ADMM For Solving Inequality Constraint Nonconvex Quadratic Programs

The convergence of alternating direction method of multipliers(ADMM)for convex problems has been observed by many researchers.However there is a lack of theory of the algorithm when the objective function is nonconvex.In this paper,based on the optimal condition of this problem,matrix splitting and iterative theory,we analyze the convergence of the ADMM for solving inequality constrained nonconvex quadratic program by the contraction mapping principle.The rest of the paper is organized as follows.In Chapter 2,we introduce the concept of saddle point system and the numerical algorithms for solving linear systems in saddle point form.Moreover,we give the related results of norm and the Lagrange's method to dive stable factorizations for symmetric indefinite matrices.Then we get the idea of alternating direction method of multipliers and the general iteration scheme.In Chapter 3,based on the splitting of matrix and the idea of Gauss-Seidel iterative method,we construct the new iterative method for solving linear system.We present the alternating direction method of multipliers for solving the nonconvex quadratic programs,and get the equivalent algorithm in matrix form to prove the convergence of the algorithm in the next chapter.In Chapter 4,by giving the results of spectral radius,the spectral norm of matrix,and the contraction mapping principle,we prove the convergence of the algorithm and we have the limit point is the solution of the stationary condition of the nonconvex quadratic programs.In Chapter 5,based on the iterative scheme defined in Chapter3 and the stability condition of the problem,we obtain the tolerance in every step and present the stopping criterion by relative and absolute tolerance.At last,we give the illustrative numerical result.