Some Research On Nonlinear Sparse Optimization Problem

In this paper,we use the sparse projection method to study the minimization problem of min {f(x):x ? Cs? B),where f:Rn? R is a continuously differentiable function and the B is a closed convex set in Rn,Cs = {x? Rn:||xIIO ? s),s ? ?1,2,...,n}.Since the sparsity constraint||x||0?s renders the feasible set Cs n B to be nonconvex,it's generally hard to reach an optimal solution efficiently,even if the objective function is convex or if the set B is just the Rn.We begin with a study of the optimality conditions of the nonlinear sparse optimization problem.Then,we use the sparse gradient projection method to find a feasible solution meeting the optimality conditions.1.In chapter 2,we first study the case where only a sparsity constraint is present and obtain a necessary optimality condition called the CW minimum.Actually,it's a stronger necessary condition.When certain symmetry assumptions(in addition to convexity and closedness)will be imposed on the set B,we give a definition of the support optimal point and its three properties and prove that it's a necessary optimality condition too.Besides,we prove that optimal solution is a strong stationary point under the assumption that the gradient function ?f is Lipschitz-continuous.2.In chapter 3,we use the sparse gradient projection method to solve the nonlinear sparse optimization problem.During the iteration,we choose a stepsize 1/L,where L>L(f)and L(f)is the Lipschitz constant of ?f.Since the set Cs?B is nonconvex,the sparse projection set ProjCs?B(xk-1/L?f(xk)is not always a singleton.Combining the algorithm for finding a sparse projection proposed by Beck and Hallak,we choose xk+1 ? ProjCs?B(xk-1/L?f(xk)Let ?Xk>k?0 generated by this algorithm,then any accumulation point of this sequence is a general stationary point and the sequence {f(xk)}k?0 is nonincreasing.