  The purpose of this paper is to establish the first and second order
necessary conditions for stochastic optimal controls in infinite dimensions.
The control system is governed by a stochastic evolution equation, in which
both drift and diffusion terms may contain the control variable and the set of
controls is allowed to be nonconvex. Only one adjoint equation is introduced to
derive the first order optimality necessary condition either by means of the
classical variational analysis approach or under some assumption which is quite
natural in the deterministic setting to guarantee the existence of optimal
controls. More importantly, in order to avoid the essential difficulty with the
well-posedness of higher order adjoint equations, using again the classical
variational analysis approach, only the first and the second order adjoint
equations are needed to formulate the second order necessary optimality
condition, in which the solutions to the second order adjoint equation are
understood in the sense of relaxed transposition.
