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This work is motivated by a conjecture of Che et al. (J Optim Theory Appl 168:475–487, 2016) which says that if the feasible region of a tensor complementarity problem is nonempty, then the corresponding optimization problem has a solution. The aim of the paper is twofold. First, we show several sufficient conditions for the solution existence of the optimization problems corresponding to polynomial complementarity problems. Consequently, some results for tensor complementarity problems are obtained. Second, we disprove the conjecture by giving a counterexample. [ABSTRACT FROM AUTHOR]

In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained. [ABSTRACT FROM AUTHOR]

Our purpose is to investigate several properties of the solution map of tensor complementarity problems. To do this, we focus on the R0-tensors and show some results on the local boundedness and the upper semicontinuity. Furthermore, by using a technique from semi-algebraic geometry, we obtain results on the finite-valuedness, the lower semicontinuity, and the local upper-Hölder stability of the map. [ABSTRACT FROM AUTHOR]