Methods for Solving Extremal Problems in Practice

During the 20 th century there has been an incredible progress in solving theoretically hard problems in practice. One of the most prominent examples is the DPLL algorithm and its derivatives to solve the Boolean satisfiability problem, which can handle instances with millions of variables and clauses in reasonable time, notwithstanding the theoretical difficulty of solving the problem. Despite this progress, there are classes of problems that contain especially hard instances, which have remained open for decades despite their relative small size. One such class is the class of extremal problems, which typically involve finding a combinatorial object under some constraints (e.g, the search for Ramsey numbers). In recent years, a number of specialized methods have emerged to tackle extremal problems. Most of these methods are applied to a specific problem, despite the fact there is a great deal in common between different problems. Following a meticulous examination of these methods, we would like to extend them to handle general extremal problems. Further more, we would like to offer ways to exploit the general structure of extremal problems in order to develop constraints and symmetry breaking techniques which will, hopefully, improve existing tools. The latter point is of immense importance in the context of extremal problems, which often hamper existing tools when there is a great deal of symmetry in the search space, or when not enough is known of the problem structure. For example, if a graph is a solution to a problem instance, in many cases any isomorphic graph will also be a solution. In such cases, existing methods can usually be applied only if the model excludes symmetries

Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Minimum time control of the rocket attitude reorientation associated with orbit dynamics

In this paper, we investigate the minimal time problem for the guidance of a rocket, whose motion is described by its attitude kinematics and dynamics but also by its orbit dynamics. Our approach is based on a refined geometric study of the extremals coming from the application of the Pontryagin maximum principle. Our analysis reveals the existence of singular arcs of higher-order in the optimal synthesis, causing the occurrence of a chattering phenomenon, i.e., of an infinite number of switchings when trying to connect bang arcs with a singular arc. We establish a general result for bi-input control-affine systems, providing sufficient conditions under which the chattering phenomenon occurs. We show how this result can be applied to the problem of the guidance of the rocket. Based on this preliminary theoretical analysis, we implement efficient direct and indirect numerical methods, combined with numerical continuation, in order to compute numerically the optimal solutions of the problem.Comment: 33 pages, 14 figure

Tropical Kraus maps for optimal control of switched systems

Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.Comment: 15 page