7 research outputs found
The Quadratic Cycle Cover Problem: special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of
node-disjoint cycles visiting all the nodes such that the total sum of
interaction costs between consecutive arcs is minimized. In this paper we study
the linearization problem for the quadratic cycle cover problem and related
lower bounds.
In particular, we derive various sufficient conditions for the quadratic cost
matrix to be linearizable, and use these conditions to compute bounds. We also
show how to use a sufficient condition for linearizability within an iterative
bounding procedure. In each step, our algorithm computes the best equivalent
representation of the quadratic cost matrix and its optimal linearizable matrix
with respect to the given sufficient condition for linearizability. Further, we
show that the classical Gilmore-Lawler type bound belongs to the family of
linearization based bounds, and therefore apply the above mentioned iterative
reformulation technique. We also prove that the linearization vectors resulting
from this iterative approach satisfy the constant value property.
The best among here introduced bounds outperform existing lower bounds when
taking both quality and efficiency into account
SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning
We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a
node-disjoint cycle cover in a directed graph with minimum interaction cost
between successive arcs. We derive several semidefinite programming (SDP)
relaxations and use facial reduction to make these strictly feasible. We
investigate a nontrivial relationship between the transformation matrix used in
the reduction and the structure of the graph, which is exploited in an
efficient algorithm that constructs this matrix for any instance of the
problem. To solve our relaxations, we propose an algorithm that incorporates an
augmented Lagrangian method into a cutting plane framework by utilizing
Dykstra's projection algorithm. Our algorithm is suitable for solving SDP
relaxations with a large number of cutting planes. Computational results show
that our SDP bounds and our efficient cutting plane algorithm outperform other
QCCP bounding approaches from the literature. Finally, we provide several
SDP-based upper bounding techniques, among which a sequential Q-learning method
that exploits a solution of our SDP relaxation within a reinforcement learning
environment
Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization
Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
On solving the MAX-SAT using sum of squares
We consider semidefinite programming (SDP) approaches for solving the maximum
satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely
known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows
the potential of SDP also for other satisfiability problems, by being
competitive with some of the best solvers in the yearly MAX-SAT competition.
Our solver combines sum of squares (SOS) based SDP bounds and an efficient
parser within a branch & bound scheme.
On the theoretical side, we propose a family of semidefinite feasibility
problems, and show that a member of this family provides the rank two
guarantee. We also provide a parametric family of semidefinite relaxations for
the MAX-SAT, and derive several properties of monomial bases used in the SOS
approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an
elegant way. Moreover, we relate our SOS-SDP relaxations for the partial
MAX-SAT to the known SAT relaxations.Comment: 26 pages, 5 figures, 8 tables, 2 appendix page
The quadratic cycle cover problem:Special cases and efficient bounds
The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore–Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account