Nonemptiness and Compactness of Solution Sets to Generalized Polynomial Complementarity Problems

Zheng, Meng-Meng; Huang, Zheng-Hai; Ma, Xiao-Xiao

doi:10.1007/s10957-020-01645-6

Cited by 11 publications

(7 citation statements)

References 32 publications

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“…To the best of our knowledge, the Karamardian-type theorem achieved by [10] can cover a lot of existing results obtained recently in TCPs, PCPs, TVIs and PVIs. However, we note that there are some papers which study the properties of solution sets of VIs and CPs by using the exceptional regularity of the involved mappings (see [26,44,45]). It is not clear whether or not the exceptional regularity of the involved mappings can lead to new results on the nonemptiness and compactness of solution sets and/or the uniqueness of solutions of WHVIs (or even WHGVIs)?…”

Section: Uniqueness Derived Under the Exceptional Regularity Conditionmentioning

confidence: 99%

Section: Reducing To Cpsmentioning

confidence: 99%

“…More recently, in [45], the authors also considered the GPCP and discussed the uniqueness of solutions to such a class of problems. One of the main result about the uniqueness given by [45,Theorem 4.8] needs to find a vector d P intpCq Ş intpC ˚q. The following example shows that all the conditions of Corollary 7.13 are true, however, we may not find such a vector d.…”

Section: Reducing To Cpsmentioning

confidence: 99%

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Unique solvability of weakly homogeneous generalized variational inequalities

Bai

Huang

Zheng

2020

Preprint

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An interesting observation is that most pairs of weakly homogeneous mappings have no strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper focuses on studying the unique solvability of the generalized variational inequality with a pair of weakly homogeneous mappings. By using a weaker condition than the strong monotonicity and some additional conditions, we achieve several results on the unique solvability of the underlying problem. These results are exported by making use of the exceptional family of elements or derived from new obtained Karamardiantype theorems or established under the exceptional regularity condition. They are new even when the problem comes down to its important subclasses studied in recent years.

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Section: Uniqueness Derived Under the Exceptional Regularity Conditionmentioning

confidence: 99%

Section: Reducing To Cpsmentioning

confidence: 99%

Section: Reducing To Cpsmentioning

confidence: 99%

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Unique solvability of weakly homogeneous generalized variational inequalities

Bai

Huang

Zheng

2020

Preprint

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“…Recently, Ling et al [27] and Zheng et al [45] considered a more general model, namely a generalized polynomial complementarity problem, denoted by G PC P(K , F, G), which is to find a vector x ∈ R n such that…”

Section: Introductionmentioning

confidence: 99%

“…We want to know whether we can remove the restriction for G PC P(K , F, G) or its special case G PC P(B, C, Λ, q). Zheng et al [45,Theorem 3.2] proved that if (A 1 , B 1 ) is an R K 0 -tensor pair and one of common degrees ind((A 1 , B 1 ) nat K (x), 0) and ind((B 1 , A 1 ) nat K * (x), 0) is nonzero, then the solution set of G PC P(K , F, G) is nonempty and compact. However, it is not easy to calculate the common degree in practice until we know the solutions of the problem.…”

Section: Introductionmentioning

confidence: 99%

Generalized Polynomial Complementarity Problems over a Polyhedral Cone

Shang

Yang

Tang

2021

J Optim Theory Appl

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The goal of this paper is to investigate a new model, called generalized polynomial complementarity problems over a polyhedral cone and denoted by GPCPs, which is a natural extension of the polynomial complementarity problems and generalized tensor complementarity problems. Firstly, the properties of the set of all R K 0 -tensors are investigated. Then, the nonemptiness and compactness of the solution set of GPCPs are proved, when the involved tensor in the leading term of the polynomial is an E R K -tensor. Subsequently, under fairly mild assumptions, lower bounds of solution set via an equivalent form are obtained. Finally, a local error bound of the considered problem is derived. The results presented in this paper generalize and improve the corresponding those in the recent literature. Keywords Generalized polynomial complementarity problemCommunicated by Alexey F. Izmailov.

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