Discrete Optimal Transport with Independent Marginals is #P-Hard

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector … Read more

The polytope of binary sequences with bounded variation

We investigate the problem of optimizing a linear objective function over the set of all binary vectors of length n with bounded variation, where the latter is defined as the number of pairs of consecutive entries with different value. This problem arises naturally in many applications, e.g., in unit commitment problems or when discretizing binary … Read more

A line search based proximal stochastic gradient algorithm with dynamical variance reduction

Many optimization problems arising from machine learning applications can be cast as the minimization of the sum of two functions: the first one typically represents the expected risk, and in practice it is replaced by the empirical risk, and the other one imposes a priori information on the solution. Since in general the first term … Read more

Adaptive Third-Order Methods for Composite Convex Optimization

In this paper we propose third-order methods for composite convex optimization problems in which the smooth part is a three-times continuously differentiable function with Lipschitz continuous third-order derivatives. The methods are adaptive in the sense that they do not require the knowledge of the Lipschitz constant. Trial points are computed by the inexact minimization of … Read more

Exploiting Prior Function Evaluations in Derivative-Free Optimization

A derivative-free optimization (DFO) algorithm is presented. The distinguishing feature of the algorithm is that it allows for the use of function values that have been made available through prior runs of a DFO algorithm for solving prior related optimization problems. Applications in which sequences of related optimization problems are solved such that the proposed … Read more

A Novel Model for Transfer Synchronization in Transit Networks and a Lagrangian-based Heuristic Solution Method

To realize the benefits of network connectivity in transfer-based transit networks, it is critical to minimize transfer disutility for passengers by synchronizing timetables of intersecting routes. We propose a mixed-integer linear programming timetable synchronization model that incorporates new features, such as dwell time determination and vehicle capacity limit consideration, which have been largely overlooked in … Read more

Stochastic trust-region and direct-search methods: A weak tail bound condition and reduced sample sizing

Using tail bounds, we introduce a new probabilistic condition for function estimation in stochastic derivative-free optimization which leads to a reduction in the number of samples and eases algorithmic analyses. Moreover, we develop simple stochastic direct-search and trust-region methods for the optimization of a potentially non-smooth function whose values can only be estimated via stochastic … Read more

Semi-infinite models for equilibrium selection

In their seminal work `A General Theory of Equilibrium Selection in Games’ (The MIT Press, 1988) Harsanyi and Selten introduce the notion of payoff dominance to explain how players select some solution of a Nash equilibrium problem from a set of nonunique equilibria. We formulate this concept for generalized Nash equilibrium problems, relax payoff dominance … Read more

Retraction based Direct Search Methods for Derivative Free Riemannian Optimization

Direct search methods represent a robust and reliable class of algorithms for solving black-box optimization problems. In this paper, we explore the application of those strategies to Riemannian optimization, wherein minimization is to be performed with respect to variables restricted to lie on a manifold. More specifically, we consider classic and line search extrapolated variants … Read more

A harmonic framework for stepsize selection in gradient methods

We study the use of inverse harmonic Rayleigh quotients with target for the stepsize selection in gradient methods for nonlinear unconstrained optimization problems. This provides not only an elegant and flexible framework to parametrize and reinterpret existing stepsize schemes, but also gives inspiration for new flexible and tunable families of steplengths. In particular, we analyze … Read more