We investigate a primal-dual (PD) method for the saddle point problem (SPP) that uses a linear approximation of the primal function instead of the standard proximal step, resulting in a linearized PD (LPD) method. For convex-strongly concave SPP, we observe that the LPD method has a suboptimal dependence on the Lipschitz constant of the primal function. To fix this issue, we combine features of Accelerated Gradient Descent with the LPD method resulting in a single-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD method achieves the optimal gradient complexity when the SPP has a semi-linear coupling function. We also present an inexact ALPD method for SPPs with a general nonlinear coupling function that maintains the optimal gradient evaluations of the primal parts and significantly improves the gradient evaluations of the coupling term compared to the ALPD method. We verify our findings with numerical experiments.

The monotone Variational Inequality (VI) is an important problem in machine learning. In numerous instances, the VI problems are accompanied by function constraints which can possibly be data-driven, making the projection operator challenging to compute. In this paper, we present novel first-order methods for function constrained VI (FCVI) problem under various settings, including smooth or nonsmooth problems with a stochastic operator and/or stochastic constraints. First, we introduce the~{\texttt{OpConEx}} method and its stochastic variants, which employ extrapolation of the operator and constraint evaluations to update the variables and the Lagrangian multipliers. These methods achieve optimal operator or sample complexities when the FCVI problem is either (i) deterministic nonsmooth, or (ii) stochastic, including smooth or nonsmooth stochastic constraints. Notably, our algorithms are simple single-loop procedures and do not require the knowledge of Lagrange multipliers to attain these complexities. Second, to obtain the optimal operator complexity for smooth deterministic problems, we present a novel single-loop Adaptive Lagrangian Extrapolation~(\texttt{AdLagEx}) method that can adaptively search for and explicitly bound the Lagrange multipliers. Furthermore, we show that all of our algorithms can be easily extended to saddle point problems with coupled function constraints, hence achieving similar complexity results for the aforementioned cases. To our best knowledge, many of these complexities are obtained for the first time in the literature.

In this paper, we give a faster width-dependent algorithm for mixed packing-covering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a $1 \eps$ approximate solution in time $O(Nw/ \varepsilon)$, where $N$ is number of nonzero entries in the constraint matrix, and $w$ is the maximum number of nonzeros in any constraint. This algorithm is faster than Nesterov's smoothing algorithm which requires $O(N\sqrt{n}w/ \eps)$ time, where $n$ is the dimension of the problem. Our work utilizes the framework of area convexity introduced in [Sherman-FOCS'17] to obtain the best dependence on $\varepsilon$ while breaking the infamous $\ell_{\infty}$ barrier to eliminate the factor of $\sqrt{n}$.