Improved lower bounds for the 2-page crossing numbers of K(m,n) and K(n) via semidefinite programming

The crossing number of a graph is the minimal number of edge crossings achievable in a drawing of the graph in the plane. The crossing numbers of complete and complete bipartite graphs are long standing open questions. In a 2-page drawing of a graph, all vertices are drawn on a circle, and no edge may … Read more

A new robust cycle-based inventory control policy

In this paper, we propose a new robust cycle-based control policy for single installation inventory models with non-stationary uncertain demand. The policy is simple, flexible, easily implementable and preliminary numerical experiments suggest that the policy has very promising empirical performance. The policy can be used both when the excess demand is backlogged as well as … Read more

Partial Smoothness,Tilt Stability, and Generalized Hessians

We compare two recent variational-analytic approaches to second-order conditions and sensitivity analysis for nonsmooth optimization. We describe a broad setting where computing the generalized Hessian of Mordukhovich is easy. In this setting, the idea of tilt stability introduced by Poliquin and Rockafellar is equivalent to a classical smooth second-order condition. Article Download View Partial Smoothness,Tilt … Read more

Subdifferential of the conjugate function in general Banach spaces

We give explicit formulas for the subdifferential set of the conjugate of non necessarily convex functions defined on general Banach spaces. Even if such a subdifferential mapping takes its values in the bidual space, we show that up to a weak** closure operation it is still described by using only elements of the initial space … Read more

Partially affine control problems: second order conditions and a well-posed shooting algorithm

This paper deals with optimal control problems for systems that are affine in one part of the control variables and nonlinear in the rest of the control variables. We have finitely many equality and inequality constraints on the initial and final states. First we obtain second order necessary and sufficient conditions for weak optimality. Afterwards, … Read more

A well-posed shooting algorithm for optimal control problems with singular arcs

In this article we establish for the first time the well-posedness of the shooting algorithm applied to optimal control problems for which all control variables enter linearly in the Hamil- tonian. We start by investigating the case having only initial-final state constraints and free control variable, and afterwards we deal with control bounds. The shooting … Read more

Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations

In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function J in the sense of strong solutions. This means that the function J growths quadratically over all feasible controls whose associated state is … Read more

A robust Kantorovich’s theorem on inexact Newton method with relative residual error tolerance

We prove that under semi-local assumptions, the inexact Newton method with a fixed relative residual error tolerance converges Q-linearly to a zero of the non-linear operator under consideration. Using this result we show that Newton method for minimizing a self-concordant function or to find a zero of an analytic function can be implemented with a … Read more

Improved Bounds for Large Scale Capacitated Arc Routing Problem

The Capacitated Arc Routing Problem (CARP) stands among the hardest combinatorial problems to solve or to find high quality solutions. This becomes even more true when dealing with large instances. This paper investigates methods to improve on lower and upper bounds of instances on graphs with over two hundred vertices and three hundred edges, dimensions … Read more

A semi-discrete in time approximation for a model first order-finite horizon mean field game problem

In this article we consider a model first order mean field game problem, introduced by J.M. Lasry and P.L. Lions. Its solution $(v,m)$ can be obtained as the limit of the solutions of the second order mean field game problems, when the \textit{noise} parameter tends to zero. We propose a semi-discrete in time approximation of … Read more