Polynomial complementarity problems
By M. Seethharama Gowda. 2016
63 citations recorded by the source corpus.
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.
Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/41e4c92887187de72de5c5d46d9c9a40075d17ec
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In Computational Mathematics
In Computational Theory and Mathematics
In Numerical Analysis
Citations in the corpus
- Z-tensors and complementarity problems (2015, background): “In [10], polynomial complementarity problems are studied, specifically extending Theorem 3.1 to polynomial functions.”