In this paper, the classical mean-variance portfolio model is modified for calculating a globally optimal portfolio under concave transaction costs. A non-decreasing concave function is employed to approximate origin transaction cost function. The resulting model is a D-C (difference of two convex functions) programming and a branch and bound algorithm is designed to solve the problem. A series of numerical experiments on the model is presented. The history data of nine stocks in Shan Xi province is used in experiments, and efficient frontiers generated from the resulting model with different limitations on investments are presented to show the effect of the model and the efficiency of the algorithm solving the model.

The D-C (difference of convex functions) programming occurs very often in economic and financial field. In this paper, a class of D-C programming problems be studied, witch objection function is the sum of a convex quadratic function and a separable concave function, the constrains is the intersection of a polytop and a rectangular. A branch and bound algorithm is given for solving it. we compare the largest distance bisection with x-subdivision in branch and bound algorithm. Numerical tests on different dimensions showed that the proposed branch and bound algorithm is efficient for solving medium-scale problems, and the efficiency of the largest distance bisection and x-subdivision is nearly.

Quasi-Newton equations play a central role in quasi-Newton methods for optimization and various quasi-Newton equations are available. This paper gives a survey on these quasi-Newton equations and studies properties of quasi-Newton methods with updates satisfying different quasi-Newton equations. These include single-step quasi-Newton equations that use only gradient information and that use both gradient and function value information in one step, and multi-step quasi-Newton equations that use the gradient information in last m steps. Main properties of quasi-Newton methods with updates satisfying different quasi-Newton equations are studied. These properties include the finite termination property, invariance, heredity of positive definite updates, consistency of search directions, global convergence and local superlinear convergence properties.

This paper concerns with the properties of Hadamard product of inverse M-matrices. Structures of tridiagonal inverse M-matrices and Hessenberg inverse M-matrices are analysed. It is proved that the product A AT satis es Willoughby's necessary conditions for being an inverse M-matrix when A is an irreducible inverse M-matrix. It is also proved that when A is either a Hessenberg inverse M-matrix or a tridiagonal inverse M-matrix then AAT is an inverse M-matrix. Based on these results, the conjecture that A?AT is an inverse M-matrix when A is an inverse M-matrix is made. Unfortunately, the conjecture is not true. Copyright

A graph G is said to be determined by its spectrum if any graph having the same spectrum as G is isomorphic to G. A T-shape is a tree with exactly one of its vertices having maximal degree 3. In this paper, we show that all T-shape trees are determined by their spectra, except for a few well-defined cases.

A continuation algorithm for the solution of Max-Bisection problems is proposed in this paper. Unlike available relaxation algorithms for Max-Bisection problems, the Max-Bisection problems are converted to an equivalent continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved using the augmented Lagrange penalty function method. The convergence property and finite termination property of the proposed algorithm are studied, and numerical experiments and comparisons on the simple circuit partitioning problems are made. Results reported in section "Numerical experiments" show the algorithm generates satisfactory solutions to test problems.

In this paper a new two-level linear relaxed bound method is proposed for solving the global solution of geometric programming problems, and its convergent properties is proved, and a numerical example is used to illustrate the effectiveness of the presented algorithm. The bound technique in this algorithm is different from the other ones. The two-level relaxed linear programming problems of geometric programming problems are given without additional new variables and constraints by making use of the linear approximation of power functions and the new formulas for product to be unequal with sum.

In This paper, we propose a new branch and bound algorithm for the solution of large scalc sepmable concave programming problems. The largest distance biseetiou (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved theft the LDB method in a normal rectangle subdivision (NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the propped branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than w-subdivision method.

This paper is concerned with filled function techniques for unconstrained global minimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and non-smooth optimization problems. These functions have either one or two adjustable parameters. Conditions on functions and on the values of parameters are given so that the constructed functions have the desired properties of filled functions.

Quasi-Newton (QN) equation plays a core role in contemporary nonlinear optimization. The usual QN equation employs only the gradients, but ignores the available function value information. In this paper, we derive a class of modified QN equations with a vector parameter which use both available gradient and function value information. The modified quasi-Newton methods maintain most properties of the usual quasi-Newton methods, meanwhile they achieve a higher-order accuracy in approximating the second-order curvature of the problem functions than the usual ones do. Numerical experiments are reported which support the theoretical analyses and show the advantages of the modified QN methods over the usual ones.