论文标题
在单纯子上获得或失去凸多元函数的视角
Gaining or Losing Perspective for Convex Multivariate Functions on a Simplex
论文作者
论文摘要
Minlo（混合企业非线性优化）通过二进制指示器变量的原点与多层之间的分离的公式在非线性组合优化中具有广泛的适用性，用于对固定成本$ c $建模与执行一组$ d $活动相关的固定成本$ c $，以及与活动级别相关的convex变量$ f $。观点放松通常用于在分支结合的上下文中解决此类模型，尤其是在$ f $是单变量的上下文中（例如，在Markowitz-Style Portfolio优化中）。但是这种放松通常需要圆锥求解器，并且通常与通用NLP软件不兼容，该软件可以适应其他类别的约束。这激发了研究较弱的放松的研究，以调查何时更简单的放松可能足够。将放松的体积（即Lebesgue度量）作为比较它们进行比较，我们将与单变量函数$ F $相关的一些结果与多变量案例提升。在此过程中，我们调查，连接和扩展了与单纯形的集成结果相关结果，其中一些我们是具体使用的，而另一些可以用于对我们的主要主题进行进一步探索。
MINLO (mixed-integer nonlinear optimization) formulations of the disjunction between the origin and a polytope via a binary indicator variable have broad applicability in nonlinear combinatorial optimization, for modeling a fixed cost $c$ associated with carrying out a set of $d$ activities and a convex variable cost function $f$ associated with the levels of the activities. The perspective relaxation is often used to solve such models to optimality in a branch-and-bound context, especially in the context in which $f$ is univariate (e.g., in Markowitz-style portfolio optimization). But such a relaxation typically requires conic solvers and are typically not compatible with general-purpose NLP software which can accommodate additional classes of constraints. This motivates the study of weaker relaxations to investigate when simpler relaxations may be adequate. Comparing the volume (i.e., Lebesgue measure) of the relaxations as means of comparing them, we lift some of the results related to univariate functions $f$ to the multivariate case. Along the way, we survey, connect and extend relevant results on integration over a simplex, some of which we concretely employ, and others of which can be used for further exploration on our main subject.
