论文标题
解决具有关节数值约束的新型二次优化问题
Solving a new type of quadratic optimization problem having a joint numerical range constraint
论文作者
论文摘要
我们提出了二次优化问题的新公式。目标函数$ f（f（x），g（x））$作为二次函数的组成$ f（z）$，带有两个$ n $ n $ - variate二次函数$ z_1 = f（x）$和$ z_2 = g（x）。此外，它与$ z_________________________________的$ z______________属于$（f，g）的关节数值范围。$，该公式非常笼统地，因为它涵盖了所有类型的二次限制的二次编程，包括不平等型，等效型，等效性型和间隔类型。更重要的是，“具有二次二次的二次制度”以及关节数值约束的组成使我们能够将现有的未解决（或未有效解决）问题提出到新模型中。在本文中，我们解决了由p $ \ actute {\ rm o} $ lik和terlaky提出的二次超曲面交叉问题（QSIC）；以及在YE和Zhang提出的二次约束上最小化二次函数的绝对值的问题（AQP）。我们表明，当$ f（z）$和联合数值范围约束均为凸时，可以通过求解SDP来获得凸优化问题的最佳值，然后从$ \ Mathcal {S} $ - 过程的新开发中获得SDP。另一方面，如果$ f（x）$和$ g（x）$的关节数值范围是非convex，则可以通过对$ [0,2π]进行分分方法来近似最佳解决方案，而相应的二次矩阵（$ f（x）$ g（x）$和$ g（x）$必须是线性依赖性的，则可以是线性依赖的。线性依赖性属性使我们能够通过基本分析相应地求解（QSIC）和（AQP）。
We propose a new formulation of quadratic optimization problems. The objective function $F(f(x),g(x))$ is given as composition of a quadratic function $F(z)$ with two $n$-variate quadratic functions $z_1=f(x)$ and $z_2=g(x).$ In addition, it incorporates with a set of linear inequality constraints in $z=(z_1,z_2)^T,$ while having an implicit constraint that $z$ belongs to the joint numerical range of $(f,g).$ The formulation is very general in the sense that it covers quadratic programming with a single quadratic constraint of all types, including the inequality-type, the equality-type, and the interval-type. Even more, the composition of "quadratic with quadratics" as well as the joint numerical range constraint all together allow us to formulate existing unsolved (or not solved efficiently) problems into the new model. In this paper, we solve the quadratic hypersurfaces intersection problem (QSIC) proposed by P$\acute{\rm o}$lik and Terlaky; and the problem (AQP) to minimize the absolute value of a quadratic function over a quadratic constraint proposed by Ye and Zhang. We show that, when $F(z)$ and the joint numerical range constraint are both convex, the optimal value of the convex optimization problem can be obtained by solving an SDP followed from a new development of the $\mathcal{S}$-procedure. The optimal solution can be approximated by conducting a bisection method on $[0,2π].$ On the other hand, if the joint numerical range of $f(x)$ and $g(x)$ is non-convex, the respective quadratic matrices of $f(x)$ and $g(x)$ must be linearly dependent. The linear dependence property enables us to solve (QSIC) and (AQP) accordingly by elementary analysis.
