An interval branch and bound method for global Robust optimization

In this paper, we design a Branch and Bound algorithm based on interval arithmetic to address nonconvex robust optimization problems. This algorithm provides the exact global solution of such difficult problems arising in many real life applications. A code was developed in MatLab and was used to solve some robust nonconvex problems with few variables. This first numerical study shows the interest of this approach providing the global solution of such difficult robust nonconvex optimization problems.

     

Heuristics and lower bounds for minimizing fuel consumption in hybrid-electrical vehicles

In hybrid electric vehicles, the electrical powertrain system has multiple energy sources that it can gather power from to satisfy the propulsion power requested by the vehicle at each instant. This paper focusses on the minimization of the fuel consumption of such a vehicle, taking advantage of the different energy sources. Based on global optimization approaches, the proposed heuristics find solutions that best split the power requested between the multi-electrical sources available. A lower bounding procedure is introduced to validate the quality of the solutions. Computational results show a significant improvement over previous results from the literature in both the computing time and the quality of the solutions.     

The small hexagon and heptagon with maximum sum of distances between vertices

The hexagon and heptagon with unit diameter and maximum sum of Euclidean distances between vertices are determined by enumerating diameter configurations, and by using a branch and cut algorithm for nonconvex quadratic programming. Lower bounds on the value on this sum are presented for polygon with a larger number of vertices.     

The small octagon with longest perimeter

The convex octagon with unit diameter and maximum perimeter is determined. This answers an open question dating from 1922. The proof uses geometric reasoning and an interval arithmetic based global optimization algorithm to solve a series of non-linear and non-convex programs involving trigonometric functions.     

Finding largest small polygons with GloptiPoly

A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n ≥ 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.     

Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

Improving Interval Analysis Bounds by Translations

We explore how a simple linear change of variable affects the inclusion functions obtained with Interval Analysis methods. Univariate and multivariate polynomial test functions are considered, showing that translation-based methods improve considerably the bounds computed by standard inclusion functions. An Interval Branch-and-Bound method for global optimization is then implemented to compare the different procedures, showing that, although with times higher than those given by Taylor forms, the number of clusters and iterations is strongly reduced.     

A metaheuristic methodology based on the limitation of the memory of interval branch and bound algorithms

Branch and Bound Algorithms based on Interval Arithmetic permit to solve exactly continuous (as well as mixed) non-linear and non-convex global optimization problems. However, their intrinsic exponential time-complexities do not make it possible to solve some quite large problems. The idea proposed in this paper is to limit the memory available during the computations of such a global optimization code in order to find some efficient feasible solutions. By this way, we introduce a metaheuristic frame to develop some new heuristic global optimization algorithms based on an exact code. We show in this paper, with a small assumption about the sorting by breadth first of elements in the data structure, that the time-complexity of such metaheuristic algorithms becomes polynomial instead of exponential for the exact code. In order to validate our metaheuristic approach, some numerical experiments about constrained global optimization problems coming from the COCONUT library were solved using a heuristic which certifies an enclosure of the global minimum value. The objective is not to solve completely the problem or find a better solution, but it is to know what is the highest precision which can be guaranteed reliably with the available memory.     

A Branch and Bound algorithm for minimizing the energy consumption of an electrical vehicle

The problem we address in this paper, is the minimization of the energy consumption of an electrical vehicle achievable on a given driving cycle. This can be formulated as an optimal control problem with a discrete switch control. In this paper, we present a new formulation of this problem by taking into account the electrical and mechanical parts of the vehicle. By using prototype optimal control shooting techniques, we did not provide efficient solutions. Consequently, we focused on an operational method based on a Branch and Bound algorithm to solve this optimal switching control problem. Some numerical experiments validate our methodology.