ALM-SQP-BHom Method For Large Scale Constrained Optimization

Nonlinear optimization problems exist widely in engineering,science,society,finance and other fields.In many practical applications these problems often need to be solved quickly,so it is particularly important to study efficient numerical solution methods.In this paper,an augmented lagrangian sequential quadratic programming block homotopy(ALM-SQP-BHom)method is proposed for large-scale nonlinear programming problems with equality and bound constraints.The classical augmented Lagrangian method(ALM)is used as the outermost algorithm framework,and the equality constraints are imposed on the objective function,so that the subproblem only contains bound constraint.The solution of the original problem is obtained by iteratively solving a series of bounded optimization subproblems and gradually updating Lagrange multiplier and penalty parameters.Sequential quadratic programming(SQP)method is used to solve bounded constraint optimization problems with fixed Lagrange multiplier and penalty parameters iteratively.However,the cost of forming and solving SQP subproblems are very high.We converted large-scale problems into small-scale problems by stochastic block coordinate descent(SBCD)method,so as to achieve the effect of dimensionality reduction on the scale of problems.It makes the whole algorithm more efficient.An accelerated proximal gradient homotopy(APG-Hom)method is used to solve the strongly convex quadratic subproblem.The APG method,as a first-order optimization method,can quickly get a good initial point,while the Hom method,as a high-order method,can get a high-precision solution.In this paper,three numerical examples are selected for numerical experiments,and the ALM-SQP-BHom method is compared with three popular large-scale nonlinear problem solver fmincon,OPTI and KNITRO.Numerical results show that our algorithm has significant advantages in solving efficiency for large-scale nonlinear constrained optimization problems.In addition,the method is also used to solve the portfolio problem with transaction cost and the rebalance portfolio problem in the financial field,and good results are obtained.