@misc{Pfetsch, author = {Pfetsch, Marc}, title = {A Branch-And-Cut for the Maximum Feasible Subsystem Problem}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8797}, number = {05-46}, abstract = {We present a branch-and-cut algorithm for the NP-hard maximum feasible subsystem problem: For a given infeasible linear inequality system, determine a feasible subsystem containing as many inequalities as possible. The complementary problem, where one has to remove as few inequalities as possible in order to render the system feasible, can be formulated as a set covering problem. The rows of this formulation correspond to irreducible infeasible subsystems, which can be exponentially many. The main issue of a branch-and-cut algorithm for MaxFS is to efficiently find such infeasible subsystems. We present three heuristics for the corresponding NP-hard separation problem and discuss further cutting planes. This paper contains an extensive computational study of our implementation on a variety of instances arising in a number of applications.}, language = {en} } @misc{Achterberg, author = {Achterberg, Tobias}, title = {SCIP - a framework to integrate Constraint and Mixed Integer Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7947}, number = {04-19}, abstract = {Constraint Programs and Mixed Integer Programs are closely related optimization problems originating from different scientific areas. Today's state-of-the-art algorithms of both fields have several strategies in common, in particular the branch-and-bound process to recursively divide the problem into smaller sub problems. On the other hand, the main techniques to process each sub problem are different, and it was observed that they have complementary strenghts. We propose a programming framework {\sffamily SCIP} that integrates techniques from both fields in order to exploit the strenghts of both, Constraint Programming and Mixed Integer Programming. In contrast to other proposals of recent years to combine both fields, {\sffamily SCIP} does not focus on easy implementation and rapid prototyping, but is tailored towards expert users in need of full, in-depth control and high performance.}, language = {en} } @misc{JokarPfetsch, author = {Jokar, Sadegh and Pfetsch, Marc}, title = {Exact and Approximate Sparse Solutions of Underdetermined Linear Equations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9488}, number = {07-05}, abstract = {In this paper, we empirically investigate the NP-hard problem of finding sparse solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a branch-and-cut approach via the maximum feasible subsystem problem to compute optimal solutions for small instances and investigate the uniqueness of the optimal solutions. We furthermore discuss five (modifications of) heuristics for this problem that appear in different parts of the literature. For small instances, the exact optimal solutions allow us to evaluate the quality of the heuristics, while for larger instances we compare their relative performance. One outcome is that the basis pursuit heuristic performs worse, compared to the other methods. Among the best heuristics are a method due to Mangasarian and a bilinear approach.}, language = {en} }