Pierre Hansen

On the Geometry of Nash Equilibria and Correlated Equilibria

Abstract.It is well known that the set of correlated equilibrium distributions of an n-player noncooperative game is a convex polytope that includes all the Nash equilibrium distributions. We demonstrate an elementary yet surprising result: the Nash equilibria all lie on the boundary of the polytope.

Finding shortest paths in the plane in the presence of barriers to travel (for any lp - norm)

Abstract This paper develops an efficient algorithm for the computation of the shortest paths between given sets of points (origins and destinations) in the plane, when these paths are constrained not to cross any of a finite set of polygonal (open or closed) barriers. It is proved that when distances are measured by an 1p - norm with 1 It is also shown that optimal solutions when distances are measured according to the rectilinear or max-norm (i.e. lp-norm with p = 1 or p = ∞) can be deduced from the results of the algorithm.

Boundary uniqueness of fuseness

It is shown that a geometrically planar fusene is uniquely determined by its boundary edge code. Surprisingly, the same conclusion is not true in general but holds for geometrically planar and non-planar fusenes with at most 25 hexagons, except for two particular cases. In addition, it is proved that two fusenes with the same boundary edge code have the same number of hexagons.

Some Further Results on Monotonicity in Globally Optimal Design

Many problems of globally optimal design have been solved by using monotonicity analysis. Extended monotonicity principles are obtained by exploiting nonstrict monotonicity of the objective function and of the constraints. Ways to detect monotonicity and branching rules using monotonicity results are also discussed. Several illustrative examples from the literature are solved in an easier way than previously.

Uniquely solvable quadratic boolean equations

Quadratic boolean equations with a unique solution are characterized. A linear-time algorithm is proposed to recognize them.

A Framework for Algorithms in Globally Optimal Design

Many problems of globally optimal design have been solved in the literature using monotonicity analysis and a variety of tests, often applied in an ad hoc way. These tests are developed here, expressed mathematically and classified according to the conclusions they yield. Moreover, many new tests, similar to those used in combinatorial optimization, are presented. Finally, a general framework is proposed in which branch-and-bound algorithms for globally optimal design problems can be expressed

Simple Polygons of Maximum Perimeter Contained in a Unit Disk

A polygon is said to be simple if the only points of the plane belonging to two of its edges are its vertices. We answer the question of finding, for a given integer n, a simple n-sided polygon contained in a disk of radius 1 that has the longest perimeter. When n is even, the optimal solution is arbitrarily close to a line segment of length 2n. When n is odd, the optimal solution is arbitrarily close to an isosceles triangle.

Industrial Applications of the Variable Neighborhood Search Metaheuristic

Variable Neighborhood Search (VNS) is a recent metaheuristic which exploits systematically the idea of change of neighborhood within the search. After recalling its basic scheme, three industrial applications are presented: a pipeline design problem, the pooling problem and the spread spectrum radar polyphase code design problem. They illustrate the simplicity, versatility and efficiency of VNS.

An Automated Procedure for Globally Optimal Design

An experimental computer system for globally optimal design, called BAGOP, is discussed. This new tool uses the computer algebra system MACSYMA to implement a variety of tests and branching rules in a flexible branch-and-bound framework. Many problems of globally optimal design have been solved by BAGOP, some of them for the first time, and most of them for the first time in an entirely automated way

Polynomial algorithms for nested univariate clustering

Clique partitioning in Euclidean space Rn consists in finding a partition of a given set of N points into M clusters in order to minimize the sum of within-cluster interpoint distances. For n = 1 clusters need not consist of consecutive points on a line but have a nestedness property. Exploiting this property, an O(N5M2) dynamic programming algorithm is proposed. A θ(N) algorithm is also given for the case M = 2.