## Positive polynomials on unbounded equality-constrained domains

Certificates of non-negativity are fundamental tools in optimization. A “certificate” is generally understood as an expression that makes the non-negativity of the function in question evident. Some classical certificates of non-negativity are Farkas Lemma and the S-lemma. The lift-and-project procedure can be seen as a certificate of non-negativity for affine functions over the union of … Read more

## Integration formulas via the Legendre-Fenchel Subdifferential of nonconvex functions

Starting from explicit expressions for the subdifferential of the conjugate function, we establish in the Banach space setting some integration results for the so-called epi-pointed functions. These results use the epsilon-subdifferential and the Legendre-Fenchel subdefferential of an appropriate weak lower semicontinuous (lsc) envelope of the initial function. We apply these integration results to the construction … Read more

## Optimization and homotopy methods for the Gibbs free energy of magmatic mixtures

In this paper we consider a mathematical model for magmatic mixtures based on the Gibbs free energy. Different reformulations of the problem are presented and some theoretical results about the existence and number of solutions are derived. Finally, two homotopy methods and a global optimization one are introduced and computationally tested. One of the homotopy … Read more

## Efficient Direct Multiple Shooting for Nonlinear Model Predictive Control on Long Horizons

We address direct multiple shooting based algorithms for nonlinear model predictive control, with a focus on problems with long prediction horizons. We describe different efficient multiple shooting variants with a computational effort that is only linear in the horizon length. Proposed techniques comprise structure exploiting linear algebra on the one hand, and approximation of derivative … Read more

## On Nesterov’s Smooth Chebyshev-Rosenbrock Function

We discuss a modification of the chained Rosenbrock function introduced by Nesterov, a polynomial of degree four of $n$ variables. Its only stationary point is the global minimizer with optimal value zero. An initial point is given such that any continuous piecewise linear descent path consists of at least an exponential number of \$0.72 \cdot … Read more

## The Reliable Hub-and-spoke Design Problem: Models and Algorithms

This paper presents a study on reliable single and multiple allocation hub-and-spoke network design problems where disruptions at hubs and the resulting hub unavailability can be mitigated by backup hubs and alternative routes. It builds nonlinear mixed integer programming models and presents linearized formulas. To solve those difficult problems, Lagrangian relaxation and Branch-and-Bound methods are … Read more

## A stochastic multiscale model for electricity generation capacity expansion

Long-term planning for electric power systems, or capacity expansion, has traditionally been modeled using simplified models or heuristics to approximate the short-term dynamics. However, current trends such as increasing penetration of intermittent renewable generation and increased demand response requires a coupling of both the long and short term dynamics. We present an efficient method for … Read more

## Distributionally robust workforce scheduling in call centers with uncertain arrival rates

Call center scheduling aims to set-up the workforce so as to meet target service levels. The service level depends on the mean rate of arrival calls, which fluctuates during the day and from day to day. The staff scheduling must adjust the workforce period per period during the day, but the flexibility in so doing … Read more

## Dependence of bilevel programming on irrelevant data

In 1997, Macal and Hurter have found that adding a constraint to the lower level problem, which is not active at the computed global optimal solution, can destroy global optimality. In this paper this property is reconsidered and it is shown that this solution remains locally optimal under inner semicontinuity of the original solution set … Read more

## On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual

Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi [K.G. Murty and S.N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39, no.2:117–129, 1987] that the strong membership problem for the copositive cone, that is deciding whether … Read more