Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints

Modern real application problems involve matrix-valued parameters, yet conventional optimizers treat them as vectors, thereby motivating matrix-aware methods that exploit input-output geometry, such as Muon which orthogonalizes the momentum matrices before parameter updates. Its empirical success raises a conceptual question: can orthogonalized momentum remain effective beyond smooth optimization? This paper studies this question for locally Lipschitz functions using a generalized derivative framework compatible with backpropagation. Our first contribution is to identify a key limitation: for every fixed momentum factor $β\in[0,1)$, Muon can fail to approach the global optimal solution of a convex Lipschitz objective from almost every initialization, when step sizes adapt to the full gradient history. The failure can occur even along bounded iterates. Our example is inspired by the one of Parshakova et al. which only covers $β\in[0,\frac{1}{2})$. Then, we show that the obstruction lies in fixed momentum rather than orthogonalization. Indeed, when the momentum factor is adaptive and approaches 1 together with a vanishing step size, Muon recovers asymptotic convergence for nonconvex nonsmooth optimization under the boundedness and regularity conditions. Moreover, we propose MAGD, which combines orthogonalized momentum with gradient, weighted based on their relative progress. MAGD retains asymptotic convergence in nonconvex settings and achieves an $O(\min\{m,n\}ε^{-2})$ rate in convex settings. A lower bound shows the optimal dimension dependence. Experiments on synthetic problems, image classification, and LLM pretraining show MAGD is a simple and practical alternative to Muon. Together, our results characterize when orthogonalized momentum fails without smoothness and how it can be made reliable and we hope that the analysis may be useful more broadly.

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems under local multiplier regularity. Both the objective and functional inequality constraints are compositions of convex Lipschitz outer functions with smooth nonlinear mappings. Unlike existing analyses requiring regularity globally or along the entire iterate sequence, we impose multiplier regularity only near the feasible set. Using an auxiliary minimax problem with a compact dual set, we derive Lyapunov estimates that identify nearly feasible iterates under feasible initialization and a sufficiently large penalty, without assuming multiplier boundedness. Local regularity then bounds the associated multipliers and makes dual truncation inactive, yielding approximate KKT certificates for the original problem. The smoothed prox-linear augmented Lagrangian method with dual regularization yields an $O(K^{-1/3})$ KKT residual rate. Without regularization, piecewise linear outer functions and additional local structural assumptions yield a local dual error bound and the sharper $O(K^{-1/2})$ rate.

In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems with exact constraints and stochastic objective evaluations. To solve this class of problems, we propose a framework of mirror descent linearized augmented Lagrangian methods that employs two-point stochastic zeroth-order gradient estimators and exploits non-Euclidean mirror descent geometry. Under mild assumptions, we establish oracle complexity guarantees for finding an $ε$-KKT point parameterized by $p \geq 2$. Under Rademacher smoothing, our analysis reveals a trade-off between the variance of the zeroth-order gradient estimators and the smoothness of the mirror map. In the high-accuracy regime, the resulting effective oracle complexity is $\mathcal{O}(p d^{2/p}ε^{-3})$ for $p \in [2,2\ln d]$ and $\mathcal{O}(\ln d\,ε^{-3})$ for $p > 2\ln d$. These bounds reduce the dimension dependence in the leading term. When $p=2$, our method recovers the Euclidean setting with an oracle complexity of $\mathcal{O}(dε^{-3})$, improving the $ε$-dependence over existing methods. Furthermore, to eliminate initial near-feasibility requirements, we introduce a multi-stage scheme that finds an $ε$-KKT point within $\mathcal{O}(1+\log\log(e/ε))$ stages while maintaining the leading-order complexity. Numerical tests on QCQPs, black-box adversarial attacks, and fairness-constrained classification demonstrate the effectiveness of our proposed method.