An approximation method for the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable

This paper mainly addresses the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable. Through an ingenious approximation mechanism, one transforms the maximization problem into a sequence of minimization problems, which can be converted into a sequence of nonlinear differential equations with constraints by variational approach. The existence and uniqueness of the solution for each equation can be demonstrated by applying the canonical duality method. Moreover, the dual transformation gives a sequence of perfect dual maximization problems. In the final analysis, one constructs the approximation of the probability density function accordingly.