This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:11731197, 2010), which confirm the theoretical results

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

In this paper, we consider the problem of approximating the longest path in a grid graph that possesses a square-free 2-factor. By (1) analyzing several characteristics of four types of cross cells and (2) presenting three cycle-merging operations and one extension operation, we propose an algorithm that finds in an n-vertex grid graph G, a long path of length at least 5/6 n + 2. Our approximation algorithm runs in quadratic time. In addition to its theoretical value, our work has also an interesting application in image-guided maze generation.

No abstract uploaded!

We present a method of finding weighted Koppelman formulas for ($p$,$q$)-forms on$n$-dimensional complex manifolds$X$which admit a vector bundle of rank$n$over$X$×$X$, such that the diagonal of$X$×$X$has a defining section. We apply the method to ℙ^{$n$}and find weighted Koppelman formulas for ($p$,$q$)-forms with values in a line bundle over ℙ^{$n$}. As an application, we look at the cohomology groups of ($p$,$q$)-forms over ℙ^{$n$}with values in various line bundles, and find explicit solutions to the $\overline{\partial}$ -equation in some of the trivial groups. We also look at cohomology groups of (0,$q$)-forms over ℙ^{$n$}×ℙ^{$m$}with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.