In this paper, we consider nonlinear optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Stochastic Sequential Quadratic Programming (TR-SSQP) method and establish its high-probability iteration complexity bounds for identifying first- and second-order ϵ -stationary points. In our algorithm, we assume that exact objective values, gradients, and Hessians are not directly accessible but can be estimated via zeroth-, first-, and second-order probabilistic oracles. Compared to existing complexity studies of SSQP methods that rely on a zeroth-order oracle with sub-exponential tail noise (i.e., light-tailed) and focus mostly on first-order stationarity, our analysis accommodates biased (also referred to as irreducible in the literature) and heavy-tailed noise in the zeroth-order oracle, and significantly extends the analysis to second-order stationarity. We show that under heavy-tailed noise conditions, our SSQP method achieves the same high-probability first-order iteration complexity bounds as in the light-tailed noise setting, while further exhibiting promising second-order iteration complexity bounds. Specifically, the method identifies a first-order ϵ -stationary point in 𝒪(ϵ ^-2) iterations and a second-order ϵ -stationary point in 𝒪(ϵ ^-3) iterations with high probability, provided that ϵ is lower bounded by a constant determined by the bias magnitude (i.e., the irreducible noise) in the estimation. We validate our theoretical findings and evaluate practical performance of our method on CUTEst benchmark test set.

In this paper, we propose a trust-region interior-point stochastic sequential quadratic programming (TR-IP-SSQP) method for solving optimization problems with a stochastic objective and deterministic nonlinear equality and inequality constraints. In this setting, exact evaluations of the objective function and its gradient are unavailable, but their stochastic estimates can be constructed. In particular, at each iteration our method builds stochastic oracles, which estimate the objective value and gradient to satisfy proper adaptive accuracy conditions with a fixed probability. To handle inequality constraints, we adopt an interior-point method (IPM), in which the barrier parameter follows a prescribed decaying sequence. Under standard assumptions, we establish global almost-sure convergence of the proposed method to first-order stationary points. We implement the method on a subset of problems from the CUTEst test set, as well as on logistic regression problems, to demonstrate its practical performance.

Reliable decision-making with streaming data requires principled uncertainty quantification of online methods. While first-order methods enable efficient iterate updates, their inference procedures still require updating proper (covariance) matrices, incurring O(d^2) time and memory complexity, and are sensitive to ill-conditioning and noise heterogeneity of the problem. This costly inference task offers an opportunity for more robust second-order methods, which are, however, bottlenecked by solving Newton systems with O(d^3) complexity. In this paper, we address this gap by studying an online Newton method with Hessian averaging, where the Newton direction at each step is approximately computed using a sketch-and-project solver with Nesterov's acceleration, matching O(d^2) complexity of first-order methods. For the proposed method, we quantify its uncertainty arising from both random data and randomized computation. Under standard smoothness and moment conditions, we establish global almost-sure convergence, prove asymptotic normality of the last iterate with a limiting covariance characterized by a Lyapunov equation, and develop a fully online covariance estimator with non-asymptotic convergence guarantees. We also connect the resulting uncertainty quantification to that of exact and sketched Newton methods without Nesterov's acceleration. Extensive experiments on regression models demonstrate the superiority of the proposed method for online inference.

Given the ubiquity of streaming data, online algorithms have been widely used for parameter estimation, with second-order methods particularly standing out for their efficiency and robustness. In this paper, we study an online sketched Newton method that leverages a randomized sketching technique to perform an approximate Newton step in each iteration, thereby eliminating the computational bottleneck of second-order methods. While existing studies have established the asymptotic normality of sketched Newton methods, a consistent estimator of the limiting covariance matrix remains an open problem. We propose a fully online covariance matrix estimator that is constructed entirely from the Newton iterates and requires no matrix factorization. Compared to covariance estimators for first-order online methods, our estimator for second-order methods is batch-free. We establish the consistency and convergence rate of our estimator, and coupled with asymptotic normality results, we can then perform online statistical inference for the model parameters based on sketched Newton methods. We also discuss the extension of our estimator to constrained problems, and demonstrate its superior performance on regression problems as well as benchmark problems in the CUTEst set.

This article presents a dynamic regret analysis for stochastic model predictive control (SMPC) in linear systems with quadratic performance index and additive and multiplicative uncertainties. Under a finite support assumption, the problem can be cast as a finite-dimensional quadratic program, but the problem becomes quickly intractable as the problem size grows exponentially in the horizon length. SMPC aims to compute approximate solutions by solving a sequence of problems with truncated prediction horizons and committing the solution in a receding-horizon fashion. While this approach is widely used in practice, its performance relative to the optimal solution is not well understood. This article reports for the first time a rigorous near-optimal performance guarantee of SMPC: Under stabilizability and detectability conditions, the dynamic regret of SMPC is exponentially small in the prediction horizon length, allowing SMPC to achieve near-optimal performance at a substantially reduced computational expense.

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O (ε^-(3α-2)/(α-1)) for finding an ε-stationary point, where α∈(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

We present an optimize-then-discretize framework for solving linear-quadratic optimal control problems (OCP) governed by time-inhomogeneous ordinary differential equations (ODEs). Our method employs a modified overlapping Schwarz decomposition based on the Pontryagin Minimum Principle, partitioning the temporal domain into overlapping intervals and independently solving Hamiltonian systems in continuous time. We demonstrate that the convergence is ensured by appropriately updating the boundary conditions of the individual Hamiltonian dynamics. The cornerstone of our analysis is to prove that the exponential decay of sensitivity (EDS) exhibited in discrete-time OCPs carries over to the continuous-time setting. Unlike the discretize-then-optimize approach, our method can flexibly incorporate different numerical integration methods for solving the resulting Hamiltonian two-point boundary-value subproblems, including adaptive-time integrators. A numerical experiment on a linear-quadratic OCP illustrates the practicality of our approach in broad scientific applications.

We consider solving nonlinear optimization problems with a stochastic objective and deterministic equality constraints, assuming that only zero-order information is available for both the objective and constraints, and that the objective is also subject to random sampling noise. Under this setting, we propose a Derivative-Free Stochastic Sequential Quadratic Programming (DF-SSQP) method. Due to the lack of derivative information, we adopt a simultaneous perturbation stochastic approximation (SPSA) technique to randomly estimate the gradients and Hessians of both the objective and constraints. This approach requires only a dimension-independent number of zero-order evaluations – as few as eight – at each iteration step. A key distinction between our derivative-free and existing derivative-based SSQP methods lies in the intricate random bias introduced into the gradient and Hessian estimates of the objective and constraints, brought by stochastic zero-order approximations. To address this issue, we introduce an online debiasing technique based on momentum-style estimators that properly aggregate past gradient and Hessian estimates to reduce stochastic noise, while avoiding excessive memory costs via a moving averaging scheme. Under standard assumptions, we establish the global almost-sure convergence of the proposed DF-SSQP method. Notably, we further complement the global analysis with local convergence guarantees by demonstrating that the rescaled iterates exhibit asymptotic normality, with a limiting covariance matrix resembling the minimax optimal covariance achieved by derivative-based methods, albeit larger due to the absence of derivative information. Our local analysis enables online statistical inference of model parameters leveraging DF-SSQP. Numerical experiments on benchmark nonlinear problems demonstrate both the global and local behavior of DF-SSQP.

We consider statistical inference of equality-constrained stochastic nonlinear optimization problems. We develop a fully online stochastic sequential quadratic programming (StoSQP) method to solve the problems, which can be regarded as applying Newton's method to the first-order optimality conditions (i.e., the KKT conditions). Motivated by recent designs of numerical second-order methods, we allow StoSQP to adaptively select any random stepsize $\bar{\alpha}_t$, as long as $\beta_t\leq \bar{\alpha}_t \leq \beta_t+\chi_t$, for some control sequences $\beta_t$ and $\chi_t=o(\beta_t)$. To reduce the dominant computational cost of second-order methods, we additionally allow StoSQP to inexactly solve quadratic programs via efficient randomized iterative solvers that utilize sketching techniques. Notably, we do not require the approximation error to diminish as iteration proceeds. For the developed method, we show that under mild assumptions (i) computationally, it can take at most $O(1/\epsilon^4)$ iterations (same as samples) to attain $\epsilon$-stationarity; (ii) statistically, its primal-dual sequence $1/\sqrt{\beta_t}\cdot (x_t - x^\star, \lambda_t - \lambda^\star)$ converges to a mean-zero Gaussian distribution with a nontrivial covariance matrix depending on the underlying sketching distribution. Additionally, we establish the almost-sure convergence rate of the iterate $(x_t, \lambda_t)$ along with the Berry-Esseen bound; the latter quantitatively measures the convergence rate of the distribution function. We analyze a plug-in limiting covariance matrix estimator, and demonstrate the performance of the method both on benchmark nonlinear problems in CUTEst test set and on linearly/nonlinearly constrained regression problems.

We consider applying stochastic approximation (SA) methods to solve nonsmooth variational inclusion problems. Existing studies have shown that the averaged iterates of SA methods exhibit asymptotic normality, with an optimal limiting covariance matrix in the local minimax sense of Hajek and Le Cam. However, no methods have been proposed to estimate this covariance matrix in a nonsmooth and potentially non-monotone (nonconvex) setting. In this paper, we study an online batch-means covariance matrix estimator introduced in Zhu et al. (2023). The estimator groups the SA iterates appropriately and computes the sample covariance among batches as an estimate of the limiting covariance. Its construction does not require prior knowledge of the total sample size, and updates can be performed recursively as new data arrives. We establish that, as long as the batch size sequence is properly specified (depending on the stepsize sequence), the estimator achieves a convergence rate of order O(root dn(-1/8+epsilon)) for any epsilon > 0, where d and n denote the problem dimensionality and the number of iterations (or samples) used. Although the problem is nonsmooth and potentially non-monotone (nonconvex), our convergence rate matches the bestknown rate for covariance estimation methods using only first-order information in smooth and strongly-convex settings. The consistency of this covariance estimator enables asymptotically valid statistical inference, including constructing confidence intervals and performing hypothesis testing.

We study online statistical inference for the solutions of stochastic optimization problems with equality and inequality constraints. Such problems are prevalent in statistics and machine learning, encompassing constrained M-estimation, physics-informed models, safe reinforcement learning, and algorithmic fairness. We develop a stochastic sequential quadratic programming (SSQP) method to solve these problems, where the step direction is computed by sequentially performing a quadratic approximation of the objective and a linear approximation of the constraints. Despite having access to unbiased estimates of population gradients, a key challenge in constrained stochastic problems lies in dealing with the bias in the step direction. As such, we apply a momentum-style gradient moving-average technique within SSQP to debias the step. We show that our method achieves global almost-sure convergence and exhibits local asymptotic normality with an optimal primal-dual limiting covariance matrix in the sense of Hájek and Le Cam. In addition, we provide a plug-in covariance matrix estimator for practical inference. To our knowledge, the proposed SSQP method is the first fully online method that attains primal-dual asymptotic minimax optimality without relying on projection operators onto the constraint set, which are generally intractable for nonlinear problems. Through extensive experiments on benchmark nonlinear problems, as well as on constrained generalized linear models and portfolio allocation problems using both synthetic and real data, we demonstrate superior performance of our method, showing that the method and its asymptotic behavior not only solve constrained stochastic problems efficiently but also provide valid and practical online inference in real-world applications.

Constrained stochastic nonlinear optimization problems have attracted significant attention for their ability to model complex real-world scenarios in physics, economics, and biology. As datasets continue to grow, online inference methods have become crucial for enabling real-time decision-making without the need to store historical data. In this work, we develop an online inference procedure for constrained stochastic optimization by leveraging a method called Sketched Stochastic Sequential Quadratic Programming (SSQP). As a direct generalization of sketched Newton methods, SSQP approximates the objective with a quadratic model and the constraints with a linear model at each step, then applies a sketching solver to inexactly solve the resulting subproblem. Building on this design, we propose a new online inference procedure called random scaling. In particular, we construct a test statistic based on SSQP iterates whose limiting distribution is free of any unknown parameters. Compared to existing online inference procedures, our approach offers two key advantages: (i) it enables the construction of asymptotically valid confidence intervals; and (ii) it is matrix-free, i.e. the computation involves only primal-dual SSQP iterates (x_t, λ_t) without requiring any matrix inversions. We validate our theory through numerical experiments on nonlinearly constrained regression problems and demonstrate the superior performance of our random scaling method over existing inference procedures.

In this work, we consider solving optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Sequential Quadratic Programming method to find both first- and second-order stationary points. Our method utilizes a random model to represent the objective function, which is constructed from stochastic observations of the objective and is designed to satisfy proper adaptive accuracy conditions with a high but fixed probability. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a quadratic approximation of the objective subject to a (relaxed) linear approximation of the problem constraints and a trust-region constraint. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature of the reduced Hessian matrix, as well as a second-order correction step to address the potential Maratos effect, which arises due to the nonlinearity of the problem constraints. Such an effect may impede the method from moving away from saddle points. Both gradient and eigen step computations leverage a novel parameter-free decomposition of the step and the trust-region radius, accounting for the proportions among the feasibility residual, optimality residual, and negative curvature. We establish global almost sure first- and second-order convergence guarantees for our method, and present computational results on CUTEst problems, regression problems, and saddle-point problems to demonstrate its superiority over existing line-search-based stochastic methods.

In this paper, we address the challenge of solving large-scale graph-structured nonlinear programs (gsNLPs) in a scalable manner. GsNLPs are problems in which the objective and constraint functions are associated with nodes on a graph and depend on the variables of adjacent nodes. This graph-structured formulation encompasses various specific instances, such as dynamic optimization, PDE-constrained optimization, multi-stage stochastic optimization, and general network optimization. By leveraging the sequential quadratic programming (SQP) framework, we propose a globally convergent overlapping graph decomposition method to solve large-scale gsNLPs under standard mild regularity conditions on the graph topology. In each iteration, we perform an overlapping graph decomposition to compute an approximate Newton direction in a parallel environment. Then, we select a suitable stepsize and update the primal-dual iterate by performing a backtracking line search on an exact augmented Lagrangian merit function. Built on the exponential decay of sensitivity of gsNLPs, we show that the approximate Newton direction is a descent direction of the augmented Lagrangian, which leads to global convergence with local linear convergence rate. In particular, global convergence is achieved for sufficiently large overlaps, and the local linear convergence rate improves exponentially in terms of the overlap size. Our results match existing state-of-the-art guarantees established for dynamic programs (which simply correspond to linear graphs). We validate the theory on two PDE-constrained problems: a semilinear elliptic problem and a boundary heating problem.

Physics-Informed Neural Networks (PINNs) represent a significant advancement in Scientific Machine Learning (SciML), which integrate physical domain knowledge into an empirical loss function as soft constraints and apply existing machine learning methods to train the model. However, recent research has noted that PINNs may fail to learn relatively complex Partial Differential Equations (PDEs). This paper addresses the failure modes of PINNs by introducing a novel, hard-constrained deep learning method – trust-region Sequential Quadratic Programming (trSQP-PINN). In contrast to directly training the penalized soft-constrained loss as in PINNs, our method performs a linear-quadratic approximation of the hard-constrained loss, while leveraging the soft-constrained loss to adaptively adjust the trust-region radius. We only trust our model approximations and make updates within the trust region, and such an updating manner can overcome the ill-conditioning issue of PINNs. We also address the computational bottleneck of second-order SQP methods by employing quasi-Newton updates for second-order information, and importantly, we introduce a simple pretraining step to further enhance training efficiency of our method. We demonstrate the effectiveness of trSQP-PINN through extensive experiments. Compared to existing hard-constrained methods for PINNs, such as penalty methods and augmented Lagrangian methods, trSQP-PINN significantly improves the accuracy of the learned PDE solutions, achieving up to 1-3 orders of magnitude lower errors. Additionally, our pretraining step is generally effective for other hard-constrained methods, and experiments have shown the robustness of our method against both problem-specific parameters and algorithm tuning parameters.

We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where at each step a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to utilize indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm must address an infeasibility issue -- the linearized equality constraints and trust-region constraints may lead to infeasible SQP subproblems. In this regard, we propose an adaptive relaxation technique to compute the trial step, consisting of a normal step and a tangential step. To control the lengths of these two steps while ensuring a scale-invariant property, we adaptively decompose the trust-region radius into two segments, based on the proportions of the rescaled feasibility and optimality residuals to the rescaled full KKT residual. The normal step has a closed form, while the tangential step is obtained by solving a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish a global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.

We study the convergence of accelerated stochastic gradient descent (SGD) for strongly convex objectives under the growth condition, which states that the variance of stochastic gradient is bounded by a multiplicative part that grows with the full gradient and a constant additive part. Through the lens of the growth condition, we investigate four widely used accelerated methods: Nesterov’s accelerated method (NAM), robust momentum method (RMM), accelerated dual averaging method (DAM+), and implicit DAM+ (iDAM+). Although these methods are known to improve the convergence rate of SGD under the condition that the stochastic gradient has bounded variance, it is not well understood how their convergence rates are affected by the multiplicative noise. In this paper, we show that these methods all converge to a neighborhood of the optimum with accelerated convergence rates (compared with SGD), even under the growth condition. In particular, NAM, RMM, and iDAM+ enjoy acceleration only with a mild multiplicative noise, whereas DAM+ enjoys acceleration, even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of DAM+ and iDAM+ to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term). We conduct numerical experiments to support our theoretical conclusions.

We propose a fast temporal decomposition procedure for solving long -horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method.

We propose a fast temporal decomposition procedure for solving long-horizon nonlinear dynamic programs. The core of the procedure is sequential quadratic programming (SQP) that utilizes a differentiable exact augmented Lagrangian as the merit function. Within each SQP iteration, we approximately solve the Newton system using an overlapping temporal decomposition strategy. We show that the approximate search direction is still a descent direction of the augmented Lagrangian provided the overlap size and penalty parameters are suitably chosen, which allows us to establish the global convergence. Moreover, we show that a unit step size is accepted locally for the approximate search direction and further establish a uniform, local linear convergence over stages. This local convergence rate matches the rate of the recent Schwarz scheme (Na et al. 2022). However, the Schwarz scheme has to solve nonlinear subproblems to optimality in each iteration, whereas we only perform a single Newton step instead. Numerical experiments validate our theories and demonstrate the superiority of our method. Funding: This work was supported by the National Science Foundation [Grant CNS-1545046] and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research [Grant DE-AC02-06CH11347].

We study nonlinear optimization problems with a stochastic objective and deterministic equality and inequality constraints, which emerge in numerous applications including finance, manufacturing, power systems and, recently, deep neural networks. We propose an active-set stochastic sequential quadratic programming (StoSQP) algorithm that utilizes a differentiable exact augmented Lagrangian as the merit function. The algorithm adaptively selects the penalty parameters of the augmented Lagrangian, and performs a stochastic line search to decide the stepsize. The global convergence is established: for any initialization, the KKT residuals converge to zero almost surely . Our algorithm and analysis further develop the prior work of Na et al. (Math Program, 2022. https://doi.org/10.1007/s10107-022-01846-z ). Specifically, we allow nonlinear inequality constraints without requiring the strict complementary condition; refine some of designs in Na et al. (2022) such as the feasibility error condition and the monotonically increasing sample size; strengthen the global convergence guarantee; and improve the sample complexity on the objective Hessian. We demonstrate the performance of the designed algorithm on a subset of nonlinear problems collected in CUTEst test set and on constrained logistic regression problems.