Let $\mathcal{P}_2(\mathbb{R}^d)$ denote the space of probability measures on $\mathbb{R}^d$ with finite second moment and $F: \mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}$ be the potential functional. We are interested in the following optimization problem

$$
\min_{m\in \mathcal{P}_2(\mathbb{R}^d)} F(m).
$$

Motivation

Let $(X,d)$ be a metric space, $F, F_n: X\to \overline{\mathbb{R}}, n=1,2,\cdots$ be functionals, suppose that $x_n\in X$ minimizes $F_n$ for each $n=1,2,\cdots$, does $\lim\limits_{n\to\infty} x_n$ (if it exists) minimize any functional $F$? And in what sense does $F_n$ converge to $F$ ensure the minimizer of $F_n$ converges to minimizer of $F$?

The KKT conditions (Karush–Kuhn–Tucker Conditions) are one of the most important results in optimization, and readers familiar with optimization will certainly not find them unfamiliar. This article will focus on introducing the motivation behind some key concepts introduced in the proof of the KKT conditions (such as the introduction of the active set and the constraint qualification), and will review the complete proof of the KKT conditions.