Monotonic and Sequential Fractional Programming for Performance - - PowerPoint PPT Presentation
Monotonic and Sequential Fractional Programming for Performance Optimization in Interference Networks Eduard Jorswieck Communications Theory 16.05.2017 Joint work with Alessio Zappone (University Cassino, Italy) Emil Bjrnsson
Eduard Jorswieck
16.05.2017
Communications Theory
Joint work with
Power
"Spectrum sharing improves the network efficiency for cellular operators," in IEEE Communications Magazine, vol. 52, no. 3, pp. 129-136, March 2014.
Processing Optimization: The Way to Balance Conflicting Metrics in 5G Systems", IEEE Signal Processing Magazine, vol. 31, no. 6, pp. 14-23, Nov. 2014.
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Wireless Networks via Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396. http://www.nowpublishers.com/article/Details/CIT-088
and that the energy demand will soon become unmanageable.
which can be reliably transmitted per Joule of consumed energy.
(1) G. Auer, V. Giannini, C. Desset, I. Godor, P. Skillermark, M. Olsson, M. Imran, D. Sabella, M. Gonzalez, O. Blume, and A. Fehske, “How much energy is needed to run a wireless network?” IEEE Wireless Communications, vol. 18, no. 5, pp. 40–49, Oct. 2011. (2) D. W. K. Ng, E. S. Lo, and R. Schober, “Energy-Efficient Resource Allocation for Secure OFDMA Systems,” IEEE Transactions on Vehicular Technology, vol. 61, no. 6, pp. 2572– 2585, July 2012.
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
EE = f(γ(p)) αp + Pc In line with the physical meaning of efficiency, the energy efficiency is defined as the system benefit-cost ratio in terms of amount of data reliably transmitted over the energy that is required to do so.
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
Observation:! always ratios!
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
Observation:! always ratios!
Fractional Programming!
Efficiency Optimization with Informed Transmitter", IEEE Trans. on Wireless Communications, vol. 11, no. 8, pp. 2946-2957, Aug. 2012.
KKT conditions are necessary and sufficient.
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
Fractional Programming Theory". Foundations and Trends in Communications and Information Theory, vol. 11, no. 3-4, June 2015, pp. 185-396.
Management Science, vol. 13, no. 7, pp. 492 - 498, March 1967
CFP by converting it into a sequence of convex problems
finding the zero of the function .
(Newton update)
12
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions on Signal Processing, vol. 64, no. 7, pp. 1668-1683, April 1, 2016.
Energy-Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
12
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions on Signal Processing, vol. 64, no. 7, pp. 1668-1683, April 1, 2016.
Energy-Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
both centralized and decentralized networks with rate and power constraints which allows encompassing some
EE is considered in the network-centric case.
modelled as rational, self-organizing agents that engage in a non-cooperative game wherein each one aims at maximizing its individual EE while targeting its own power and rate constraint.
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Networks: How Many Antennas Do We Need?," in IEEE Journal on Selected Areas in Communications, vol. 31, no. 2, pp. 160-171, February 2013.
3rd ed. New York, NY, USA: Springer, 2006.
Lemma 1: Let F be the following matrix with spectral radius . The solutions to global and minimum EE exist if and only if
First, for one block N=1:
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
First, for one block N=1:
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
First, for one block N=1:
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Does not result in concave denominator and not in concave fractional program
approximate problems Pl with objectives fl such that the following properties hold
and if the optimization variables converges, too, then to a point fulfilling the KKT conditions.
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for nonconvex mathematical programs,” Operations Research, vol. 26, no. 4, pp. 681–683, July–Aug. 1978.
spectrum balancing in multi-user DSL networks,” in Proc. IEEE Int. Conf.
with and use
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm Fractional program Lower Bound max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm Fractional program Lower Bound max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm Fractional program Lower Bound max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Concave fractional program Dinkelbach Algorithm Fractional program Lower Bound max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
f`(x) ≤ f(x) f`(xn) = f(xn) f 0
`(xn)
= f 0(xn)
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Algorithm 1 monotonically increases the GEE value and converges to a point fulfilling the KKT conditions of the
approximation framework,” IEEE Trans. Signal Process., vol. 65, no. 13, pp. 3313–3328, Jul. 2017.
Antenna Systems with Artifical Noise and Statistical CSI", IEEE Journal on
Underlay and Overlay Device-to-Device Communications and Cognitive Radio Systems", IEEE Trans. on Signal Processing, vol. 65, no. 4, pp. 1026-1042, Feb. 2017.
Bidirectional Massive MIMO Relay Beamforming", IEEE Signal Processing Letters, to appear 2017
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Lemma 2: If then with
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Proposition: The game admits a nonempty set of GNE points. Proposition: The game admits a unique GNE point, which can be obtained by starting from any feasible power vector and iteratively updating the transmit powers according to Lemma 2. Lemma 3: The solution from CFP with
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
Coordinated case:
bounds to objective function and the rate constraints Distributed case:
condition for uniqueness
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
massive MIMO system, K=5, S=1, M=50
and MRC, Rayleigh fading with pass loss, link budget as in LTE
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
assisted multi-cell network, S=3, N=16, K=3, M=3
kHz, Rayleigh fading with pass loss, link budget as in LTE
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions
29
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions on Signal Processing, vol. 64, no. 7, pp. 1668-1683, April 1, 2016.
Energy-Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
29
Efficient Power Control: A Look at 5G Wireless Technologies," in IEEE Transactions on Signal Processing, vol. 64, no. 7, pp. 1668-1683, April 1, 2016.
Energy-Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
power control problems, two approaches are presented.
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June1, 1 2017.
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
be expressed as a monotonic optimisation problems in canonical form. GEE WMEE
Concave fractional program Dinkelbach Algorithm max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Concave fractional program Dinkelbach Algorithm max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Monotonic Programming
max
p∈P K
X
k=1
log2(1 + γk) −λj(µkpk + Ψk)
Concave fractional program Dinkelbach Algorithm Fractional program max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K max
x
f(x) g(x) s.t. hk(x) ≤ 0 ∀k = 1, ..., K
concave / convex arbitrary / convex
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Monotonic Programming
max
p∈P K
X
k=1
log2(1 + γk) −λj(µkpk + Ψk)
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Massive MIMO uplink as before Achieved GEE for K=3
Efficient Power Control and Receiver Design in Wireless Networks," in IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 2844-2859, June 1, 1 2017.
Convergence behavior of the BRB algorithm in the last iteration of Dinkelbach algorithm
convex programming problems
an iterative superlinear convergent Dinkelbach algorithm
with more objectives and other constraints can benefit