The Regular Deformation Of Feasible Set Of Quadratic Constrained Optimization Problems

In 2006, the boundary moving combined homotopy method for solving non-convex programming is given by Yu Bo and Shang Yu feng and the existence and convergence of the homotopy path is proved under some weak conditions. The conditions needed is weaker and the homotopy is easier to be constructed than the modified combined homotopy under quasi-normal cone condition and pseudo-cone condition. Moreover, it need not to choose the start point inside the interior part of the feasible set, so the method is more convenient to be implemented.In boundary moving combined homotopy method for solving the problem with only inequality constraints, to ensure its global convergence, it is needed to construct a class of constraint shifting functions with the following two properties:a) Positive independence of active constraints; b) Starting feasible set satisfies normal cone condition. In this paper, We propose the theoretical analysis of constructing the constraint shifting functions and a formal algorithm. From the view of algorithm, this paper also gives a judgment condition, which is equivalent to the positive independence of vector groups, and an algorithm which is designed in order to judge the positive independence. Some specific examples are given to verify the feasibility of the algorithm.It is needed to calculate the tangent vector in estimated phase so as to determine the estimate direction while numerically tracking homotopy path with Euler-Newton method. By modifying the original one in the book named the theory and method of nonlinear nu-merical analysis, we proposed a new column primary QR decomposition algorithm which is not required to calculate the minimum exchange number of k1, k2,…, kn according toits ascending order in this paper.moreover, algorithm implementation in the end, we can get an explicit which is easy to store and solve with back substitution method. Compared with the original algorithm, it is more convenient of the programming and of high efficiency and more practical application.