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Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce…
Higher-order tensor methods were recently proposed for minimizing smooth convex and nonconvex functions. Higher-order algorithms accelerate the convergence of the classical first-order methods thanks to the higher-order derivatives used in…
We consider minimizing the composite function that consists of a strongly convex function and a convex function. The fast dual proximal gradient (FDPG) method decreases the dual function with a rate $O(1/k^2)$, leading to a rate $O(1/k)$…
This paper is devoted to the design of efficient primal-dual algorithm (PDA) for solving convex optimization problems with known saddle-point structure. We present a new PDA with larger acceptable range of parameters and correction, which…
The linear primal-dual hybrid gradient (PDHG) method is a first-order method that splits convex optimization problems with saddle-point structure into smaller subproblems. Unlike those obtained in most splitting methods, these subproblems…
In this paper, we propose a variance-reduced primal-dual algorithm with Bregman distance for solving convex-concave saddle-point problems with finite-sum structure and nonbilinear coupling function. This type of problems typically arises in…
We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different…
First-order methods (FOMs) have been widely used for solving large-scale problems. A majority of existing works focus on problems without constraint or with simple constraints. Several recent works have studied FOMs for problems with…
This paper introduces a smoothed proximal Lagrangian method for minimizing a nonconvex smooth function over a convex domain with additional explicit convex nonlinear constraints. Two key features are 1) the proposed method is single-looped,…
This paper proposes a novel first-order algorithm that solves composite nonsmooth and stochastic convex optimization problem with function constraints. Most of the works in the literature provide convergence rate guarantees on the…
Conditional Gradient algorithms (aka Frank-Wolfe algorithms) form a classical set of methods for constrained smooth convex minimization due to their simplicity, the absence of projection steps, and competitive numerical performance. While…
We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness…
It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable,…
Convex optimization models find interesting applications, especially in signal/image processing and compressive sensing. We study some augmented convex models, which are perturbed by strongly convex functions, and propose a dual gradient…
We consider Lagrangian duality based approaches to design and analyze algorithms for online energy-efficient scheduling. First, we present a primal-dual framework. Our approach makes use of the Lagrangian weak duality and convexity to…
This work focuses on a class of general decentralized constraint-coupled optimization problems. We propose a novel nested primal-dual gradient algorithm (NPGA), which can achieve linear convergence under the weakest known condition, and its…
Dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure. Although the Augmented Lagrangian is computationally more stable than the ordinary Lagrangian, the…
In this paper we consider a class of convex conic programming. In particular, we first propose an inexact augmented Lagrangian (I-AL) method that resembles the classical I-AL method for solving this problem, in which the augmented…
In this paper, we propose a new primal-dual algorithm for minimizing $f(x) + g(x) + h(Ax)$, where $f$, $g$, and $h$ are proper lower semi-continuous convex functions, $f$ is differentiable with a Lipschitz continuous gradient, and $A$ is a…
Convex quadratic programming (QP) is an essential class of optimization problems with broad applications across various fields. Traditional QP solvers, typically based on simplex or barrier methods, face significant scalability challenges.…