3,425 research outputs found
Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems
Optimization methods are at the core of many problems in signal/image
processing, computer vision, and machine learning. For a long time, it has been
recognized that looking at the dual of an optimization problem may drastically
simplify its solution. Deriving efficient strategies which jointly brings into
play the primal and the dual problems is however a more recent idea which has
generated many important new contributions in the last years. These novel
developments are grounded on recent advances in convex analysis, discrete
optimization, parallel processing, and non-smooth optimization with emphasis on
sparsity issues. In this paper, we aim at presenting the principles of
primal-dual approaches, while giving an overview of numerical methods which
have been proposed in different contexts. We show the benefits which can be
drawn from primal-dual algorithms both for solving large-scale convex
optimization problems and discrete ones, and we provide various application
examples to illustrate their usefulness
A path following algorithm for the graph matching problem
We propose a convex-concave programming approach for the labeled weighted
graph matching problem. The convex-concave programming formulation is obtained
by rewriting the weighted graph matching problem as a least-square problem on
the set of permutation matrices and relaxing it to two different optimization
problems: a quadratic convex and a quadratic concave optimization problem on
the set of doubly stochastic matrices. The concave relaxation has the same
global minimum as the initial graph matching problem, but the search for its
global minimum is also a hard combinatorial problem. We therefore construct an
approximation of the concave problem solution by following a solution path of a
convex-concave problem obtained by linear interpolation of the convex and
concave formulations, starting from the convex relaxation. This method allows
to easily integrate the information on graph label similarities into the
optimization problem, and therefore to perform labeled weighted graph matching.
The algorithm is compared with some of the best performing graph matching
methods on four datasets: simulated graphs, QAPLib, retina vessel images and
handwritten chinese characters. In all cases, the results are competitive with
the state-of-the-art.Comment: 23 pages, 13 figures,typo correction, new results in sections 4,5,
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
On complexity of optimized crossover for binary representations
We consider the computational complexity of producing the best possible
offspring in a crossover, given two solutions of the parents. The crossover
operators are studied on the class of Boolean linear programming problems,
where the Boolean vector of variables is used as the solution representation.
By means of efficient reductions of the optimized gene transmitting crossover
problems (OGTC) we show the polynomial solvability of the OGTC for the maximum
weight set packing problem, the minimum weight set partition problem and for
one of the versions of the simple plant location problem. We study a connection
between the OGTC for linear Boolean programming problem and the maximum weight
independent set problem on 2-colorable hypergraph and prove the NP-hardness of
several special cases of the OGTC problem in Boolean linear programming.Comment: Dagstuhl Seminar 06061 "Theory of Evolutionary Algorithms", 200
Scalable Robust Kidney Exchange
In barter exchanges, participants directly trade their endowed goods in a
constrained economic setting without money. Transactions in barter exchanges
are often facilitated via a central clearinghouse that must match participants
even in the face of uncertainty---over participants, existence and quality of
potential trades, and so on. Leveraging robust combinatorial optimization
techniques, we address uncertainty in kidney exchange, a real-world barter
market where patients swap (in)compatible paired donors. We provide two
scalable robust methods to handle two distinct types of uncertainty in kidney
exchange---over the quality and the existence of a potential match. The latter
case directly addresses a weakness in all stochastic-optimization-based methods
to the kidney exchange clearing problem, which all necessarily require explicit
estimates of the probability of a transaction existing---a still-unsolved
problem in this nascent market. We also propose a novel, scalable kidney
exchange formulation that eliminates the need for an exponential-time
constraint generation process in competing formulations, maintains provable
optimality, and serves as a subsolver for our robust approach. For each type of
uncertainty we demonstrate the benefits of robustness on real data from a
large, fielded kidney exchange in the United States. We conclude by drawing
parallels between robustness and notions of fairness in the kidney exchange
setting.Comment: Presented at AAAI1
SiGMa: Simple Greedy Matching for Aligning Large Knowledge Bases
The Internet has enabled the creation of a growing number of large-scale
knowledge bases in a variety of domains containing complementary information.
Tools for automatically aligning these knowledge bases would make it possible
to unify many sources of structured knowledge and answer complex queries.
However, the efficient alignment of large-scale knowledge bases still poses a
considerable challenge. Here, we present Simple Greedy Matching (SiGMa), a
simple algorithm for aligning knowledge bases with millions of entities and
facts. SiGMa is an iterative propagation algorithm which leverages both the
structural information from the relationship graph as well as flexible
similarity measures between entity properties in a greedy local search, thus
making it scalable. Despite its greedy nature, our experiments indicate that
SiGMa can efficiently match some of the world's largest knowledge bases with
high precision. We provide additional experiments on benchmark datasets which
demonstrate that SiGMa can outperform state-of-the-art approaches both in
accuracy and efficiency.Comment: 10 pages + 2 pages appendix; 5 figures -- initial preprin
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