Related papers: Convergence analysis of primal-dual augmented Lagr…
Mathematical optimization is the workhorse behind several aspects of modern robotics and control. In these applications, the focus is on constrained optimization, and the ability to work on manifolds (such as the classical matrix Lie…
Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible…
We propose a new bundle-based augmented Lagrangian framework for solving constrained convex problems. Unlike the classical (inexact) augmented Lagrangian method (ALM) that has a nested double-loop structure, our framework features a…
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…
This paper addresses a class of general nonsmooth and nonconvex composite optimization problems subject to nonlinear equality constraints. We assume that a part of the objective function and the functional constraints exhibit local…
In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the…
This paper analyzes the contraction of the primal-dual gradient optimization via contraction theory in the context of discrete-time updating dynamics. The contraction theory based on Riemannian manifolds is first established for convergence…
We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums…
In this work, we revisit a classical incremental implementation of the primal-descent dual-ascent gradient method used for the solution of equality constrained optimization problems. We provide a short proof that establishes the linear…
A stochastic linear quadratic (LQ) optimal control problem with a pointwise linear equality constraint on the terminal state is considered. A strong Lagrangian duality theorem is proved under a uniform convexity condition on the cost…
Necessary optimality conditions in Lagrangian form and the sequential minimization framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local…
This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and…
We study the problem of computing an optimal large language model (LLM) policy for the constrained alignment problem, where the goal is to maximize a primary reward objective while satisfying constraints on secondary utilities. Despite the…
Many algorithms in verification and automated reasoning leverage some form of duality between proofs and refutations or counterexamples. In most cases, duality is only used as an intuition that helps in understanding the algorithms and is…
We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
In this paper we consider a non-monotone (mixed) variational inequality model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle-point system for the VI model in question,…
We propose a new first-order primal-dual optimization framework for a convex optimization template with broad applications. Our optimization algorithms feature optimal convergence guarantees under a variety of common structure assumptions…
Augmented Lagrangian Methods (ALMs) are widely employed in solving constrained optimizations, and some efficient solvers are developed based on this framework. Under the quadratic growth assumption, it is known that the dual iterates and…
Based on the complete-lattice approach, a new Lagrangian duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued versions for the known scalar formulas involving infimum and supremum…