In this paper, we consider a family of neural networks for solving nonlinear complementarity problems (NCP). The neural networks are constructed from the merit functions based on three classes of NCP-functions: the generalized natural residual function and its two symmetrizations. In this paper, we first characterize the stationary points of the induced merit functions. Growth behavior of the complementarity functions is also described, as this will play an important role in describing the level sets of the merit functions. In addition, the stability of the steepest descent-based neural network model for NCP is analyzed. We provide numerical simulations to illustrate the theoretical results, and also compare the proposed neural networks with existing neural networks based on other well-known NCP-functions. Numerical results indicate that the performance of the neural network is better when the parameter <i>p</i> associated

No abstract uploaded!

No abstract uploaded!

Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function . We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., <i>f</i>(<i>t</i>) = <i>t</i> ² <i>t</i> ³ 1/<i>t</i> <i>t</i> ^1/2, |<i>t</i>|, and [<i>t</i>]₊. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.

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.