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The purpose of this paper is to establish first and second order necessaryoptimality conditions for optimal control problems of stochastic evolutionequations with control and state constraints. The control acts both in thedrift and diffusion terms and the control region is a nonempty closed subset ofa separable Hilbert space. We employ some classical set-valued analysis toolsand theories of the transposition solution of vector-valued backward stochasticevolution equations and the relaxed-transposition solution of operator-valuedbackward stochastic evolution equations to derive these optimality conditions.The correction part of the second order adjoint equation, which does not appearin the first order optimality condition, plays a fundamental role in the secondorder optimality condition.

First and Second Order Necessary Optimality Conditions for Controlled Stochastic Evolution Equations with Control and State Constraints