Extended mean-field control under constraints: The generalized Fritz-John conditions and Lagrangian method

Abstract

This paper studies mean-field control with joint law dependence under dynamic expectation constraints and/or dynamic state-control-law constraints. We pioneer the establishment of the stochastic maximum principle (SMP) and the derivation of the backward SDE (BSDE) from the perspective of constrained optimization using the method of Lagrangian multipliers. We first propose to embed the constrained mean-field control (C-MFC) with joint-law dependence into some abstract optimization problems with constraints on Banach spaces, for which we develop the generalized Fritz-John (FJ) optimality conditions. We then prove the stochastic maximum principle (SMP) for C-MFC by transforming the FJ conditions into an equivalent stochastic first-order condition associated with a general type of constrained forward-backward SDEs (FBSDEs). Contrary to the existing literature, we treat the McKean-Vlasov SDE as an infinite-dimensional equality constraint such that the BSDE induced by the FJ first-order optimality conditions can be interpreted as the generalized Lagrange multiplier. We also employ the methodology to stochastic control and mean field game problems under dynamic constraints.