Study On Algorithms For Some Class Of Equations With Box Constraint

Many practical problems can be transformed into equations with box constraint, such as nonlinear finite element problems, nonlinear fracture problems, electrical problems, power system computation and so on. Equations with box constraint are widely applied in defense science and technology, engineering technology, economic and financial fields. Therefore, the research of it has important theoretical and practical value. This dissertation mainly focus on some class of equations with box constraint. The main research works and results are as follows:1. A trust region method for semismooth equations with box constraint is proposed and applied to complementary problems. The main feature of this method is summarized as follows. Firstly, the trust region is used rectangular and a modified conjugate direction method is used to solve the subproblem; Secondly, a new active set is defined and a non-monotonic technique is used. The non-monotonic technique are achieved by choosing the direction, i.e., two directions d1 and d2 are defined, in each iteration, one of directions d1 and d2 is only accepted. If d1 is accepted, the value of objective function may be increasing. This method has good numerical results, and the global convergence and local quadratic convergence are established under certain assumptions.2. A feasible decomposition method for equations with nonnegativity constraint is pro-posed. Firstly, for the convex quadratic programming with nonnegative constraints, we dis-cuss some propositions of solution when the number of variables is 2; Secondly, a convex quadratic programming with nonnegative constraints is defined as subproblem of equations with box constraint. Thirdly, solving the subproblem by decomposition method, solutions of equations with nonnegativity constraint can be obtained. In addition, this method ia applied to complementarity problems. Finally, under certain assumptions, the global convergence is proved.3. In this chapter, equations with box constraint are transformed into a minimization problem with box constraint. There are many methods for the minimization problem with box constraint, such as Newton method, gradient projection method, trust region method and so on. After studying these methods, we find that quadratic convergence of majority algorithms is based on the solvability of the equations. But whether the equations are solvable is previously unknown. To this end, a algorithm with fast quadratic convergence rates is proposed. The local convergence does not depend on the solvability of the equations. The proposed algorithm is applied to a class of stochastic linear complementarity problems. Both theoretical analysis and numerical results show the efficiency of the proposed algorithm.4. A feasible smooth method based on Barzilai-Borwein method is proposed for ERM of stochastic linear complementarity problem. The significant advantage of our method are that: 1) we partition the index set into three mutually sets; 2) we define direction in every mutually set; 3) every iteration is feasible. From the numerical results, it can be seen our method is very encouraging.5. Since Non-negative Matrix Factorization (NMF) was first proposed over a decade ago, it has attracted much attention, particularly when applied to numerous data analysis problems. Most of the existing algorithms for NMF are based on multiplicative iterative and alternating least squares algorithms. However, algorithms based on the optimization method are few, especially in the case where two variables are derived at the same time. In this chapter, we propose a non-monotone projection gradient method for NMF and establish the convergence results of our algorithm. Experimental results show that our algorithm converges to better solutions than popular multiplicative update based algorithms.6. Non-negative Matrix Factorization(NMF) is a method to obtain a representation of data using non-negativity constraints. A popular approach is alternating non-negative least squares (ANLS). As is well known, if the sequence generated by ANLS has at least one limit point, then the limit point is a stationary point of NMF. However, no evdience has shown that the sequence generated by ANLS has at least one limit point. In order to overcome this shortcoming, we propose two modified strategies for ANLS in this paper. The two modified strategies can ensure the sequence generated by ANLS has at least one limit point, and this limit point is a stationary point of NMF. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.