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Stochastic MPECs have found increasing relevance for modeling a broad rangeof settings in engineering and statistics. Yet, there seem to be no efficientfirst/zeroth-order schemes equipped with non-asymptotic rate guarantees forresolving even deterministic variants of such problems. We consider SMPECswhere the parametrized lower-level equilibrium problem is given by adeterministic/stochastic VI problem whose mapping is strongly monotone. Wedevelop a zeroth-order implicit algorithmic framework by leveraging a locallyrandomized spherical smoothing scheme. We present schemes for single-stage andtwo-stage stochastic MPECs when the upper-level problem is either convex ornonconvex. (I). Single-stage SMPECs: In convex regimes, our proposed inexactschemes are characterized by a complexity in upper-level projections,upper-level samples, and lower-level projections of$\mathcal{O}(\tfrac{1}{\epsilon^2})$, $\mathcal{O}(\tfrac{1}{\epsilon^2})$, and$\mathcal{O}(\tfrac{1}{\epsilon^2}\ln(\tfrac{1}{\epsilon}))$ , respectively.Analogous bounds for the nonconvex regime are$\mathcal{O}(\tfrac{1}{\epsilon})$, $\mathcal{O}(\tfrac{1}{\epsilon^2})$, and$\mathcal{O}(\tfrac{1}{\epsilon^3})$, respectively . (II). Two-stage SMPECs: Inconvex regimes, our proposed inexact schemes have a complexity in upper-levelprojections, upper-level samples, and lower-level projections of$\mathcal{O}(\tfrac{1}{\epsilon^2}),\mathcal{O}(\tfrac{1}{\epsilon^2})$, and$\mathcal{O}(\tfrac{1}{\epsilon^2}\ln(\tfrac{1}{\epsilon}))$ while thecorresponding bounds in the nonconvex regime are$\mathcal{O}(\tfrac{1}{\epsilon})$, $\mathcal{O}(\tfrac{1}{\epsilon^2})$, and$\mathcal{O}(\tfrac{1}{\epsilon^2}\ln(\tfrac{1}{\epsilon}))$ , respectively .In addition, we derive statements for exact as well as acceleratedcounterparts. We also provide a comprehensive set of numerical results forvalidating the theoretical findings.

Complexity guarantees for an implicit smoothing-enabled method for stochastic MPECs