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

Consensus optimization has received considerable attention in recent years. A number of decentralized algorithms have been proposed for convex consensus optimization. However, to the behaviors or consensus nonconvex optimization, our understanding is more limited. When we lose convexity, we cannot hope that our algorithms always return global solutions though they sometimes still do. Somewhat surprisingly, the decentralized consensus algorithms, DGD and Prox-DGD, retain most other properties that are known in the convex setting. In particular, when diminishing (or constant) step sizes are used, we can prove convergence to a (or a neighborhood of) consensus stationary solution under some regular assumptions. It is worth noting that Prox-DGD can handle nonconvex nonsmooth functions if their proximal operators can be computed. Such functions include SCAD, MCP, and lq quasinorms, q ∈ [0, 1). Similarly, Prox-DGD can take the constraint to a nonconvex set with an easy projection. To establish these properties, we have to introduce a completely different line of analysis, as well as modify existing proofs that were used in the convex setting.

The trust-region problem,which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact semi-definite linear programming relaxation (SDP-relaxation) and strong duality. Unfortunately, such properties do not, in general, hold for an extended trust-region problem having extra linear constraints. This paper shows that two useful and powerful features of the classical trust-region problem continue to hold for an extended trust-region problem with linear inequality constraints under a new dimension condition. First, we establish that the class of extended trust-region problems has an exact SDP-relaxation, which holds without the Slater constraint qualification. This is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range-convexity whenever the dimension condition is fulfilled. Second, we show that the dimension condition together with the Slater condition ensures that a set of combined first and second-order Lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust-region problem and consequently for strong duality. Through simple examples we also provide an insightful account of our development from SDP-relaxation to strong duality. Finally, we show that the dimension condition is easily satisfied for the extended trust-region model that arises from the reformulation of a robust least squares problem (LSP) as well as a robust second order cone programming model problem (SOCP) as an equivalent semi-definite linear programming problem. This leads us to conclude that, under mild assumptions, solving a robust LSP or SOCP under matrix-norm uncertainty or polyhedral uncertainty is equivalent to solving a semi-definite linear programming problem and so, their solutions can be validated in polynomial time.

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The free boundary value problem in obstacle problem for von Krmn equations is studied. By using the method of complementarity analysis, Rockafellar's theory of duality is generalized to the nonlinear variational problems and a complementarity theory of obstacle problem for von Krmn plates is established. We prove that the uniqueness and existence of solution directly depend on a complementary gap function. Moreover, a generalized dual extreme principle is established. We prove that the nonlinear primal variational inequality problem is eventually equivalent to a semi-quadratic dual optimization problem defined on a statically admissible space. This equivalence can be used to develop an effective numerical method for solving nonlinear free boundary value problems.