7,502 research outputs found
An Algorithm for Global Maximization of Secrecy Rates in Gaussian MIMO Wiretap Channels
Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap
channels is considered. While this channel has attracted a significant
attention recently and a number of results have been obtained, including the
proof of the optimality of Gaussian signalling, an optimal transmit covariance
matrix is known for some special cases only and the general case remains an
open problem. An iterative custom-made algorithm to find a globally-optimal
transmit covariance matrix in the general case is developed in this paper, with
guaranteed convergence to a \textit{global} optimum. While the original
optimization problem is not convex and hence difficult to solve, its minimax
reformulation can be solved via the convex optimization tools, which is
exploited here. The proposed algorithm is based on the barrier method extended
to deal with a minimax problem at hand. Its convergence to a global optimum is
proved for the general case (degraded or not) and a bound for the optimality
gap is given for each step of the barrier method. The performance of the
algorithm is demonstrated via numerical examples. In particular, 20 to 40
Newton steps are already sufficient to solve the sufficient optimality
conditions with very high precision (up to the machine precision level), even
for large systems. Even fewer steps are required if the secrecy capacity is the
only quantity of interest. The algorithm can be significantly simplified for
the degraded channel case and can also be adopted to include the per-antenna
power constraints (instead or in addition to the total power constraint). It
also solves the dual problem of minimizing the total power subject to the
secrecy rate constraint.Comment: accepted by IEEE Transactions on Communication
On barrier and modified barrier multigrid methods for 3d topology optimization
One of the challenges encountered in optimization of mechanical structures,
in particular in what is known as topology optimization, is the size of the
problems, which can easily involve millions of variables. A basic example is
the minimum compliance formulation of the variable thickness sheet (VTS)
problem, which is equivalent to a convex problem. We propose to solve the VTS
problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\
Polyak and later studied by Ben-Tal and Zibulevsky and others. The most
computationally expensive part of the algorithm is the solution of linear
systems arising from the Newton method used to minimize a generalized augmented
Lagrangian. We use a special structure of the Hessian of this Lagrangian to
reduce the size of the linear system and to convert it to a form suitable for a
standard multigrid method. This converted system is solved approximately by a
multigrid preconditioned MINRES method. The proposed PBM algorithm is compared
with the optimality criteria (OC) method and an interior point (IP) method,
both using a similar iterative solver setup. We apply all three methods to
different loading scenarios. In our experiments, the PBM method clearly
outperforms the other methods in terms of computation time required to achieve
a certain degree of accuracy
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
A decomposition procedure based on approximate newton directions
The efficient solution of large-scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner. We describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems. We also study how to correct these directions using (decomposable) preconditioned conjugate gradient methods to ensure local convergence in all cases. The choice of appropriate preconditioners results in a natural manner from the structure in the problem. Finally, we conduct computational experiments to compare the resulting procedures with direct methods, as well as to study the impact of different preconditioner choices
A sequential semidefinite programming method and an application in passive reduced-order modeling
We consider the solution of nonlinear programs with nonlinear
semidefiniteness constraints. The need for an efficient exploitation of the
cone of positive semidefinite matrices makes the solution of such nonlinear
semidefinite programs more complicated than the solution of standard nonlinear
programs. In particular, a suitable symmetrization procedure needs to be chosen
for the linearization of the complementarity condition. The choice of the
symmetrization procedure can be shifted in a very natural way to certain linear
semidefinite subproblems, and can thus be reduced to a well-studied problem.
The resulting sequential semidefinite programming (SSP) method is a
generalization of the well-known SQP method for standard nonlinear programs. We
present a sensitivity result for nonlinear semidefinite programs, and then
based on this result, we give a self-contained proof of local quadratic
convergence of the SSP method. We also describe a class of nonlinear
semidefinite programs that arise in passive reduced-order modeling, and we
report results of some numerical experiments with the SSP method applied to
problems in that class
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