@misc{Helmberg, author = {Helmberg, Christoph}, title = {A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6527}, number = {01-26}, abstract = {The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 0-1 programming problems by odd-cycle inequalities. We also consider separating odd-cycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for max-cut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.}, language = {en} } @misc{Steinbach, author = {Steinbach, Marc}, title = {Tree-Sparse Convex Programs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6340}, number = {01-08}, abstract = {Dynamic stochastic programs are prototypical for optimization problems with an inherent tree structure inducing characteristic sparsity patterns in the KKT systems of interior methods. We propose an integrated modeling and solution approach for such tree-sparse programs. Three closely related natural formulations are theoretically analyzed from a control-theoretic viewpoint and compared to each other. Associated KKT solution algorithms with linear complexity are developed and comparisons to other interior approaches and related problem formulations are discussed.}, language = {en} } @misc{HelmbergKiwiel, author = {Helmberg, Christoph and Kiwiel, K.C.}, title = {A Spectral Bundle Method with Bounds}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4256}, number = {SC-99-37}, abstract = {Semidefinite relaxations of quadratic 0-1 programming or graph partitioning problems are well known to be of high quality. However, solving them by primal-dual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equality-constrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. Encouraging preliminary computational results are reported.}, language = {en} } @misc{Helmberg, author = {Helmberg, Christoph}, title = {Semidefinite Programming}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4382}, number = {SC-99-49}, abstract = {Due to its many applications in control theory, robust optimization, combinatorial optimization and eigenvalue optimization, semidefinite programming had been in wide spread use even before the development of efficient algorithms brought it into the realm of tractability. Today it is one of the basic modeling and optimization tools along with linear and quadratic programming. Our survey is an introduction to semidefinite programming, its duality and complexity theory, its applications and algorithms.}, language = {en} } @misc{HarksHeinzPfetschetal., author = {Harks, Tobias and Heinz, Stefan and Pfetsch, Marc and Vredeveld, Tjark}, title = {Online Multicommodity Routing with Time Windows}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9654}, number = {07-22}, abstract = {We consider a multicommodity routing problem, where demands are released \emph{online} and have to be routed in a network during specified time windows. The objective is to minimize a time and load dependent convex cost function of the aggregate arc flow. First, we study the fractional routing variant. We present two online algorithms, called Seq and Seq\$^2\$. Our first main result states that, for cost functions defined by polynomial price functions with nonnegative coefficients and maximum degree~\$d\$, the competitive ratio of Seq and Seq\$^2\$ is at most \$(d+1)^{d+1}\$, which is tight. We also present lower bounds of \$(0.265\,(d+1))^{d+1}\$ for any online algorithm. In the case of a network with two nodes and parallel arcs, we prove a lower bound of \$(2-\frac{1}{2} \sqrt{3})\$ on the competitive ratio for Seq and Seq\$^2\$, even for affine linear price functions. Furthermore, we study resource augmentation, where the online algorithm has to route less demand than the offline adversary. Second, we consider unsplittable routings. For this setting, we present two online algorithms, called U-Seq and U-Seq\$^2\$. We prove that for polynomial price functions with nonnegative coefficients and maximum degree~\$d\$, the competitive ratio of U-Seq and U-Seq\$^2\$ is bounded by \$O{1.77^d\,d^{d+1}}\$. We present lower bounds of \$(0.5307\,(d+1))^{d+1}\$ for any online algorithm and \$(d+1)^{d+1}\$ for our algorithms. Third, we consider a special case of our framework: online load balancing in the \$\ell_p\$-norm. For the fractional and unsplittable variant of this problem, we show that our online algorithms are \$p\$ and \$O{p}\$ competitive, respectively. Such results where previously known only for scheduling jobs on restricted (un)related parallel machines.}, language = {en} } @misc{SagnolBlancoSauvage, author = {Sagnol, Guillaume and Blanco, Marco and Sauvage, Thibaut}, title = {Approximation Hierarchies for the cone of flow matrices}, issn = {1438-0064}, doi = {10.1016/j.endm.2018.02.002}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68424}, abstract = {Let \$G\$ be a directed acyclic graph with \$n\$ arcs, a source \$s\$ and a sink \$t\$. We introduce the cone \$K\$ of flow matrices, which is a polyhedral cone generated by the matrices \$1_P 1_P^T \in R^{n\times n}\$, where \$1_P\in R^n\$ is the incidence vector of the \$(s,t)\$-path \$P\$. Several combinatorial problems reduce to a linear optimization problem over \$K\$. This cone is intractable, but we provide two convergent approximation hierarchies, one of them based on a completely positive representation of \$K\$. We illustrate this approach by computing bounds for a maximum flow problem with pairwise arc-capacities.}, language = {en} }