An approximation method for the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable
An approximation method for the optimization of $p$-th moment of $\mathbb{R}^n$-valued random variable
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发表时间：
2016-07
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通讯作者：
Xiaojun Lu;Yanhua Wu
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作者：
Xiaojun Lu;Yanhua Wu
本文主要研究$\mathbb{R}^n$值随机变量的$p$阶矩的优化问题。通过巧妙的逼近机制，将最大化问题转化为一系列最小化问题，通过变分方法可以将其转化为一系列带约束的非线性微分方程。每个方程解的存在性和唯一性可以通过应用正则对偶方法来证明。此外，对偶变换给出了一系列完美的对偶最大化问题。在最后的分析中，人们相应地构造概率密度函数的近似。
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.