convex if log f is convex.

Special Cases of Log-Convexity and Log-Concavity

For any positive powers, x^a is log-convex when a ≤ 0 and log-concave when a ≥ 0.
Numerous common probability density functions are log-concave. For example, the normal distribution defined as:
- f(x) = (1 / ((2π)^(n/2)) * (det Σ)^(1/2)) * exp{(-1/2) * ((x - x̄)^T Σ^(-1) (x - x̄))}
The cumulative Gaussian distribution function, denoted as Φ(x), is also log-concave, expressed by the integral:
- Φ(x) = ∫(from -∞ to x) (1 / √(2π)) * e^(-u^2/2) du.

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Description

This quiz explores how simple convex functions can be derived using various operations that maintain their convexity. Key operations include nonnegative weighted sums, compositions with affine functions, pointwise maxima, and more. Test your understanding of these concepts and their applications in convex analysis.