凸一元函数分段线性欠估计的损益透视
Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions
摘要：
我们研究MINLO（混合整数非线性优化）配方的析取$x\in\{0\}\cup[\ell,u]$，其中$z$是一个二元指标的$x\in[\ell,u]$（$0 \leq \ell <u$），和$y$“捕获”$f(x)$，这是假定为凸和积极的域$[\ell,u]$，但否则$y=0$时，$x=0$。该模型在非线性组合优化中非常有用，其中存在在操作范围$[\ell,u]$中的水平$x$处操作活动的固定成本，然后存在进一步的（凸）可变成本$f(x)$。特别是，我们研究了松弛相关的一个自然的分段线性下估计$f$，通过选择$f$线性化点的透视变换。使用三维体积（在$(x,y,z)$）作为一个凸松弛的紧密性的措施，我们调查作为$f$，$\ell$，$u$的函数的松弛质量，和线性化点的选择。对凸幂函数$f(x):=x^p$，$p>1$进行了详细的研究.
英文摘要：
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of $x\in[\ell,u]$ ($0 \leq \ell <u$), and $y$ "captures" $f(x)$, which is assumed to be convex and positive on its domain $[\ell,u]$, but otherwise $y=0$ when $x=0$. This model is very useful in nonlinear combinatorial optimization, where there is a fixed cost of operating an activity at level $x$ in the operating range $[\ell,u]$, and then there is a further (convex) variable cost $f(x)$. In particular, we study relaxations related to the perspective transformation of a natural piecewise-linear under-estimator of $f$, obtained by choosing linearization points for $f$. Using 3-d volume (in $(x,y,z)$) as a measure of the tightness of a convex relaxation, we investigate relaxation quality as a function of $f$, $\ell$, $u$, and the linearization points chosen. We make a detailed investigation for convex power functions $f(x):=x^p$, $p>1$.
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