Research On A Class Of Greedy Simplex Methods For Double Sparse Constrained Optimization Problems

The double sparse constraint optimization problem divides the decision variables into two parts,and requires the two parts of the decision variables to satisfy a certain degree of sparsity at the same time.This kind of problems are widely used in image denoising,machine learning,signal and image processing,gene selection and other fields.In view of the wide application of the double sparse constraint optimization problem and the shortcomings of existing algorithm research,a greedy simplex algorithm for solving double sparse constrained optimization problems is studied,in this thesis.The main research results are as follows:The necessary optimality conditions for the double sparse constraint optimization problem are studied.Firstly,this thesis proposes the CW(Coordinate Wise)optimality condition and the basic feasibility condition for the double sparse constraint optimization problem,and the properties of the CW optimality condition are discussed.Secondly,based on the established basic feasibility conditions,the relationship between the CW optimal solution and the L-stable point is analyzed,which clarifies that the CW optimal solution is more extensive than the L-stable point.Based on the CW optimality condition,the greedy simplex algorithm for solving the double sparse constraint optimization problem is designed,and the feasibility and convergence of the algorithm are discussed.According to the design idea of the algorithm,it is proved that the proposed algorithm is feasibleis,and under some assumptions,the global convergence result of the algorithm is given.The accumulation point of the iterative point sequence generated by the algorithm satisfies the CW optimality condition.Further,if the gradient of the objective function satisfies the Lipschitz condition,then the convergent point of the iterative point sequence also converges to the L-stable point.According to the proposed greedy simplex algorithm,thesis uses the Mathlab language to compile the code to verify the feasibility of the algorithm,and conduct preliminary numerical experiments of the classical mathematical model.The preliminary numerical results show that the greedy simplex algorithm proposed in this thesis is feasible.