求非线性方程组和优化问题全部解的胞腔方法

求非线性方程组和优化问题全部解的胞腔方法张讲社，游兆永，徐宗本（西安交通大学数学系）ＡＣＥＬＬＭＥＴＨＯＤＦＯＲＦＩＮＤＩＮＧＡＬＬＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＥＱＵＡＴＩＯＮＳＡＮＤＯＰＴＩＭＩＺＡＴＩＯＮＰＲＯＢＬＥＭＳ￥ＺｈａｎｇＪ...     

求整体优化问题全部解的胞腔排除法

其中X~0为一各边平行于坐标轴的立方体(以下将这类立方体简称为胞腔),X~0的最大边长度称为网径,记为W(X~0).在本文中,X(?)X~0表示任一胞腔,f(X)={f(x):x∈X}表示f在X上的值域,f在X~0上的整体极小值记为f~*,f在X~0上全部整体极小点集合记为M.以下恒假定M仅由有限个孤立点组成,且M中所有点含于X~O内部.于是,(?)x~*∈M,有     

解非线性方程组的一类算法

§1.引言 求解线性方程组 a_i~Tx=b_i,i=1,2,…,n,(1.1)其中a_1,a_2,…,a_n线性无关. 设y~((1))为初值,U~((1))为任意非奇异n阶矩阵,我们用如下方法求解方程组(1.1). 先考虑前k-1个方程组成的亚定方程组 a_i~Tx=b_i,i=1,2,…,k-1.设{U~((k))}={a_1,a_2,…,a_(k-1)},这里{U~((k))}表示由U~((k))的列组成的子空间.显然,rank(U~((k)))=n-b+1.若y~((k))是相应的亚定方程的一个特解,则将其看作方程组     

一类非线性椭圆方程组很弱解的正则性

本文在可控增长条件（１．２）－（１．４）下,对一类非线性椭圆方程组（１．１）改进其很弱解偏微商的可积性,使其为经典意义下的弱解。     

一类非线性抛物型方程组解的整体存在及爆破

研究了一类带有非线性边界条件的非线性抛物型方程组解的整体存在及解在有限时刻爆破问题.通过构造方程组的上、下解.得到了解整体存在及解在有限时刻爆破的充分条件.对指数型反应项和边界流采用了常微分方程方法构造其上下解,而其它例如第一特征值等方法运用于该方程就比较困难.     

Nonlinear optimization exclusion tests for finding all solutions of nonlinear equations

Exclusion tests are a well known tool in the area of interval analysis for finding the zeros of a function over a compact domain. Recently, K. Georg developed linear programming exclusion tests based on Taylor expansions. In this paper, we modify his approach by choosing another objective function and using nonlinear constraints to make the new algorithm converges faster than the algorithm in [K. Georg, A new exclusion test, J. Comput. Appl. Math. 152 (2003) 147–160]. In this way, we reduce the number of subinterval in each level. The computational complexity for the new tests are investigated. Also, numerical results and comparisons will be presented.     

An efficient algorithm for finding all solutions of separable systems of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of nonlinear equations with separable mappings. This algorithm is based on interval analysis, the dual simplex method, the contraction method, and a special technique which makes the algorithm not require large memory space and not require copying tableaus. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 2000 nonlinear equations in acceptable computation time. AMS subject classification (2000)  65H10, 65G10     

Finding all solutions of nonlinearly constrained systems of equations

A new approach is proposed for finding all-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in BB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.     

LP narrowing: A new strategy for finding all solutions of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of n nonlinear equations. This algorithm is based on interval analysis and a new strategy called LP narrowing. In the LP narrowing strategy, boxes (n-dimensional rectangles in the solution domain) containing no solution are excluded, and boxes containing solutions are narrowed so that no solution is lost by using linear programming techniques. Since the LP narrowing is very powerful, all solutions can be found very efficiently. By numerical examples, it is shown that the proposed algorithm could find all solutions of systems of 5000-50,000 nonlinear equations in practical computation time.     

A cell exclusion algorithm for determining all the solutions of a nonlinear system of equations

A new group of methods named cell exclusion algorithms (CEAs) is developed for finding all the solutions of a nonlinear system of equations. These types of algorithms, different in principle from those of homotopy, interval and cell-mapping-dynamical-analysis approaches, are based on cellular discretization and the use of a certain simple necessity test of the solutions. The main advantages of the algorithms are their simplicity, reliability, and general applicability. Having all features of interval techniques (but without using interval arithmetic) and with complexity O(log(1/)), the algorithms improve significantly on both the interval algorithms and the cell mapping techniques. Theoretical analysis and numerical simulations both demonstrate that CEAs are very efficient.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

带有热源项的非线性扩散方程的精确解

讨论了带有热源项的非线性扩散方程.通过一种直接简洁的方法得到了几种精确解.该方法可用于更高阶演化方程的求解问题.     

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.