26,236 research outputs found
New solution approaches for the quadratic assignment problem
MSc., Faculty of Science, University of the Witwatersrand, 2011A vast array of important practical problems, in many di erent elds, can be modelled and
solved as quadratic assignment problems (QAP). This includes problems such as university
campus layout, forest management, assignment of runners in a relay team, parallel and
distributed computing, etc. The QAP is a di cult combinatorial optimization problem
and solving QAP instances of size greater than 22 within a reasonable amount of time
is still challenging. In this dissertation, we propose two new solution approaches to the
QAP, namely, a Branch-and-Bound method and a discrete dynamic convexized method.
These two methods use the standard quadratic integer programming formulation of the
QAP. We also present a lower bounding technique for the QAP based on an equivalent
separable convex quadratic formulation of the QAP. We nally develop two di erent new
techniques for nding initial strictly feasible points for the interior point method used in
the Branch-and-Bound method. Numerical results are presented showing the robustness
of both methods
Continuous optimization methods for convex mixed-integer nonlinear programming
The topic of this dissertation is the design of fast branch-and-bound algorithms that use intelligently adapted approaches from continuous optimization for solving convex mixed-integer nonlinear programming problems. This class of optimization problems is NP-hard and polynomial-time algorithms for these problems are therefore unlikely to exist (unless P=NP). The importance of this class is highlighted by the fact that many real-world applications can be modeled as a (convex) mixed-integer nonlinear programming problem. Currently, there are several standard techniques such as outer approximation that are used within recent state-of-the-art software. Although all these algorithms include sophisticated
improvements such as primal heuristics and effective preprocessing, they do not take into account the large gap between the algorithmic performance of NLP and IP solvers. While NLP solvers are well-engineered for large-scale problems, MIP problems of similar sizes are by far harder to solve in practice. Therefore, when using NLP techniques within MIP solvers, these NLP algorithms have to be adjusted to handle small-size instances effectively.
Taking this problem into account, we present three branch-and-bound algorithms, based on a former work by Buchheim et al. (2012) on unconstrained convex quadratic integer programming problems. The main strategies used within this branch-andbound framework include extensive preprocessing and fast incremental computations, aiming at a very fast enumeration of the nodes. The first algorithm we present is designed to solve convex quadratic mixed-integer programming problems with linear inequality constraints and is based on a new feasible active set algorithm applied to the dual of the continuous relaxation. This active set algorithm is tailored for the continuous problem and fully exploits its structure. Furthermore, a warmstarting procedure is used to reduce the number of active set iterations per node. The second algorithm we introduce is an approach called quadratic outer approximation for solving box-constrained convex mixed-integer nonlinear programming problems. It extends the classical outer approximation by using quadratic underestimators leading to a faster convergence in practice. Finally, the last algorithm we devise is aimed at a class of mean-risk portfolio optimization problems that can be modeled as convex mixed-integer programming problems with a single linear budget constraint. For this application we propose a branch-and-bound scheme using a modified Frank-Wolfe type algorithm to solve the node relaxations. Similarly to the branch-and-bound algorithms mentionded above we exploit the simplicity of the relaxations to enumerate the nodes as quickly as possible rather than focussing on strong dual bounds.
We implemented all three algorithms and compared their performance with several state-of-the art approaches. Our extensive computational studies show that all new approaches presented in this thesis are able to effectively solve large classes of real-world instances
Exact methods for nonlinear combinatorial optimization
We consider combinatorial optimization problems with nonlinear objective functions.
Solution approaches for this class of problems proposed so far are either
highly problem-specific or they apply generic algorithms for constrained nonlinear
optimization, which often does not yield satisfactory results in practice.
Our aim is to develop, implement and experimentally evaluate exact algorithms
that address the nonlinearity of the objective function and at the same time exploit
the underlying combinatorial structure of the problem. To this end we follow
two approaches. The first combines good polyhedral descriptions of the objective
function and the feasible set in a branch and cut-algorithm. The second approach
is based on Lagrangean decomposition. By decomposing the original problem into
an unconstrained nonlinear problem and a linear combinatorial problem, we are
able to compute strong dual bounds for the optimal value. The computation of
lower bounds is then embedded into a branch and bound-algorithm. For many
applications there already exist efficient algorithms for the combinatorial subproblem,
thus an important aspect of this thesis is the study of the corresponding
unconstrained nonlinear subproblems.
Both approaches have the advantage that they can easily be adapted to a wide
range of nonlinear combinatorial problems.We devise both polyhedral and decomposition-
based algorithms for submodular applications from wireless network design
and portfolio optimization and evaluate their performance experimentally.
Exploiting the equivalence between unconstrained binary quadratic optimization
and the maximum cut problem gives rise to a branch and cut-algorithm for
quadratic combinatorial problems which we use to compute optimal layouts of
tanglegrams, an application from computational biology. Additionally we study
the effect of quadratic reformulation of linear constraints, both theoretically and
experimentally. The last class of nonlinear combinatorial problems we consider
are two-scenario problems. Here we propose a new technique to compute lower
bounds in the unconstrained subproblem of the decomposition. Our computational
study of the two-scenario minimum spanning tree problem shows that
the new Lagrangean decomposition-based algorithm is able to solve significantly
larger instances than the standard linearization approach
Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference
We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF
inference problems. The core of our method is a very efficient bounding
procedure, which combines scalable semidefinite programming (SDP) and a
cutting-plane method for seeking violated constraints. In order to further
speed up the computation, several strategies have been exploited, including
model reduction, warm start and removal of inactive constraints.
We analyze the performance of the proposed method under different settings,
and demonstrate that our method either outperforms or performs on par with
state-of-the-art approaches. Especially when the connectivities are dense or
when the relative magnitudes of the unary costs are low, we achieve the best
reported results. Experiments show that the proposed algorithm achieves better
approximation than the state-of-the-art methods within a variety of time
budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 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
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
A fast branch-and-bound algorithm for non-convex quadratic integer optimization subject to linear constraints using ellipsoidal relaxations
We propose two exact approaches for non-convex quadratic integer minimization subject to linear constraints where lower bounds are computed by considering ellipsoidal relaxations of the feasible set. In the first approach, we intersect the ellipsoids with the feasible linear subspace. In the second approach we penalize exactly the linear constraints. We investigate the connection between both approaches theoretically. Experimental results show that the penalty approach significantly outperforms CPLEX on problems with small or medium size variable domains. © 2015 Elsevier B.V. All rights reserved
Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem
Alternating current optimal power flow (AC OPF) is one of the most
fundamental optimization problems in electrical power systems. It can be
formulated as a semidefinite program (SDP) with rank constraints. Solving AC
OPF, that is, obtaining near optimal primal solutions as well as high quality
dual bounds for this non-convex program, presents a major computational
challenge to today's power industry for the real-time operation of large-scale
power grids. In this paper, we propose a new technique for reformulation of the
rank constraints using both principal and non-principal 2-by-2 minors of the
involved Hermitian matrix variable and characterize all such minors into three
types. We show the equivalence of these minor constraints to the physical
constraints of voltage angle differences summing to zero over three- and
four-cycles in the power network. We study second-order conic programming
(SOCP) relaxations of this minor reformulation and propose strong cutting
planes, convex envelopes, and bound tightening techniques to strengthen the
resulting SOCP relaxations. We then propose an SOCP-based spatial
branch-and-cut method to obtain the global optimum of AC OPF. Extensive
computational experiments show that the proposed algorithm significantly
outperforms the state-of-the-art SDP-based OPF solver and on a simple personal
computer is able to obtain on average a 0.71% optimality gap in no more than
720 seconds for the most challenging power system instances in the literature
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