| The research of convex optimization models and algorithms is an important issue in the current optimization field,which is widely used in economic balance,image processing,transportation,machine learning and other fields.Equality constrained quadratic convex optimization(ECQCO)is an important class of convex optimization problem.Since the objective function of ECQCO is a quadratic convex function,it has strong convex property,which is beneficial for the algorithm improvement.The classical algorithms for solving this model include Augmented Lagrangian method(ALM),Alternating direction method of multipliers(ADMM),Proximal point algorithm(PPA)and Dual ascent method(DAM),et al.Among them,DAM is an effective method to solve this problem.Its basic idea is to minimize the original variables and update the dual variables to solve the optimal solution of the dual problem.He and Zhang modified the original algorithm and proposed a new Dual ascent method(NDAM).It is suggested that the logarithmic quadratic proximal(LQP)is applied for regularizing the subproblems,Since the non-negative orthant constraint x ?0 is penalized into the objective function,the subproblems turn out to be unconstrained nonlinearly equations.Its global convergence can be proven under weaker assumptions.In order to ensure the convergence of the algorithm and improve the efficiency,many researchers have added the LQP regularization term to the subproblems.This thesis is also based on this research.Firstly,the thesis introduces a modified self-adaptive Dual ascent method(MDAM),which has a great correlation with the algorithm we proposed later.Then,the LQP term is applied to regularize the subproblems of MDAM and a modified self-adaptive dual ascent method with LQP term(DAMLQP)is proposed.Furthermore,the DAMLQP algorithm is extended to solve convex optimization problems with two block variables.Under certain conditions,the convergence of two proposed methods can be guaranteed better and the subproblems can be solved in parallel when parallel computation devices are available,thus the computation time in one iteration could be greatly reduced.In the part of the theoretical analysis,this thesis uses the knowledge of variational inequalities to prove the convergence of the two proposed algorithms.Finally,in the part of numerical experiments,We may conclude from the numerical results that for solving synthetic problem,the DAMLQP algorithm outperforms CVX markedly,its performance is competitive compared with Customized proximal point algorithm(CPPA).Moreover,the DAMLQP algorithm is preferable with larger dimension n and smaller tolerance tol.Therefore,the DAMLQP algorithm is more suitable to solve large-scale problems in high-precision settings and has obvious computational performance advantages,making it more suitable to solve practical problems. |
