Nonsmooth Version Research Articles

A nonsmooth variant of the Mangasarian-Fromovitz constraint qualification

A unified view on constraint qualifications for nonsmooth equality and inequality constrained programs is presented. A fairly general constraint qualification for programs involving B-differential functions is given. Further specification to piecewise differentiable equality constraints and locally Lipschitz continuous inequality constraints yields a nonsmooth version of the Mangasarian-Fromovitz constraint qualification.

A nonsmooth version of Newton's method

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-deri...

A nonsmooth version of Newton's method

Newton's method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized Jacobian instead of the derivative. This extension includes the B-derivative version of Newton's method as a special case. Convergence theorems are proved under the condition of semismoothness. It is shown that the gradient function of the augmented Lagrangian forC2-nonlinear programming is semismooth. Thus, the extended Newton's method can be used in the augmented Lagrangian method for solving nonlinear programs.

The connection between the maximum principle and dynamic programming in stochastic control

There are usually two ways to study optimal stochastic control problems: Pontryagin's maximum principle and Bellman's dynamic programming, involving an adjoint process ψ and the value function V, respectively. The classical result on the connection between the maximum principle and dynamic programming is known as ψ(t)=V x(t,◯(t)) where ◯(∣) is the optimal path. In this paper we establish a nonsmooth version of the classical result by employing the notions of super_ and sub_differential introduced by Crandall and Lions. Thus the illusory assumption that V is differentiate is dispensed with.