Numerical Optimization for the Location of Wastewater Outfalls

In this paper we solve a constrained optimal control problem related to the location of the wastewater outfalls in a sewage disposal system. This is a problem where the control is the position and the constraints are non-convex and pointwise, which makes difficult its resolution. We discretize the problem by means of a characteristics-Galerkin method and we use three algorithms for the numerical resolution of the discretized optimization problem: an interior point algorithm, the Nelder-Mead simplex method and a duality method. Finally, we compare the numerical results obtained by applying the described methods for a realistic problem posed in the ría of Vigo (Galicia, Spain).     

Multi-objective Meta-heuristics for the Traveling Salesman Problem with Profits

We introduce and test a new approach for the bi-objective routing problem known as the traveling salesman problem with profits. This problem deals with the optimization of two conflicting objectives: the minimization of the tour length and the maximization of the collected profits. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined or one of the objectives has been treated as a constraint. The purpose of our study is to find solutions to this problem using the notion of Pareto optimality, i.e. by searching for efficient solutions and constructing an efficient frontier. We have developed an ejection chain local search and combined it with a multi-objective evolutionary algorithm which is used to generate diversified starting solutions in the objective space. We apply our hybrid meta-heuristic to synthetic data sets and demonstrate its effectiveness by comparing our results with a procedure that employs one of the best single-objective approaches.      

小波伽辽金方法应用于变系数波动方程

我们考虑问题K（x）uxx=ua.0＜X〈1，t≥0，其中K（x）≥a≥0，u（0，t）=g，ix（0，t）=0．这是一个不适当的方程，因为当解存在时在边界g上一个小的扰动将对它的解造成很大的改变．我们考虑存在解u（x，·）∈L^2（R）用小波伽辽金方法和Meyer多分辨分析去滤掉高频部分，从而在尺度空间Vj上得到适定的近似解．我们也可以得到问题的准确解与它在Vj上的正交投影之间的误差估计．     

Shape identification of anisotropic diffraction gratings for TM-polarized electromagnetic waves

This paper concerns the shape identification problem of anisotropic periodic structures which are known as diffraction gratings. We study the so-called Factorization method as a tool for reconstructing the anisotropic periodic media from measured spectral data involving scattered electromagnetic waves in TM modes. This sampling method provides a simple criterion to compute a picture of shape of diffraction gratings in a rapid way. We propose a rigorous analysis for the method as well as numerical experiments to examine its performance.     

无约束优化问题的一个下降方法

本文研究了无约束优化问题.利用当前和前面迭代点的信息以及曲线搜索技巧产生新的迭代点,得到了一个新的求解无约束优化问题的下降方法.在较弱条件下证明了算法具有全局收敛性.当目标函数为一致凸函数时,证明了算法具有线性收敛速率.初步的数值试验表明算法是有效的.     

A GLOBALal DERIVTIVE-FREE DESCENT METHOD FOR NONLINEAR COMPLEMENTARITY PROBLEMS

1. IntroductionConsider the nonlinear complementarity problem (NCP for short), which is to findan x E M" such thatwhere F: Wu - ac and the inequalities are taken componentwise. This problem havemany important applications in various fields. [13, 7, 5].Due to the less storage in computation, derivative--free descent method, which meansthe search direction used does not involye the Jacobian matrix of F, is popular infinding solutions of nonlinear complementarity Problems. We briefly view som…     

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank–Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.     

An iterative method for pricing American options under jump-diffusion models

We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou?s and Merton?s jump-diffusion models show that the resulting iteration converges rapidly.     

Optimization for Inconsistent Split Feasibility Problems

The split feasibility problem deals with finding a point in a closed convex subset of the domain space of a linear operator such that the image of the point under the linear operator is in a prescribed closed convex subset of the image space. The split feasibility problem and its variants and generalizations have been widely investigated as a means for resolving practical inverse problems in various disciplines. Many iterative algorithms have been proposed for solving the problem. This article discusses a split feasibility problem which does not have a solution, referred to as an inconsistent split feasibility problem. When the closed convex set of the domain space is the absolute set and the closed convex set of the image space is the subsidiary set, it would be reasonable to formulate a compromise solution of the inconsistent split feasibility problem by using a point in the absolute set such that its image of the linear operator is closest to the subsidiary set in terms of the norm. We show that the problem of finding the compromise solution can be expressed as a convex minimization problem over the fixed point set of a nonexpansive mapping and propose an iterative algorithm, with three-term conjugate gradient directions, for solving the minimization problem.     

二次规划逆问题的牛顿方法

针对二次规划逆问题,将其表达为带有互补约束的锥约束优化问题.借助于对偶理论,将问题转化为变量更少的线性互补约束非光滑优化问题.通过扰动的方法求解转化后的问题并证明了收敛性.采用非精确牛顿法求解扰动问题,给出了算法的全局收敛性与局部二阶收敛速度.最后通过数值实验验证了该算法的可行性.     

三维Laplace方程Cauchy问题的小波解(英文)

本文研究了三维Laplace方程的柯西问题.该问题是不适定的,即其解(若存在)不连续依赖原始数据.利用Meyer小波和小波Galerkin方法,获得了在L2范数意义下的稳定小波逼近解,并且给出分辨率水平的选取方法.     

Image collection planning for KOrea Multi-Purpose SATellite-2

This paper studies an image collection planning problem for a Korean satellite, KOMPSAT-2 (KOrea Multi-Purpose SATellite-2). KOMPSAT-2 has the mission goal of maximizing image acquisition in time and quality requested by customers and operates under several complicating conditions. One of the characteristics in KOMPSAT-2 is its strip mode operation, in which segments of continuous-observation areas with known sizes are captured one at a time. In this paper, we regard the segment as a group of adjoining geographical square regions (scenes), whose size must also be determined. Thus, the problem involves the determination of proper segment lengths as well as an image collection schedule. We present a binary integer programming model for this problem in a multi-orbit long-term planning environment and provide a heuristic solution approach based on the Lagrangian relaxation and subgradient methods. We also present the results of our computational experiment based on randomly generated data.     

Strong convergence of a general viscosity explicit rule for the sum of two monotone operators in Hilbert spaces

In this paper, we study a general viscosity explicit rule for approximating the solutions of the variational inclusion problem for the sum of two monotone operators. We then prove its strong convergence under some new conditions on the parameters in the framework of Hilbert spaces. As applications, we apply our main result to the split feasibility problem and the LASSO problem. We also give some numerical examples to support our main result. The results presented in this paper extend and improve the corresponding results in the literature.     

求解带Toeplitz矩阵的线性互补问题的一类预处理模系矩阵分裂迭代法

针对系数矩阵为对称正定Toeplitz矩阵的线性互补问题,本文提出了一类预处理模系矩阵分裂迭代方法.先通过变量替换将线性互补问题转化为一类非线性方程组,然后选取Strang或T.Chan循环矩阵作为预优矩阵,利用共轭梯度法进行求解.我们分析了该方法的收敛性.数值实验表明,该方法是高效可行的.     

A compromise solution for the multiobjective stochastic linear programming under partial uncertainty

This paper solves the multiobjective stochastic linear program with partially known probability. We address the case where the probability distribution is defined by crisp inequalities. We propose a chance constrained approach and a compromise programming approach to transform the multiobjective stochastic linear program with linear partial information on probability distribution into its equivalent uniobjective problem. The resulting program is then solved using the modified L-shaped method. We illustrate our results by an example.     

A novel neural network based on NCP function for solving constrained nonconvex optimization problems

This article presents a novel neural network (NN) based on NCP function for solving nonconvex nonlinear optimization (NCNO) problem subject to nonlinear inequality constraints. We first apply the p‐power convexification of the Lagrangian function in the NCNO problem. The proposed NN is a gradient model which is constructed by an NCP function and an unconstrained minimization problem. The main feature of this NN is that its equilibrium point coincides with the optimal solution of the original problem. Under a proper assumption and utilizing a suitable Lyapunov function, it is shown that the proposed NN is Lyapunov stable and convergent to an exact optimal solution of the original problem. Finally, simulation results on two numerical examples and two practical examples are given to show the effectiveness and applicability of the proposed NN. © 2015 Wiley Periodicals, Inc. Complexity 21: 130–141, 2016     

A moving pseudo‐boundary method of fundamental solutions for void detection

We propose a new moving pseudo‐boundary method of fundamental solutions (MFS) for the determination of the boundary of a void. This problem can be modeled as an inverse boundary value problem for harmonic functions. The algorithm for imaging the interior of the medium also makes use of radial polar parametrization of the unknown void shape in two dimensions. The center of this radial polar parametrization is considered to be unknown. We also include the contraction and dilation factors to be part of the unknowns in the resulting nonlinear least‐squares problem. This approach addresses the major problem of locating the pseudo‐boundary in the MFS in a natural way, because the inverse problem in question is nonlinear anyway. The feasibility of this new method is illustrated by several numerical examples. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

New intergrid transfer operator in multigrid method for P1-nonconforming finite element method

For a second-order elliptic boundary value problem, We develop an intergrid transfer operator in multigrid method for the P₁-nonconforming finite element method. This intergrid transfer operator needs smaller computation than previous intergrid transfer operators. Multigrid method with this operator converges well.     

A Globally Derivative-Free Descent Method for Nonlinear Complementarity Problems

Based on a class of functions, which generalize the squared Fischer-Burmeister NCP function and have many desirable properties as the latter function has, we reformulate nonlinear complementarity problem (NCP for short) as an equivalent unconstrained optimization problem, for which we propose a derivative-free descent method in monotone case. We show its global convergence under some mild conditions. If $F$, the function involved in NCP, is $R_0$-function, the optimization problems has bounded level sets. A local property of the merit function is discussed. Finally,we report some numerical results.     

张量鲁棒主成分分析的增广拉格朗日交替方向方法

张量的鲁棒主成分分析是将未知的一个低秩张量与一个稀疏张量从已知的它们的和中分离出来.因为在计算机视觉与模式识别中有着广阔的应用前景,该问题在近期成为学者们的研究热点.本文提出了一种针对张量鲁棒主成分分析的新的模型,并给出交替方向极小化的求解算法,在求解过程中给出了两种秩的调整策略.针对低秩分量本文对其全部各阶展开矩阵进行低秩矩阵分解,针对稀疏分量采用软阈值收缩的策略.无论目标低秩张量为精确低秩或近似低秩,本文所提方法均可适用.本文对算法给出了一定程度上的收敛性分析,即算法迭代过程中产生的任意收敛点均满足KKT条件.如果目标低秩张量为精确低秩,当迭代终止时可对输出结果进行基于高阶奇异值分解的修正.针对人工数据和真实视频数据的数值实验表明,与同类型算法相比,本文所提方法可以得到更好的结果.