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

Stochastic MPECs have found increasing relevance for modeling a broad range of settings in engineering and statistics. Yet, there seem to be no efficient first/zeroth-order schemes equipped with non-asymptotic rate guarantees for resolving even deterministic variants of such problems. We consider MPECs where the parametrized lower-level equilibrium problem is given by a deterministic/stochastic VI problem whose mapping is strongly monotone, uniformly in upper-level decisions. We develop a zeroth-order implicit algorithmic framework by leveraging a locally randomized spherical smoothing scheme. We make three sets of contributions: (i) Convex settings. When the implicit problem is convex and the lower-level decision is obtainable by inexactly solving a strongly monotone stochastic VI to compute an $epsilon$-solution, we derive iteration complexity guarantees of $mathcal{O}left(tfrac{L_0^2n^2}{epsilon^2}right)$ (upper-level) and $mathcal{O}left(tfrac{L_0^2 n^2}{epsilon^2} ln(tfrac{L_0 n}{epsilon})right)$ (lower-level); (ii) Exact oracles and accelerated schemes. When the lower-level problem can be resolved exactly, employing accelerated schemes, the complexity improves to $mathcal{O}(tfrac{1}{epsilon})$ and $mathcal{O}(tfrac{1}{epsilon^{2+delta}})$, respectively. Notably, this guarantee extends to stochastic MPECs with equilibrium constraints imposed in an almost sure sense; (iii) Nonconvex regimes. When the implicit problem is not necessarily convex and the lower-level problem can be inexactly resolved via a stochastic approximation framework, computing an $epsilon$-stationary point is equipped with complexity bounds of $mathcal{O}left(tfrac{L_0^2n^2}{epsilon}right)$ (upper-level) and $mathcal{O}left(tfrac{L_0^6n^6}{epsilon^3}right)$ (lower-level). We also provide numerical results for validating the theoretical findings in this work.