The problems studied in this paper are a class of monotone constrained variational inequalities VI (S, f) in which S is a convex set with some linear constraints. By introducing Lagrangian multipliers to the linear constraints, such problems can be solved by some projection type prediction-correction methods. We focus on the mapping f that does not have an explicit form. Therefore, only its function values can be employed in the numerical methods. The number of iterations is significantly dependent on a parameter that balances the primal and dual variables. To overcome potential difficulties, we present a self-adaptive prediction-correction method that adjusts the scalar parameter automatically. Convergence of the proposed method is proved under mild conditions. Preliminary numerical experiments including some traffic equilibrium problems indicate the effectiveness of the proposed methods.

The well-known logarithmic-quadratic proximal (LQP) method has motivated a number of efficient numerical algorithms for solving nonlinear complementarity problems (NCPs). In this paper, we aim at improving one of them, i.e., the LQP-based interior prediction-correction method proposed in [He, Liao and Yuan, J. Comp. Math., 2006, 24(1): 33―44], via identifying more appropriate step-sizes in the correction steps. Preliminary numerical results for solving some NCPs arising in traffic equilibrium problems are reported to verify the theoretical assertions.

For the split feasibility problem, we propose a new type of solution method by introducing a new searching direction with fixed stepsize. Its global convergence is proved under a suitable condition. Preliminary numerical experiments show the efficiency of the proposed method.

We study core stability and some related properties of flow games defined on simple networks (all edge capacities are equal) from an algorithmic point of view. We first present a sufficient and necessary condition that can be tested efficiently for a simple flow game to have a stable core. We also prove the equivalence of the properties of core largeness, extendability, and exactness of simple flow games and provide an equivalent graph theoretic characterization which allows us to decide these properties in polynomial time.

Parametric search is a useful tool in geometric optimization. Invented by Nimrod Megiddo in 1983, it has been widely used in computational geometry. Unfortunately, this technique has rarely been used in the combinatorial optimization community in China. In this paper, we introduce parametric search via three new geometric optimization applications.

A wireless sensor network usually consists of a large number of sensor nodes deployed in a field. One of the major communication operations is to broadcast a message from one node to the rest of the others. In this paper, we adopt the conflict-free communication model and study how to compute a transmission schedule that determines when and where a node should forward the message so that all nodes could receive the message in minimum time. We give two approximation algorithms for this NP-hard problem that have better theoretically guaranteed performances than the existing algorithms. The proposed approach could be applied to some other similar problems.

This paper is concerned with the optimal control of jump type stochastic differential equations associated with (general) Lévy generators. The maximum principle is formulated for the solutions of the equations, which is inspired by N. C. Framstad, B. Øsendal and A. Sulem [J. Optim. Theory Appl., 2004, 121: 77―98] (and a continuation, J. Bennett and J. -L. Wu [Front. Math. China, 2007, 2(4): 539―558]). The result is then applied to optimization problems in financial models driven by Lévy-type processes.

M. Bordemann has studied non-associative algebras with nondegenerate associative bilinear forms. In this paper, we focus on pseudo-Riemannian bilinear forms and study pseudo-Riemannian Leibniz algebras, i.e., Leibniz algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We give the notion and some properties of T^∗-extensions of Leibniz algebras. In addition, we introduce the definition of equivalence and isometrical equivalence for two T^∗-extensions of a Leibniz algebra, and give a sufficient and necessary condition for the equivalence and isometrical equivalence.

For a positive integer n, a finite p-group G is called an M_n-group, if all subgroups of index pⁿ of G are metacyclic, but there is at least one subgroup of index p^n−1 that is not. A classical result in p-group theory is the classification of M₁-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result.

This paper focuses on developing fast numerical algorithms for selection of a kernel optimal for a given training data set. The optimal kernel is obtained by minimizing a cost functional over a prescribed set of kernels. The cost functional is defined in terms of a positive semi-definite matrix determined completely by a given kernel and the given sampled input data. Fast computational algorithms are developed by approximating the positive semi-definite matrix by a related circulant matrix so that the fast Fourier transform can apply to achieve a linear or quasi-linear computational complexity for finding the optimal kernel. We establish convergence of the approximation method. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.

In this paper, we consider the stochastic Korteweg-de Vries–Benjamin–Ono equation with white noise. Using Fourier restriction norm method and some basic inequalities, we obtain a local existence and uniqueness result for the solution of this problem. We also get global existence of the L²(R) solution.

Let G be a finite group, and let π_e(G) be the spectrum of G, that is, the set of all element orders of G. In 1987, Shi Wujie put forward the following conjecture. If G is a finite group and M is a non-abelian simple group, then G≌M if and only if G=M and π_e(G)=π_e(M). In this short paper, we prove that if G is a finite group, then G≌M if and only if G=M and π_e(G)=π_e(M), where M=D_n(2) and n is even.