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We study monotone function-constrained variational inequalities (FCVIs) whose feasible region is the intersection of a projection-friendly set and several convex function constraints; both the operator and the constraint functions may be smooth, nonsmooth, and/or stochastic. Computing the projection operator is challenging for FCVIs. We introduce the adaptive operator extrapolation (AdOpEx) method, which employs an operator extrapolation on the Karush–Kuhn–Tucker operator of the FCVI in a smooth deterministic setting. Because this operator is not uniformly Lipschitz continuous in the Lagrange multipliers, we employ an adaptive two-timescale algorithm leading to bounded multipliers and achieving the optimal [Formula: see text] convergence rate. For the nonsmooth and stochastic VIs, we introduce design changes to the AdOpEx method and propose a novel P-OpEx method that takes a partial extrapolation. It converges at the rate of [Formula: see text] when both the operator and constraints are stochastic or nonsmooth. This method has suboptimal dependence on the noise and Lipschitz constants of function constraints. We propose a constraint extrapolation approach leading to the OpConEx method that improves this dependence by an order of magnitude. All our algorithms also extend to solving saddle point problems with jointly convex function constraints that couple the primal and dual variables. Within this structured class of problems, our methods preserve their respective complexity guarantees, establishing what we believe to be the first such comprehensive complexity results. Funding: D. Boob and M. Khalafi were partially supported by the National Science Foundation [Grants 2245705 and 2340858]. D. Boob was partially supported by the Office of Naval Research [Grant N000142412749]. Q. Deng was partially supported by the National Natural Science Foundation of China [Grants 12571325 and 72394364/72394360] and the Natural Science Foundation of Shanghai [Grant 24ZR1421300]. 

Abstract We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method. 

Abstract Emerging evidence points out that the responses of soil organic carbon (SOC) to nitrogen (N) addition differ along the soil profile, highlighting the importance of synthesizing results from different soil layers. Here, using a global meta‐analysis, we found that N addition significantly enhanced topsoil (0–30 cm) SOC by 3.7% (±1.4%) in forests and grasslands. In contrast, SOC in the subsoil (30–100 cm) initially increased with N addition but decreased over time. The model selection analysis revealed that experimental duration and vegetation type are among the most important predictors across a wide range of climatic, environmental, and edaphic variables. The contrasting responses of SOC to N addition indicate the importance of considering deep soil layers, particularly for long‐term continuous N deposition. Finally, the lack of depth‐dependent SOC responses to N addition in experimental and modeling frameworks has likely resulted in the overestimation of changes in SOC storage under enhanced N deposition. 

Individuals infected by human immunodeficiency virus (HIV) are under oxidative stress due to the imbalance between reactive oxygen species (ROS) production and elimination. This paper presents a mathematical model with the cytotoxic T lymphocytes (CTL) immune response to examine the role of ROS in the dynamics of HIV infection. We classify the equilibria of the model and study the stability of these equilibria. Numerical simulations show that incorporating ROS and CTL immune response into the model leads to very rich dynamics, including bistable phenomena and periodic solutions. Although the current antiretroviral therapy can suppress viral load to the undetectable level, it cannot eradicate the virus. A high level of ROS may be a factor for HIV persistence in patients despite suppressive therapy. These results suggest that oxidative damage and anti-oxidant therapy should be considered in the study of HIV infection and treatment.