| Sparse constraint optimization problem refers to a kind of important nonconvex optimization problem with sparse constraints.It is widely applied to the compression perception,artificial intelligence,signal processing and many other aspects.Usually,non-convex properties of sparse constraint makes the convex optimization theory cannot be applied to deal with this problem.However,the special structure that sparse sets can be decomposed into union of finite subspace provides a new idea for our research.This paper takes the optimization problem with mixed sparse constraints as the research object.Two kinds of first order necessary conditions are given.On the one hand,under the Robinson regular condition,starting that regular normal cone is a negative cone of the linearized tangent cone,the upper estamator or equivalent expression of the regular normal cone of the feasible set is obtained by using the properties of the negative cone,the tangent cone and the normal cone in the variational analysis.Furthermore,the strong first order necessary condition of the optimization problem is given.On the other hand,using the Ekeland’s variation principle,an algorithm for subdifferentiation,Hoffman lemma and other knowledge,a sufficient condition is obtained for the mixed regular/subregular and mixed direction regular/subregular of the set-value mapping.This paper proves that the local optimal solution of the optimization problem contains some auxiliary mapping that does not subregular.Moreover,the M optimality conditions for the optimization problem is given. |
