It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of FischerBurmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions

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Recently Tseng (Math Program 83:159185, 1998) extended a class of merit functions, proposed by Luo and Tseng (<i>A new class of merit functions for the nonlinear complementarity problem</i>, in Complementarity and Variational Problems: State of the Art, pp. 204225, 1997), for the nonlinear complementarity problem (NCP) to the semidefinite complementarity problem (SDCP) and showed several related properties. In this paper, we extend this class of merit functions to the second-order cone complementarity problem (SOCCP) and show analogous properties as in NCP and SDCP cases. In addition, we study another class of merit functions which are based on a slight modification of the aforementioned class of merit functions. Both classes of merit functions provide an error bound for the SOCCP and have bounded level sets.

Let each point of a homogeneous Poisson process in ℝ^{$d$}independently be equipped with a random number of stubs (half-edges) according to a given probability distribution$μ$on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution$μ$. Leaving aside degenerate cases, we prove that for any$μ$there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on$μ$for the absence and presence of infinite components.

In a uniform random recursive$k$-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to$n$, chooses$k$uniform random parents from among the nodes of smaller index. If$S$_{$n$}is the shortest path distance from node$n$to the root, then we determine the constant$σ$such that$S$_{$n$}/log$n$→$σ$in probability as$n$→∞. We also show that max_{1≤$i$≤$n$}$S$_{$i$}/log$n$→$σ$in probability.