Related papers: Convergence analysis of primal-dual augmented Lagr…
We develop a Lagrange multiplier theory for nonconvex set-valued optimization problems under Lipschitz-type regularity conditions. Instead of classical continuous linear functionals, we introduce closed convex processes -- set-valued…
The authors study an infinite dimensional duality theory finalized to obtain the existence of a strong duality between a convex optimization problem connected with the management of vaccinations and its Lagrange dual. Specifically, the…
We study the Bregman Augmented Lagrangian method (BALM) for solving convex problems with linear constraints. For classical Augmented Lagrangian method, the convergence rate and its relation with the proximal point method is well-understood.…
First-order methods have been studied for nonlinear constrained optimization within the framework of the augmented Lagrangian method (ALM) or penalty method. We propose an improved inexact ALM (iALM) and conduct a unified analysis for…
Optimization problems with convex quadratic cost and polyhedral constraints are ubiquitous in signal processing, automatic control and decision-making. We consider here an enlarged problem class that allows to encode logical conditions and…
Continuous time primal-dual gradient dynamics that find a saddle point of a Lagrangian of an optimization problem have been widely used in systems and control. While the global asymptotic stability of such dynamics has been well-studied, it…
A large number of objectives have been proposed to train latent variable generative models. We show that many of them are Lagrangian dual functions of the same primal optimization problem. The primal problem optimizes the mutual information…
Stochastic gradient methods (SGMs) have been widely used for solving stochastic optimization problems. A majority of existing works assume no constraints or easy-to-project constraints. In this paper, we consider convex stochastic…
In this paper, we aim at unifying, simplifying and improving the convergence rate analysis of Lagrangian-based methods for convex optimization problems. We first introduce the notion of nice primal algorithmic map, which plays a central…
This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous…
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…
Lagrangian decomposition (LD) is a relaxation method that provides a dual bound for constrained optimization problems by decomposing them into more manageable sub-problems. This bound can be used in branch-and-bound algorithms to prune the…
The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying…
We consider convex optimization problems with prioritized equality constraints, which may be infeasible. In many applications, such as network optimization and image reconstruction, it is often desirable to compute solutions that satisfy…
In this paper, we study saddle point (SP) problems, focusing on convex-concave optimization involving functions that satisfy either two-sided quadratic functional growth (QFG) or two-sided quadratic gradient growth (QGG)--novel conditions…
In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting…
We analyze the convergence behaviour of a recently proposed algorithm for regularized estimation called Dual Augmented Lagrangian (DAL). Our analysis is based on a new interpretation of DAL as a proximal minimization algorithm. We…
We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is…
We propose smoothed primal-dual algorithms for solving stochastic and smooth nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single stochastic gradient based on one…
We provide a correction to the sufficient conditions under which closed-form expressions for the optimal Lagrange multiplier are provided in arXiv:2112.13138 [math.OC]. We first present a simple counterexample where the original conditions…