In this paper, we focus on fractional programming problems that minimize the ratio of two indefinite quadratic functions subject to two quadratic constraints. Utilizing the relationship between fractional programming and parametric programming, we transform the original problem into a univariate nonlinear equation. To evaluate the function in the equation, we need to solve a problem of minimizi...

We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key...

In this paper we present a new method to find simple or multiple roots of functions in a finite interval. In this method using bisection method we can find an interval such that this function is one to one on it, thus we can transform problem of finding roots in this interval into an ordinary differential equation with boundary conditions. By solving this equation using collocation method we ca...

Recently, in [Found. Comput. Math., 7(2) (2007), 245–269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466–488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles N in the output partition of such a method is ...

In this paper, we present a randomized algorithm to compute the bisection width of cubic and 4-regular graphs. The analysis of the proposed algorithms on random graphs provides asymptotic upper bounds for the bisection width of random cubic and random 4-regular graphs with n vertices, giving upper bounds of 0:174039n for random cubic, and of 0:333333n for random 4-regular. We also obtain asympt...

In this paper, we study the problem of minimizing the ratio of two quadratic functions subject to a quadratic constraint. First we introduce a parametric equivalent of the problem. Then a bisection and a generalized Newton-based method algorithms are presented to solve it. In order to solve the quadratically constrained quadratic minimization problem within both algorithms, a semidefinite optim...

In this paper, we develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes. Adaptive finite element methods (AFEMs) based on local mesh refinement are an important and efficient approach when the solution is non-smooth. An optimality theory on AFEM for linear elliptic equations can be found in Nochet...

Recently, the globally uniquely solvable (GUS) property of the linear transformation M ∈ Rn×n in the second-order cone linear complementarity problem (SOCLCP) receives much attention and has been studied substantially. Yang and Yuan [30] contributed a new characterization of the GUS property of the linear transformation, which is formulated by basic linear-algebra-related properties. In this pa...