5,914 research outputs found
Conditional Gradient Algorithms for Rank-One Matrix Approximations with a Sparsity Constraint
The sparsity constrained rank-one matrix approximation problem is a difficult
mathematical optimization problem which arises in a wide array of useful
applications in engineering, machine learning and statistics, and the design of
algorithms for this problem has attracted intensive research activities. We
introduce an algorithmic framework, called ConGradU, that unifies a variety of
seemingly different algorithms that have been derived from disparate
approaches, and allows for deriving new schemes. Building on the old and
well-known conditional gradient algorithm, ConGradU is a simplified version
with unit step size and yields a generic algorithm which either is given by an
analytic formula or requires a very low computational complexity. Mathematical
properties are systematically developed and numerical experiments are given.Comment: Minor changes. Final version. To appear in SIAM Revie
On convergence of the maximum block improvement method
Abstract. The MBI (maximum block improvement) method is a greedy approach to solving optimization problems where the decision variables can be grouped into a finite number of blocks. Assuming that optimizing over one block of variables while fixing all others is relatively easy, the MBI method updates the block of variables corresponding to the maximally improving block at each iteration, which is arguably a most natural and simple process to tackle block-structured problems with great potentials for engineering applications. In this paper we establish global and local linear convergence results for this method. The global convergence is established under the Lojasiewicz inequality assumption, while the local analysis invokes second-order assumptions. We study in particular the tensor optimization model with spherical constraints. Conditions for linear convergence of the famous power method for computing the maximum eigenvalue of a matrix follow in this framework as a special case. The condition is interpreted in various other forms for the rank-one tensor optimization model under spherical constraints. Numerical experiments are shown to support the convergence property of the MBI method
Multicast Multigroup Beamforming under Per-antenna Power Constraints
Linear precoding exploits the spatial degrees of freedom offered by
multi-antenna transmitters to serve multiple users over the same frequency
resources. The present work focuses on simultaneously serving multiple groups
of users, each with its own channel, by transmitting a stream of common symbols
to each group. This scenario is known as physical layer multicasting to
multiple co-channel groups. Extending the current state of the art in
multigroup multicasting, the practical constraint of a maximum permitted power
level radiated by each antenna is tackled herein. The considered per antenna
power constrained system is optimized in a maximum fairness sense. In other
words, the optimization aims at favoring the worst user by maximizing the
minimum rate. This Max-Min Fair criterion is imperative in multicast systems,
where the performance of all the receivers listening to the same multicast is
dictated by the worst rate in the group. An analytic framework to tackle the
Max-Min Fair multigroup multicasting scenario under per antenna power
constraints is therefore derived. Numerical results display the accuracy of the
proposed solution and provide insights to the performance of a per antenna
power constrained system.Comment: Presented in IEEE ICC 2014, Sydney, AUS. arXiv admin note:
substantial text overlap with arXiv:1406.755
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