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In this paper, we consider nonconvex optimization problems with nonsmooth nonconvex objective function and nonlinear equality constraints. We assume that both the objective function and the functional constraints can be separated into 2…
In this paper, we consider the linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose an inexact augmented Lagrangian (IAL) framework for…
(Renegar, 2016) introduced a novel approach to transforming generic conic optimization problems into unconstrained, uniformly Lipschitz continuous minimization. We introduce {\it radial transformations} generalizing these ideas, equipped…
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 consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from…
In this work we present two particular cases of the general duality result for linear optimisation problems over signed measures with infinitely many constraints in the form of integrals of functions with respect to the decision variables…
This article studies convex duality in stochastic optimization over finite discrete-time. The first part of the paper gives general conditions that yield explicit expressions for the dual objective in many applications in operations…
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in…
We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions…
Motivated by variational models in continuum mechanics, we introduce a novel algorithm to perform nonsmooth and nonconvex minimizations with linear constraints in Euclidean spaces. We show how this algorithm is actually a natural…
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…
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…
We are interested in solving convex optimization problems with large numbers of constraints. Randomized algorithms, such as random constraint sampling, have been very successful in giving nearly optimal solutions to such problems. In this…
We consider a general multi-agent convex optimization problem where the agents are to collectively minimize a global objective function subject to a global inequality constraint, a global equality constraint, and a global constraint set.…
This paper studies the continuous-time dynamics generated by control-theoretic Lagrangian methods for equality-constrained optimization. In particular, we consider dynamics induced by proportional-integral and feedback linearization…
This paper considers stochastic optimization problems with weakly convex objective and constraint functions. We propose Prox-PEP, a proximal method equipped with quadratic subproblems. To handle nonlinear equality constraints, we employ an…
Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work…
In recent years, constrained optimization has become increasingly relevant to the machine learning community, with applications including Neyman-Pearson classification, robust optimization, and fair machine learning. A natural approach to…
This paper studies the continuous-time dynamics of primal-dual algorithms for linearly constrained convex optimization problems and provides a quantitative convergence analysis using the Lyapunov functions. With the growing prevalence of…
In this paper, we focus on nonlinear infinite-norm minimization problems that have many applications, especially in computer science and operations research. We set a reliable Lagrangian dual aproach for solving this kind of problems in…