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Abstract In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization withnon-Lipschitzianvalue functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a$$(T+1)$$ $(T+1)$-stage stochastic MINLP satisfyingL-exact Lipschitz regularization withd-dimensional state spaces, to obtain an$$\varepsilon $$ $\epsilon$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$ $O\left({\left(\frac{2LT}{\epsilon }\right)}^d\right)$, and is lower bounded by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$ $O\left({\left(\frac{\mathrm{LT}}{4\epsilon }\right)}^d\right)$for the general case or by$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$ $O\left({\left(\frac{\mathrm{LT}}{8\epsilon }\right)}^{d/2-1}\right)$for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity dependspolynomiallyon the number of stages. We further show that the iteration complexity dependslinearlyonT, if all the state spaces are finite sets, or if we seek a$$(T\varepsilon )$$ $\left(T\epsilon \right)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale withT. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization. 

Abstract This paper describes the atmospheric component of the US Department of Energy's Energy Exascale Earth System Model (E3SM) version 3. Significant updates have been made to the atmospheric physics compared to earlier versions. Specifically, interactive gas chemistry has been implemented, along with improved representations of aerosols and dust emissions. A new stratiform cloud microphysics scheme more physically treats ice processes and aerosol‐cloud interactions. The deep convection parameterization has been largely improved with sophisticated microphysics for convective clouds, making model convection sensitive to large‐scale dynamics, and incorporating the dynamical and physical effects of organized mesoscale convection. Improvements in aerosol wet removal processes and parameter re‐tuning of key aerosol and cloud processes have improved model aerosol radiative forcing. The model's vertical resolution has increased from 72 to 80 layers with the extra eight layers added in the lower stratosphere to better simulate the Quasi‐Biennial Oscillation. These improvements have enhanced E3SM's capability to couple aerosol, chemistry, and biogeochemistry and reduced some long‐standing biases in simulating tropical variability. Compared to its predecessors, the model shows a much stronger signal for the Madden‐Julian Oscillation, Kelvin waves, mixed Rossby‐gravity waves, and eastward inertia‐gravity waves. Aerosol radiative forcing has been considerably reduced and is now better aligned with community best estimates, leading to significantly improved skill in simulating historical temperature records. Its simulated mean‐state climate is largely comparable to E3SMv2, but with some notable degradation in shortwave cloud radiative effect, precipitable water, and surface wind stress, which will be addressed in future updates.