We show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the ...${\mathbb{}}^{}$${\mathbb{}}^{\mathrm{}}$${}^{}$

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, ...

We establish a general condition on the cost function to obtain uniqueness and Monge solutions in the multi-marginal optimal transport problem, under the assumption that a given collection of the marginals are absolutely continuous with respect to ...

The Frank–Wolfe (FW) method, which implements efficient linear oracles that minimize linear approximations of the objective function over a fixed compact convex set, has recently received much attention in the optimization and machine learning ...

We study piecewise affine policies for multi-stage adjustable robust optimization (ARO) problems with non-negative right-hand side uncertainty. First, we construct new dominating uncertainty sets and show how a multi-stage ARO problem can be ...

We investigate the concept of adjustability—the difference in objective values between two types of dynamic robust optimization formulations: one where (static) decisions are made before uncertainty realization, and one where uncertainty is ...

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer ...${}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$${}_{\mathrm{}}$

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with ...

This note provides a counterexample to a theorem announced in the last part of the paper (Vicente and Custódio Math Program 133:299–325, 2012). The counterexample involves an objective function $f:\mathbb{R}\to \mathbb{R}$ which satisfies all the assumptions required by ...${{}_{}}_{\mathbb{}}$${}_{}$${}_{}$${}_{}{}_{}$${}_{}$

We consider the oracle complexity of computing an approximate stationary point of a Lipschitz function. When the function is smooth, it is well known that the simple deterministic gradient method has finite dimension-free oracle complexity. ...

Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We introduce a ...${}_{\mathrm{}}$${}_{}$

In this paper, a special case of the generalized 4-block n-fold IPs is investigated, where $B_i=B$ and B has a rank at most 1. Such IPs, called almost combinatorial 4-block n-fold IPs, include the generalized n-fold IPs as a subcase. We are ...$$$\mathbf{}$$\mathbf{}$${}_{}$$$$\mathbf{}$$$${}_{}$$\mathcal{}$$\mathcal{}{}^{\mathrm{}\mathrm{}}{\stackrel{}{}}^{\mathrm{}}$$\stackrel{}{}$${}_{}$$$

We introduce a Bi-level OPTimization (BiOPT) framework for minimizing the sum of two convex functions, where one of them is smooth enough. The BiOPT framework offers three levels of freedom: (i) choosing the order p of the proximal term; (ii) ...$\mathrm{}$$\mathrm{}$$\mathcal{}{}^{\mathrm{}}$$\mathrm{}$

The network pricing problem (NPP) is a bilevel problem, where the leader optimizes its revenue by deciding on the prices of certain arcs in a graph, while expecting the followers (also known as the commodities) to choose a shortest path based on ...

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction ...

We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve ...

The truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation with minimum error measured by a unitarily invariant norm, has been applied to many domains such as biology, healthcare, among others, where ...

The minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $f(·)$ due to an ordering $\sigma$ of the items (...${}_{}{}_{}{}_{}$${}_{}$$$$\mathrm{}\frac{\mathrm{}{}_{}}{\mathrm{}}$${}_{}\frac{}{{}_{}}$$\mathrm{}{}_{}$$\mathrm{}\frac{\mathrm{}}{\mathrm{}}$$$$\frac{\mathrm{}}{\mathrm{}}$