论文标题
解决一类分数半无限多项式编程问题
On Solving a Class of Fractional Semi-infinite Polynomial Programming Problems
论文作者
论文摘要
在本文中，我们研究了一类分数半官方多项式编程（FSIPP）问题，其中目的是凸多项式和凹形多项式的一部分，并且约束由无限的许多凸多项式不平等等式组成。为了解决这样的问题，我们首先将其重新将其重新定义为一对原始和双圆锥优化问题，如果我们可以将平方的结构带入圆锥约束，则将其减少到半标准编程（SDP）问题。为此，我们为凸的半无限编程问题提供了特征性的锥体约束资格，以确保二元性二元性以及在双重问题中实现解决方案，这是其自身利益的。在此框架中，我们首先提出了SDP松弛的层次结构，该层次是FSIPP问题的渐近收敛，其指数集由有限的多项式不平等定义。接下来，我们研究了四例FSIPP问题，可以将它们简化为单个SDP问题或有限的SDP问题序列，其中至少可以提取一个最小化器。然后，我们将此方法应用于四种相应的多目标案例以找到有效的解决方案。
In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.
