The conjugate (or Legendre-Fenchel transform) of a convex function $f$ is a mathematical operation that produces a new function $f^*$ based on $f$. This conjugate function plays an important role in optimization, duality theory, and variational analysis.

Definition of the Conjugate:

Given a convex function $f: \mathbb{R}^n \to \mathbb{R} \cup \{+\infty\}$, its conjugate (or Legendre-Fenchel transform) is defined as:

$f^*(y) = \sup_{x \in \mathbb{R}^n} \left( \langle x, y \rangle - f(x) \right)$

where:

• $\langle x, y \rangle$ is the dot product of $x$ and $y$.
• $f(x)$ is the original convex function.
• $f^*(y)$ is the conjugate of $f$.

Interpretation:

The conjugate function $f^*(y)$ can be thought of as the "best" value of the linear function $\langle x, y \rangle$ minus $f(x)$, for each value of $y$. In optimization problems, the conjugate is often used in the context of duality, where the original problem is referred to as the "primal problem," and the conjugate corresponds to the "dual problem."

Properties:

1. Convexity: If $f$ is convex, then $f^*$ is also convex.
2. Lower Semicontinuity: If $f$ is lower semicontinuous, then $f^*$ is also lower semicontinuous.
3. Duality: In optimization, the dual function is the conjugate of the Lagrangian of the primal problem.

Example:

Let $f(x) = \frac{1}{2} \|x\|^2$, the quadratic function. Then, its conjugate is:

$f^*(y) = \sup_x \left( \langle x, y \rangle - \frac{1}{2} \|x\|^2 \right)$

This can be solved by finding the value of $x$ that maximizes the expression. It turns out that the conjugate of this quadratic function is:

$f^*(y) = \frac{1}{2} \|y\|^2$

Summary:

The conjugate of a convex function transforms the original problem into a new one, often revealing useful properties about the dual problem. It is a powerful tool in optimization theory and is particularly useful for solving constrained optimization problems.

Would you like a detailed example of the conjugate in action, or have any specific questions related to this topic?

Related Questions:

1. What are the properties of the Legendre transform for non-convex functions?
2. How does the conjugate relate to dual optimization problems?
3. Can the conjugate of a non-differentiable convex function be computed?
4. How does the concept of the conjugate extend to functions on vector spaces?
5. What is the relationship between the primal and dual functions in convex optimization?

Tip:

When computing the conjugate of a function, it can often be helpful to use differentiation techniques (if the function is differentiable) or other maximization strategies (like completing the square) to simplify the supremum calculation.