31 research outputs found
Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming
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.Comment: 7 page
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). 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 , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving 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
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 in, e.g., 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, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We exploit this result both within a branch-and-cut algorithm and within an SDP-based branch-and-bound algorithm. Experimental results for bipartite crossing minimization show that this approach clearly outperforms 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 in, e.g., 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, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We exploit this result both within a branch-and-cut algorithm and within an SDP-based branch-and-bound algorithm. Experimental results for bipartite crossing minimization show that this approach clearly outperforms other methods
ON THE COMPLEXITY OF SEMIDEFINITE PROGRAMS ARISING IN POLYNOMIAL OPTIMIZATION
In this paper we investigate matrix inequalities which hold irrespective of the size of the matrices involved, and explain how the search for such inequalities can be implemented as a semidefinite program (SDP). We provide a comprehensive discussion of the time complexity of these SDPs
Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization
Alternating direction methods of multipliers (ADMMs) are popular approaches
to handle large scale semidefinite programs that gained attention during the
past decade. In this paper, we focus on solving doubly nonnegative programs
(DNN), which are semidefinite programs where the elements of the matrix
variable are constrained to be nonnegative. Starting from two algorithms
already proposed in the literature on conic programming, we introduce two new
ADMMs by employing a factorization of the dual variable.
It is well known that first order methods are not suitable to compute high
precision optimal solutions, however an optimal solution of moderate precision
often suffices to get high quality lower bounds on the primal optimal objective
function value. We present methods to obtain such bounds by either perturbing
the dual objective function value or by constructing a dual feasible solution
from a dual approximate optimal solution. Both procedures can be used as a
post-processing phase in our ADMMs.
Numerical results for DNNs that are relaxations of the stable set problem are
presented. They show the impact of using the factorization of the dual variable
in order to improve the progress towards the optimal solution within an
iteration of the ADMM. This decreases the number of iterations as well as the
CPU time to solve the DNN to a given precision. The experiments also
demonstrate that within a computationally cheap post-processing, we can compute
bounds that are close to the optimal value even if the DNN was solved to
moderate precision only. This makes ADMMs applicable also within a
branch-and-bound algorithm.Comment: 24 pages, 8 figure
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