5,426 research outputs found
Second-order optimality conditions for interval-valued functions
This work is included in the search of optimality conditions for solutions to the scalar
interval optimization problem, both constrained and unconstrained, by means of
second-order optimality conditions. As it is known, these conditions allow us to reject
some candidates to minima that arise from the first-order conditions. We will define
new concepts such as second-order gH-derivative for interval-valued functions,
2-critical points, and 2-KKT-critical points. We obtain and present new types of
interval-valued functions, such as 2-pseudoinvex, characterized by the property that
all their second-order stationary points are global minima. We extend the optimality
criteria to the semi-infinite programming problem and obtain duality theorems.
These results represent an improvement in the treatment of optimization problems
with interval-valued functions.Funding for open access publishing: Universidad de Cádiz/CBUA. The research has been supported by MCIN through
grant MCIN/AEI/PID2021-123051NB-I00
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
Trading Performance for Stability in Markov Decision Processes
We study the complexity of central controller synthesis problems for
finite-state Markov decision processes, where the objective is to optimize both
the expected mean-payoff performance of the system and its stability.
We argue that the basic theoretical notion of expressing the stability in
terms of the variance of the mean-payoff (called global variance in our paper)
is not always sufficient, since it ignores possible instabilities on respective
runs. For this reason we propose alernative definitions of stability, which we
call local and hybrid variance, and which express how rewards on each run
deviate from the run's own mean-payoff and from the expected mean-payoff,
respectively.
We show that a strategy ensuring both the expected mean-payoff and the
variance below given bounds requires randomization and memory, under all the
above semantics of variance. We then look at the problem of determining whether
there is a such a strategy. For the global variance, we show that the problem
is in PSPACE, and that the answer can be approximated in pseudo-polynomial
time. For the hybrid variance, the analogous decision problem is in NP, and a
polynomial-time approximating algorithm also exists. For local variance, we
show that the decision problem is in NP. Since the overall performance can be
traded for stability (and vice versa), we also present algorithms for
approximating the associated Pareto curve in all the three cases.
Finally, we study a special case of the decision problems, where we require a
given expected mean-payoff together with zero variance. Here we show that the
problems can be all solved in polynomial time.Comment: Extended version of a paper presented at LICS 201
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