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We study the problem of minimizing a sum of local objective convex functions over a network of processors/agents. This problem naturally calls for distributed optimization algorithms, in which the agents cooperatively solve the problem…
This paper studies generalized semi-infinite programs (GSIPs) given by polynomials. We propose a hierarchy of polynomial optimization relaxations to solve them. They are based on Lagrange multiplier expressions and polynomial extensions.…
We present a new algorithm for solving optimization problems with objective functions that are the sum of a smooth function and a (potentially) nonsmooth regularization function, and nonlinear equality constraints. The algorithm may be…
We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest…
We consider convex optimization problems with the objective function having Lipshitz-continuous $p$-th order derivative, where $p\geq 1$. We propose a new tensor method, which closes the gap between the lower…
Linear optimization is many times algorithmically simpler than non-linear convex optimization. Linear optimization over matroid polytopes, matching polytopes and path polytopes are example of problems for which we have simple and efficient…
A polyhedral convex set optimization problem is given by a set-valued objective mapping from the $n$-dimensional to the $q$-dimensional Euclidean space whose graph is a convex polyhedron. This problem can be seen as the most elementary…
This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization…
Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from…
We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation…
We propose a hierarchical architecture for efficiently computing high-quality solutions to structured mixed-integer programs (MIPs). To reduce computational effort, our approach decouples the original problem into a higher level problem and…
We develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear…
For a large class of optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs), we establish a dichotomy on the number of levels of the Lasserre hierarchy of semi-definite programs…
Standard gradient descent methods are susceptible to a range of issues that can impede training, such as high correlations and different scaling in parameter space.These difficulties can be addressed by second-order approaches that apply a…
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…
We revisit the problem of minimizing a given polynomial $f$ on the hypercube $[-1,1]^n$. Lasserre's hierarchy (also known as the moment- or sum-of-squares hierarchy) provides a sequence of lower bounds $\{f_{(r)}\}_{r \in \mathbb N}$ on the…
In this paper, we design and analyze a new family of adaptive subgradient methods for solving an important class of weakly convex (possibly nonsmooth) stochastic optimization problems. Adaptive methods that use exponential moving averages…
Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…
This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex…