6 research outputs found
Worst-case convergence analysis of inexact gradient and Newton methods through semidefinite programming performance estimation
We provide new tools for worst-case performance analysis of the gradient (or
steepest descent) method of Cauchy for smooth strongly convex functions, and
Newton's method for self-concordant functions, including the case of inexact
search directions. The analysis uses semidefinite programming performance
estimation, as pioneered by Drori and Teboulle [Mathematical Programming,
145(1-2):451-482, 2014], and extends recent performance estimation results for
the method of Cauchy by the authors [Optimization Letters, 11(7), 1185-1199,
2017]. To illustrate the applicability of the tools, we demonstrate a novel
complexity analysis of short step interior point methods using inexact search
directions. As an example in this framework, we sketch how to give a rigorous
worst-case complexity analysis of a recent interior point method by Abernethy
and Hazan [PMLR, 48:2520-2528, 2016].Comment: 22 pages, 1 figure. Title of earlier version was "Worst-case
convergence analysis of gradient and Newton methods through semidefinite
programming performance estimation
Weighted Sum Rate Maximization for Downlink OFDMA with Subcarrier-pair based Opportunistic DF Relaying
This paper addresses a weighted sum rate (WSR) maximization problem for
downlink OFDMA aided by a decode-and-forward (DF) relay under a total power
constraint. A novel subcarrier-pair based opportunistic DF relaying protocol is
proposed. Specifically, user message bits are transmitted in two time slots. A
subcarrier in the first slot can be paired with a subcarrier in the second slot
for the DF relay-aided transmission to a user. In particular, the source and
the relay can transmit simultaneously to implement beamforming at the
subcarrier in the second slot. Each unpaired subcarrier in either the first or
second slot is used for the source's direct transmission to a user. A benchmark
protocol, same as the proposed one except that the transmit beamforming is not
used for the relay-aided transmission, is also considered. For each protocol, a
polynomial-complexity algorithm is developed to find at least an approximately
optimum resource allocation (RA), by using continuous relaxation, the dual
method, and Hungarian algorithm. Instrumental to the algorithm design is an
elegant definition of optimization variables, motivated by the idea of
regarding the unpaired subcarriers as virtual subcarrier pairs in the direct
transmission mode. The effectiveness of the RA algorithm and the impact of
relay position and total power on the protocols' performance are illustrated by
numerical experiments. The proposed protocol always leads to a maximum WSR
equal to or greater than that for the benchmark one, and the performance gain
of using the proposed one is significant especially when the relay is in close
proximity to the source and the total power is low. Theoretical analysis is
presented to interpret these observations.Comment: 8 figures, accepted and to be published in IEEE Transactions on
Signal Processing. arXiv admin note: text overlap with arXiv:1301.293
Improving Complexity of Structured Convex Optimization Problems Using Self-concordant Barriers
The purpose of this paper is to provide improved complexity results for several classes of structured convex optimization problems using to the theory of self-concordant functions developed in [11]. We describe the classical short-step interior-point method and optimize its parameters in order to provide the best possible iteration bound. We also discuss the necessity of introducing two parameters in the definition of self-concordancy and which one is the best to fix. A lemma from [3] is improved, which allows us to review several classes of structured convex optimization problems and improve the corresponding complexity results