This paper establishes the theoretical foundations of the online scaled gradient methods (OSGM), a framework that utilizes online learning to adapt stepsizes and provably accelerate first-order methods. OSGM quantifies the effectiveness of a stepsize by a feedback function motivated from a convergence measure and uses the feedback to adjust the stepsize through an online learning … Read more
The proximal alternating predictor-corrector (PAPC) method is a widely used first-order algorithm for solving convex-concave saddle-point problems involving both smooth and nonsmooth components. Unlike the primal-dual hybrid gradient (PDHG) method, which incorporates an extrapolation step with parameter $\theta \in (0,1]$ to improve convergence, the existing convergence analysis of PAPC has been limited to the case … Read more
The development of a multi-cut stochastic approximation (SA) method for solving stochastic convex composite optimization (SCCO) problems has remained an open challenge. The difficulty arises from the fact that the stochastic multi-cut model, constructed as the pointwise maximum of individual stochastic linearizations, provides a biased estimate of the objective function, with the error being uncontrollable. … Read more
We study inverse optimization (IO), where the goal is to use a parametric optimization program as the hypothesis class to infer relationships between input-decision pairs. Most of the literature focuses on learning only the objective function, as learning the constraint function (i.e., feasible regions) leads to nonconvex training programs. Motivated by this, we focus on … Read more
This paper introduces cuHALLaR, a GPU-accelerated implementation of the HALLaR method proposed in Monteiro et al. 2024 for solving large-scale semidefinite programming (SDP) problems. We demonstrate how our Julia-based implementation efficiently uses GPU parallelism through optimization of simple, but key, operations, including linear maps, adjoints, and gradient evaluations. Extensive numerical experiments across three problem classes—maximum … Read more
In nonsmooth optimization, a negative subgradient is not necessarily a descent direction, making the design of convergent descent methods based on zeroth-order and first-order information a challenging task. The well-studied bundle methods and gradient sampling algorithms construct descent directions by aggregating subgradients at nearby points in seemingly different ways, and are often complicated or lack … Read more
We propose AdaNAG, an adaptive accelerated gradient method based on Nesterov’s accelerated gradient method. AdaNAG is line-search-free, parameter-free, and achieves the accelerated convergence rates \( f(x_k) – f_\star = \mathcal{O}\left(1/k^2\right) \) and \( \min_{i\in\left\{1,\dots, k\right\}} \|\nabla f(x_i)\|^2 = \mathcal{O}\left(1/k^3\right) \) for \( L \)-smooth convex function \( f \). We provide a Lyapunov analysis for … Read more
We propose new adaptive strategies to compute stepsizes for the steepest descent method to solve unconstrained nonlinear multiobjective optimization problems without employing any linesearch procedure. The resulting algorithms can be applied to a wide class of nonconvex unconstrained multi-criteria optimization problems satisfying a global Lipschitz continuity condition imposed on the gradients of all objectives. In … Read more
Symmetry is a recurring feature in algorithms for monotone operator theory and convex optimization, particularly in problems involving the sum of two operators, as exemplified by the Peaceman–Rachford splitting scheme. However, in more general settings—such as composite optimization problems with three convex functions or structured convex-concave saddle-point formulations—existing algorithms often exhibit inherent asymmetry. In particular, … Read more
The Augmented Lagrangian Method (ALM), firstly proposed in 1969, remains a vital framework in large-scale constrained optimization. This paper addresses a linearly constrained composite convex minimization problem and presents a general proximal ALM that incorporates both Nesterov acceleration and relaxed acceleration, while enjoying a proximal-indefinite term. Under mild assumptions (potentially without requiring prior knowledge of … Read more