Convergence Analysis of an Infeasible Interior Point Algorithm Based on a Regularized Central Path for Linear Complementarity Problems

Most existing interior-point methods for a linear complementarity problem (LCP) require the existence of a strictly feasible point to guarantee that the iterates are bounded. Based on a regularized central path, we present an infeasible interior-point algorithm for LCPs without requiring the strict feasibility condition. The iterates generated by the algorithm are bounded when the problem is a P _* LCP and has a solution. Moreover, when the problem is a monotone LCP and has a solution, we prove that the convergence rate is globally linear and it achieves `-feasibility and `-complementarity in at most O(n ² ln(1/`)) iterations with a properly chosen starting point.     

Neural Network Models for Finline Discontinuities

The radial basis network is used as the finline discontinuities electromagnetic artifical neural network(EMANN) models. EM software analysis is employed to characterize finline discontinuities. EMANN models are then trained using physical parameters and frequency as inputs and equivalent electric circuit element parameters of finline discontinuities as outputs. Once trained , the EMANN models can simulate equivalent electric circuit element parameters of finline step, notch and strip very fast and efficiently.     

On Logarithmic Smoothing of the Maximum Function

We consider the maximum function f resulting from a finite number of smooth functions. The logarithmic barrier function of the epigraph of f gives rise to a smooth approximation g of f itself, where >0 denotes the approximation parameter. The one-parametric family g converges – relative to a compact subset – uniformly to the function f as tends to zero. Under nondegeneracy assumptions we show that the stationary points of g and f correspond to each other, and that their respective Morse indices coincide. The latter correspondence is obtained by establishing smooth curves x() of stationary points for g , where each x() converges to the corresponding stationary point of f as tends to zero. In case of a strongly unique local minimizer, we show that the nondegeneracy assumption may be relaxed in order to obtain a smooth curve x().     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC<Superscript>1</Superscript> minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.     

求解非线性规划的原始对偶内点法(英文)

In this paper, we propose a primal-dual interior point method for solving general constrained nonlinear programming problems. To avoid the situation that the algorithm we use may converge to a saddle point or a local maximum, we utilize a merit function to guide the iterates toward a local minimum. Especially, we add the parameter ε to the Newton system when calculating the decrease directions. The global convergence is achieved by the decrease of a merit function. Furthermore, the numerical results confirm that the algorithm can solve this kind of problems in an efficient way.     

具有电子旋转方向的旋转间断的稳定性

用一维混合粒子编码研究了具有电子旋转方向的对称和非对称旋转间断的稳定性。发现在相对窄的过渡层厚度下,具有电子旋转方向的旋转间断是不稳定的,它趋向于发展成离子旋转方向的旋转间断和一些MHD波。这个发展过程部分地受到电子温度的限制。当过渡层厚度增加到足够宽时,电子旋转方向的旋转间断变成稳定的。本文简单地讨论了导致这种旋转间断不稳定的可能原因。     

The affine Plateau problem

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions.

     

A compact high‐order HLL solver for nonlinear hyperbolic systems

We present a new finite‐volume method for calculating complex flows on non‐uniform meshes. This method is designed to be highly compact and to accurately capture all discontinuities that may arise within the solution of a nonlinear hyperbolic system. In the first step, we devise a fourth‐degree Hermite polynomial to interpolate the solution. The coefficients defining this polynomial are calculated by using a least‐square method. To introduce monotonicity conditions within the procedure, two constraints are added into the least‐square system. Those constraints are derived by locally matching the high‐order Hermite polynomial with a low‐order TVD polynomial. To emulate these constraints only in regions of discontinuities, data‐depending weights are defined; these weights are based upon normalized indicators of smoothness of the solution and are parameterized by an O(1) quantity. The reconstruction so generated is highly compact and is fifth‐order accurate when the solution is smooth; this reconstruction becomes first order in regions of discontinuities. In the second step, this reconstruction is inserted in an HLL approximate Riemann solver. This solver is designed to correctly capture all discontinuities that may arise into the solution. To this aim, we introduce the contribution of a possible contact discontinuity into the HLL Riemann solver. Thus, a spatially fifth‐order non‐oscillatory method is generated. This method evolves in time the solution and its first derivative. In a one‐dimensional context, a linear spectral analysis and extensive numerical experiments make it possible to assess the robustness and the advantages of the method in computing multi‐scale problems with embedded discontinuities. Copyright © 2008 John Wiley & Sons, Ltd.     

欧拉常数γ的几种表示

调和级数∑n=1^∞1/n是发散的，而极限lim n→∞（∑k=1^∞1/k-lnn）却是收敛的，其极限值称为欧拉常数γ，本文给出了欧拉常数γ的几个有趣的级数表示和积分表示．