Angelika Wiegele

Regularization Methods for Semidefinite Programming

We introduce a new class of algorithms for solving linear semidefinite programming (SDP) problems. Our approach is based on classical tools from convex optimization such as quadratic regularization and augmented Lagrangian techniques. We study the theoretical properties and we show that practical implementations behave very well on some instances of SDP having a large number of constraints. We also show that the “boundary point method” from Povh, Rendl, and Wiegele [Computing, 78 (2006), pp. 277-286] is an instance of this class.

A Boundary Point Method to Solve Semidefinite Programs

We investigate the augmented Lagrangian penalty function approach to solve semidefinite programs. It turns out that this method generates iterates which lie on the boundary of the cone of semidefinite matrices which are driven to the affine subspace described by the linear equations defining the semidefinite program. We provide some computational experience with this method and show in particular, that it allows to compute the theta number of a graph to reasonably high accuracy for instances which are beyond reach by other methods.

Exact Algorithms for the Quadratic Linear Ordering Problem

The quadratic linear ordering problem naturally generalizes various optimization problems such as bipartite crossing minimization or the betweenness problem, which includes linear arrangement. These problems have important applications, e.g., in automatic graph drawing and computational biology. We present a new polyhedral approach to the quadratic linear ordering problem that is based on a linearization of the quadratic objective function. Our main result is a reformulation of the 3-dicycle inequalities using quadratic terms. After linearization, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We use this result both within a branch-and-cut algorithm and within a branch-and-bound algorithm based on semidefinite programming. Experimental results for bipartite crossing minimization show that this approach clearly outperforms other methods.

A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations

In this paper we present a method for finding exact solutions of the Max-Cut problem max xTLxsuch that xi¾? { i¾? 1,1}n. We use a semidefinite relaxation combined with triangle inequalities, which we solve with the bundle method. This approach is due to [12] and uses Lagrangian duality to get upper bounds with reasonable computational effort. The expensive part of our bounding procedure is solving the basic semidefinite programming relaxation of the Max-Cut problem. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to unconstrained quadratic 0-1 problems and to instances of the graph bisection problem. The experiments show, that our method nearly always outperforms all other approaches. Our algorithm, which is publicly accessible through the Internet, can solve virtually any instance with about 100 variables in a routine way.

Semidefinite relaxations for non-convex quadratic mixed-integer programming

We present semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems. These relaxations yield tight bounds and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded. In this case, the problem contains an infinite number of linear constraints; these constraints are separated dynamically. We use this approach as a bounding routine in an SDP-based branch-and-bound framework. In case of a convex objective function, the new SDP bound improves the bound given by the continuous relaxation of the problem. Numerical experiments show that our algorithm performs well on various types of non-convex instances.

Solving k-Way Graph Partitioning Problems to Optimality: The Impact of Semidefinite Relaxations and the Bundle Method

This paper is concerned with computing global optimal solutions for maximum k-cut problems. We improve on the SBC algorithm of Ghaddar, Anjos and Liers in order to compute such solutions in less time. We extend the design principles of the successful BiqMac solver for maximum 2-cut to the general maximum k-cut problem. As part of this extension, we investigate different ways of choosing variables for branching. We also study the impact of the separation of clique inequalities within this new framework and observe that it frequently reduces the number of subproblems considerably. Our computational results suggest that the proposed approach achieves a drastic speedup in comparison to SBC, especially when k=3. We also made a comparison with the orbitopal fixing approach of Kaibel, Peinhardt and Pfetsch. The results suggest that, while their performance is better for sparse instances and larger values of k, our proposed approach is superior for smaller k and for dense instances of medium size. Furthermore, we used CPLEX for solving the ILP formulation underlying the orbitopal fixing algorithm and conclude that especially on dense instances the new algorithm outperforms CPLEX by far.

Computational Approaches to Max-Cut

Max-Cut is one of the most studied combinatorial optimization problems because of its wide range of applications and because of its connections with other fields of discrete mathematics (see, e.g., the book by Deza and Laurent [10]). Like other interesting combinatorial optimization problems, Max-Cut is very simple to state.

Building a Quantum Network: How to Optimize Security and Expenses

Quantum key distribution (QKD) is regarded as a key-technology for the upcoming decades. Its practicability has been demonstrated through various experimental implementations. Wide-area QKD networks are a natural next step and should inherit the selling point of provable security. However, most research in QKD focuses on point-to-point connections, leaving end-to-end security to the trustworthiness of intermediate repeater nodes, thus defeating any formal proof of security: why bother outwitting QKD, if the repeater node is an easy prey, and an equally valuable target? We discuss methods of designing QKD networks with provable end-to-end security at provably optimized efforts. We formulate two optimization problems, along with investigations of computational difficulty: First, what is the minimal cost for a desired security? Second, how much security is achievable under given (budget-)constraints? Both problems permit applications of commercial optimization software, so allow taking a step towards an economic implementation of a globally spanning QKD network.

Using a Conic Bundle Method to Accelerate Both Phases of a Quadratic Convex Reformulation

We present algorithm MIQCR-CB that is an advancement of MIQCR. MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem (SDP); in the second phase, the equivalent formulation is solved by a standard solver. Because the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers even for medium-sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving (SDP) that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method, which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm. First, we apply MIQCR-CB to th...

Using a factored dual in augmented Lagrangian methods for semidefinite programming

Abstract In the context of augmented Lagrangian approaches for solving semidefinite programming problems, we investigate the possibility of eliminating the positive semidefinite constraint on the dual matrix by employing a factorization. Hints on how to deal with the resulting unconstrained maximization of the augmented Lagrangian are given. We further use the approximate maximum of the augmented Lagrangian with the aim of improving the convergence rate of alternating direction augmented Lagrangian frameworks. Numerical results are reported, showing the benefits of the approach.