Polynomial complementarity problems
Polynomial complementarity problems
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发表时间：
2016-09
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通讯作者：
M. Gowda
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作者：
M. Gowda
给定欧几里得n-空间上的多项式映射f和向量q，多项式互补问题PCP（f，q）是寻找非负向量x使得y=f（x）+q非负且正交于x的非线性互补问题。如果多项式映射是齐次的，则称之为张量互补问题。本文建立了多项式互补问题PCP（f，q）与张量互补问题PCP（f*，0）之间的联系，其中f* 是f分解为齐次多项式映射之和的首项.例如，我们证明了，当零是PCP（f*，0）的唯一解且min{x，f*（x）}在原点的局部（拓扑）度非零时，PCP（f，q）对每个q都有非空紧解集。作为结果，我们建立Karamanshan型结果多项式互补问题。通过确定一个m阶n维张量A及其相应的齐次多项式F（x）：= Ax^{m-1}，我们将我们的结果与张量互补问题联系起来。这些结果表明，在适当的条件下，PCP（F+P，q）对所有次小于m-1的多项式映射P和所有向量q都有非空紧解集，从而大大改进了只考虑PCP（F，q）型问题的张量互补结果.本文引入了R_0张量的次数的概念，证明了R张量的次数为1。我们通过构造基于矩阵的张量来说明我们的结果。
Given a polynomial map f on the Euclidean n-space and a vector q, the polynomial complementarity problem, PCP(f,q), is the nonlinear complementarity problem of finding a nonnegative vector x such that y=f(x)+q is nonnegative and orthogonal to x. It is called a tensor complementarity problem if the polynomial map is homogeneous. In this paper, we establish results connecting the polynomial complementarity problem PCP(f,q) and the tensor complementarity problem PCP(f*,0), where f* is the leading term in the decomposition of f as a sum of homogeneous polynomial maps. We show, for example, that PCP(f,q) has a nonempty compact solution set for every q when zero is the only solution of PCP(f*,0)and the local (topological) degree of min{x,f*(x)} at the origin is nonzero. As a consequence, we establish Karamardian type results for polynomial complementarity problems. By identifying a tensor A of order m and dimension n with its corresponding homogeneous polynomial F(x):= Ax^{m-1}, we relate our results to tensor complementarity problems. These results show that under appropriate conditions, PCP(F+P,q) has a nonempty compact solution set for all polynomial maps P of degree less than m-1 and for all vectors q, thereby substantially improving the existing tensor complementarity results where only problems of the type PCP(F,q) are considered. We introduce the concept of degree of an R_0-tensor and show that the degree of an R-tensor is one. We illustrate our results by constructing matrix based tensors.