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Lagrangian-based methods are classical methods for solving convex optimization problems with equality constraints. We present novel prediction-correction frameworks for such methods and their variants, which can achieve $O(1/k)$ non-ergodic…
In the past years, augmented Lagrangian methods have been successfully applied to several classes of non-convex optimization problems, inspiring new developments in both theory and practice. In this paper we bring most of these recent…
Non-convex functional constrained optimization problems have gained substantial attention in machine learning and data science, addressing broad requirements that typically go beyond the often performance-centric objectives. An influential…
Trajectory optimization is an efficient approach for solving optimal control problems for complex robotic systems. It relies on two key components: first the transcription into a sparse nonlinear program, and second the corresponding solver…
This paper considers the analysis of continuous time gradient-based optimization algorithms through the lens of nonlinear contraction theory. It demonstrates that in the case of a time-invariant objective, most elementary results on…
This paper proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a \textit{fixed} time from any given initial condition for unconstrained optimization, constrained…
In this paper, we consider the minimization of a nonsmooth nonconvex objective function $f(x)$ over a closed convex subset $\mathcal{X}$ of $\mathbb{R}^n$, with additional nonsmooth nonconvex constraints $c(x) = 0$. We develop a unified…
We propose a modified primal-dual method for general convex optimization problems with changing constraints. We obtain properties of Lagrangian saddle points for these problems which enable us to establish convergence of the proposed…
In this paper, a projected primal-dual gradient flow of augmented Lagrangian is presented to solve convex optimization problems that are not necessarily strictly convex. The optimization variables are restricted by a convex set with…
This paper describes new results linking constrained optimization theory and nonlinear contraction analysis. Generalizations of Lagrange parameters are derived based on projecting system dynamics on the tangent space of possibly…
A novel Follow-the-Perturbed-Leader type algorithm is proposed and analyzed for solving general long-term constrained optimization problems in an online manner, where the target and constraint functions are oblivious adversarially generated…
This paper studies equality-constrained minimization problems through the lens of feedback control. We introduce a unified control-theoretic framework by showing that a PID feedback law acting on the dual variable induces the PID…
Foundations of a new projection-based model reduction approach for convection dominated nonlinear fluid flows are summarized. In this method the evolution of the flow is approximated in the Lagrangian frame of reference. Global basis…
From the perspective of control theory, the gradient descent optimization methods can be regarded as a dynamic system where various control techniques can be designed to enhance the performance of the optimization method. In this paper, we…
In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a…
Conservation principles like conservation of charge or energy provide a natural way to couple and constrain different physical variables. In this letter, we propose a dynamical system model that exploits these constraints for solving…
Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics (Aug-PDGD)…
This paper is concerned with augmented Lagrangian methods for the treatment of fully convex composite optimization problems. We extend the classical relationship between augmented Lagrangian methods and the proximal point algorithm to the…
We study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term. We introduce an auxiliary variable to…
For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods sequentially minimize a Lagrangian function subject to linearized constraints. These methods converge rapidly near a solution but may not be…