A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising

On the cone eigenvalue complementarity problem for higher-order tensors

In this paper, we consider the tensor generalized eigenvalue complementarity problem (TGEiCP), which is an interesting generalization of matrix eigenvalue complementarity problem (EiCP). First, we given an affirmative result showing that TGEiCP is solvable and has at least one solution under some reasonable assumptions. Then, we introduce two optimization reformulations of TGEiCP, thereby beneficially establishing an upper bound of cone eigenvalues of tensors. Moreover, some new results concerning the bounds of number of eigenvalues of TGEiCP further enrich the theory of TGEiCP. Last but not least, an implementable projection algorithm for solving TGEiCP is also developed for the problem under consideration. As an illustration of our theoretical results, preliminary computational results are reported.Comment: 26 pages, 2 figures, 3 table

Complementarity, impatience, and the resilience of natural-resource-dependent economies

We study how human preferences aect the resilience of economies that depend on more than one type of natural resources. In particular, we analyze whether the degree of substitutability of natural resources in consumer needs may give rise to multiple steady states and path dependence even when resources are managed optimally. This is a major shift in the interpretation and analysis of resilience, from viewing (limited) resilience as an objective property of the economy-environment system to acknowledging its partially subjective, preference-based character. We nd that society tends to be less willing to buer exogenous shocks if resource goods are complements in consumption than if they are substitutes. Hence, the stronger the complementarity between the various types of resource goods, the less resilient the economy is.Resilience; substitutes and complements; discounting; multiple steady states; natural resources; path dependence; regime shifts; tipping points