a b s t r a c tIn this paper, we present a projection method to solve monotone nonlinear equations with convex constraints. This method can be viewed as an extension of CG_DESCENT method which is one of the most effective conjugate gradient methods for solving unconstrained optimization problems. Because of derivative-free and low storage, the proposed method can be used to solve large-scale nonsmooth monotone nonlinear equations. Its global convergence is established under some appropriate conditions. Preliminary numerical results show that the proposed method is effective and promising. Moreover, we also successfully use the proposed method to solve the sparse signal reconstruction in compressive sensing.

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In this paper, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz's function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function, respectively. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.

In this paper, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued maps are introduced. Necessary and sufficient conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.