# 方差最小化的凸性

$$X$$是一個離散隨機變量，其值為$$x_n$$，概率為$$1 / N$$表示$$n = 1，\ ldots，N$$。我想在優化問題中設置$$x_n$$值。我的目標是在滿足一組約束的同時最大程度地減少方差。

$$">開始{array} {ll}＆\ min \ limits _ {\ {x_n \} _ {n = 1} ^ N} {\ operatorname {Var}（X）} \\＆\ text {st} \ \ ldots \ end {array}$$

My questions are:

1. The variance is not convex, so minimization is hard, right? Is there a common method for this?
2. Is there any convex function which results in a low variance after being minimized?

It holds $$\begin{array}{rcl} \operatorname V(x) &= &\dfrac1N\left\| x-\dfrac{e^\top x}{N} e \right\|^2 \\ & = & \dfrac1N\left(x^\top x+\dfrac{(e^\top x)^2 e^\top e}{N^2}-2\dfrac{(e^\top x)^2}N\right) \\ & = & \dfrac{x^\top x}{N} - \dfrac{(e^\top x)^2}{N^2}. \end{array}$$ So you are minimizing the $$\ell^2$$-norm of an affine expression which is known to be convex.

The problem $$\begin{array}{lcl} \min & \dfrac{\|x-e u\|}{N} & \\ \mbox{s.t.} & \dfrac{e^\top x}{N} - u & = & 0 \\ \end{array}$$ provides a nice interpretation since $$u$$ is the average. Note the problem tries to make all the $$x$$ equal to the average value.

Alternatively the last problem can be stated as $$\begin{array}{lcl} \min & \dfrac{s}{N}&\\ \mbox{s.t.} & \dfrac{e^\top x}{N} - u &=0 \\ &(s;x-e u) &\in Q. \\ \end{array}$$ where $$Q$$ is a quadratic cone. This provides another convexity proof because the quadratic cone is convex. Hence, the problem can be solved using SOCP also known as conic quadratic optimization.