Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through the use of the perspective function in higher dimensional space—the so-called extended formulation of the convex hull of a disjunctive convex set. However, there are a number of challenges in using this characterization in computation which prevents its wide-spread use, including issues that arise because of the functional form of the perspective function. In this paper, we propose an explicit algebraic representation of a fairly large class of nonlinear disjunctive convex sets using the perspective function that addresses this latter computational challenge. This explicit representation can be used to generate (tighter) algebraic reformulations for a variety of different problems containing disjunctive convex sets, and we report illustrative computational results using this representation for several nonlinear disjunctive problems.

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使用透视函数的非线性析取凸集的计算有用的代数表示

非线性析取凸集自然而然地出现在许多离散连续优化问题的制定或求解方法中。通常，寻求分离凸集的紧密代数表示，其中最紧密的这种表达涉及分离凸集的凸包的表征。在最一般的情况下，这可以通过在更高维度的空间中使用透视函数来明确表示，这是析取凸集的凸包的所谓扩展公式。但是，在计算中使用这种特征会阻止其广泛使用，但存在许多挑战，其中包括由于透视函数的功能形式而引起的问题。在本文中，我们提出使用透视函数解决相当大的一类非线性析取凸集的显式代数表示形式，以解决后者的计算难题。此显式表示可用于为包含析取凸集的各种不同问题生成（更紧密的）代数公式，并且我们使用此表示形式对几个非线性析取问题进行说明性计算结果。