Adaptive Projected Subgradient Method andits Applications to Robust Signal Processing

(Invited Paper)

Isao Yamada, Konstantinos Slavakis, Masahiro Yukawa and Renato L. G. CavalcanteDept. of Communications and Integrated Systems, Tokyo Institute of Technology, Tokyo 152-8552 JAPAN

Emails: {isao, slavakis,masahiro,renato}@comm.ss.titech.ac.jp

Abstract-The adaptive projected subgradient method of- filtering algorithms. Indeed, by designing a certain sequencefers a unified mathematical perspective for the adaptive (set- of convex objectives, a variety of adaptive filtering algorithmsmembership / set-theoretic) filtering schemes. In this paper, are derived in a unified manner as simple examples of thewe introduce an overview of its recent theoretical advancesand successful applications to robust signal processing problems adapt2ve projected subgradient method.including the stereo acoustic echo canceling, the MAI suppression In this paper, we introduce basic idea2 of the APSM andin DS/CDMA receivers, and the robust adaptive beamforming its successful applications to robust signal processing problemswith array antenna systems. including the stereo acoustic echo canceling [17,18], the MAI

suppression in DS/CDMA receivers [19-21], and the robustI. INTRODUCTION Capon beamforming with array antenna systems [22].

A common strategy among many schemes ([1-7] and refer- 1* ADAPTIVE PROJECTED SUBGRADIENT METHODences therein) in adaptive set-membership filtering / adaptiveset-theoretic filtering is the iterative approximation of an esti- A. Preliminariesmandum as a point in the intersection of possibly time-varying Let 'H be a (possibly infinite dimensional) real Hilbert spacefamily of closed convex sets. Such closed convex sets are equipped with an inner product (x y), Vx, y e -, and its in-defined, to restrict candidates of the estimandum, with mea- 1 2surements of signals or with a priori knowledge'. The great duced normVx (,

G) 1/ V E XH. A set C C is called

flexibility of closed convex sets brings significant benefits to convex If Vp, y e C, Vv e(, 1), Cx + (1m-rv)y eC. Forus in realizing advanced adaptive filtering in accordance with any nonempty closed convex set C Ct, the metric projectionthe intended applications (see the Set-Membership Normalized Pch th - C maps

m iX to the unique vector Pc(x) C C

LMS (SM-NLMS) [4], the frequency-domain Set-Membership such .that d(x, C) :.minyi cXai - ZX Pc(z) ANormalized LMS (F-SM-NLMS) [7] and the Adaptive Parallel function

GX H -* IR iS saId to be convex if V,y CXH and

Subgradient Projection (Adaptive PSP)[5]). Vv C (0,1), 9(vx + (1 - v)y) < v/(z) + (1 - v)9(y). IfThe above mentioned strategy has obvious commonality (9 is continuous and convex, lev<0o := {x G - 9<(x) <

with algorithmic solutions to the convex feasibility problems 0} is closed convex and the subdifferential of 9 at y (the(see for example [8-12] and references therein), hence the set of all the subgradients of 9 at y) satisfies 0&(y)goal is achieved mathematically by minimizing a family of s 'H (x - y, s) + ((y) < 8(x),Vx e XH} + 0. By thisnonnegative convex objectives; e.g., distances to closed convex definition, we have 0 E 0((y) X /3(y) minxr- 9(x).sets, in a real Hilbert space. (Note: the complex adaptive The convex function 9 'H -* R has a unique subgradient atfiltering problems can be formulated naturally in a real space). Yo 'H bfu iS differentiable at y This unique subgradi{t iOn the other hand, an inherent difference between the nothing but the gradient V@(y), i.e., 0(3(y) = (y)

scenarios for the convex feasibility problems and those for B. Adaptive Projected Subgradient Methodthe adaptive filtering problems must be that any algorithmfor the convex feasibility problem is based on infinitely many A primitive idea of the set-membership adaptive filtering /uses of information on a specific nonnegative convex objective the set-theoretic adaptive filtering is found in the Normalizedbut any algorithm for the adaptive filtering problems can not. Least Mean Squares (NLMS)[1]. The NLMS approximates it-This is because the convex objective for the adaptive filtering eratively a possibly time-varying estimandum, which is highlyproblems keeps changing due to the (possibly nonstationary) expected to belong to (or expected to exist closely to) theinput and noise processes. intersection of time-varying sequence of hyperplanes in a

The Adaptive Projected Subgradient Method (APSM) [13- vector space, by computing the metric projection onto each15] was developed as an efficient algorithm for asymptotic hyperplane in sequential order3. Obviously, at each time, theminimization of a certain sequence of nonnegative convex task of the NLMS is to minimize a nonnegative convexfunctions. The APSM is a natural extension of the Polyak's function defined as the distance to the current hyperplane.subgradient algorithm [16], for nonsmooth convex optimiza- This simple strategy of the NLMS has been proven verytion problem with a fixed target value, to the case where the successful through extensive real-world applications dealingconvex objective itself keeps changing in the whole process. with stationary I nonstationary random signals. The excellent

It isreveledtatte APM offrs ausefl matematcal For complete discussion and many examples of the APSM, see [15]foundation of a wide range of the projection based adaptive appeared in a more mathematical publication.

3To accelerate the speed of convergence of the NLMS, the APA [2, 3]1Similar idea is found in a fundamental principle in Husserl's phenomenol- introduces the metric projection onto a linear variety defined as the intersection

ogy [Edmund Husserl (1859-1938)]. of certain number of hyperplanes.

0-7803-9390-2/06/$20.00 ©2006 IEEE 269 ISCAS 2006

robust performance of the NLMS was reaffirmed through H°- subgradient projection (Adaptive PSP) [5] as its special ex-theory [23]. The APSM is a natural generalization of the amples. The Adaptive PSP uses weighted average of multipleNLMS and designed to minimize asymptotically a sequence subgradient projections to keep low computational cost of theof more general nonnegative convex functions than the simple NLMS as well as to achieve fast and stable convergence evendistance to hyperplane. The next theorem presents the APSM in severely noisy environment.and its elementary but useful properties. Example 2: (Embedded constrained version) Let V := v +

Theorem 1: (APSM [13-15]) Let 9k 7H-X > [0, ox) (Vk C M be a nonempty linear variety, where v C XY, M C XH isN) be a sequence ofcontinuous convex functions andK C XH a a closed subspace of XY, and 9k : V ` [0, ox) a continuousnonempty closed convex set. For an arbitrarily given ho e K, convex function. Then T4k : XH h H-* 9k (Pv (h)) C [0, xo)theAPSMproducesasequence (hk)kN C K by satisfies TIk(h) = k(h), Vh C V and PM (9J(PV(h))) C

(Ik(h), Vh C (, V9 C 0&k(h). Application of (1) tokJPK (h Aket(hk)Ek(Ih))if9hk (hk) #0 1k and K V yields a new algorithm for asymptotichk+1 :h (hk J minimization of (9k)kcN over V:hk otherwise, E(h)P(E'(h))

~hk Ak k O/(hkwhere (9/(hk) e 0&k(hk) and0 <.k < 2. hk±1.m P (e,(hk)) 2 if 9k(hk)

Then the sequence (hk)kcN satisfies the followings4. hk otherwise,(a) (Monotone approximation) Suppose that where ho e V, (9(hk) e 0(9k(hk) and 0 < Ak < 2. This

hk X Qk := {h C K 9k(h) =e}W 0, algorithm ensures (hk)kcN c V. The constrained NLMS [6] isan example of the embedded constrained version for 9k (h)

where 9k := infheK 9k(h). Then, by using VAk1 e d(h, Vk) where Vk is a hyperplane.

,2 (i- <E, (h))). we have III. APPLICATIONS

(Vh*(k) e Qk) 1hk+l - h*(k) 11 < 1hk - h*(k) A. Stereophonic Acoustic Echo Canceling(b) (Boundedness, Asymptotic optimality) Suppose :~N0 e Multi-microphone systems have been in the spotlight in

N s.t. 8*) = 0 Vk > No and Q := nk>o Qk + place of single-microphone systems in many applications such0, n ( N is b as hands-free mobile telephony and teleconferencing, since[ 2Then(h-82] (0

s bounded If we use-VApro the performance of single-microphone systems is generally[EJ, 2- E2] C (0, 2), we have kliM k(hk) = 0 pro- limited. To realize high quality in the communication, it isvided that (9/ (hk))kN is bounded. D hence a central burden to develop an efficient and stable echo

Example 1: Let S(k) C X (k1k C Z) and K C -, be canceling scheme. Main difficulties of multi-channel acoustic

nodconvex sets. Define a sequence of convex echo cancellation (MAEC) are as follows: (i) strong cross-1 (k S S

correlation among signals observed at the microphones infunctions by /9k(h) := L E w)d(hk, 5,j) )d(hI, S(k)) the transmission room causes slow convergence [25], (ii) a

k tCIk large number of filter taps per each microphone [(# of loud-Vk E N, where Z E ikM(k) 1 {w(k)}tck c (0, 1], if Lk := speakers) x (a few thousand)] are required for sufficient echo

biw(k)d(hk, S(k)) + 0, and 9k(h) := 0 otherwise. cancellation due to long acoustic impulse response stemming1 Jk)

k \ from slow propagation of sounds, (iii) there may exist highThen we have 9i)k(hk) = L7 ZitCE-Ik y2tthk- PS(k) (hk)) background noise and (iv) speech signals have generally highC 09k(hk) if Lk :t 0, and e (hk) = 0 C a&k(h) auto-correlation and nonstationarity. Therefore, the classicalotherwise. By applying (1) to 9) and K c IRN, we deduce a algorithms, NLMS, APA and RLS, are not suitable for MAECscheme: due to (i).(iv) for NLMS, (ii).(iii) for APA, and (ii).(iv) for

RLS.hk±1 := PK Ihk + I'k (k) (hk) -hk Recently, an efficient technique named pairwise optimalk+ 1 kP hk)Pkweight realization (POWER) has been proposed [26], which

tC-Tk realizes reasonable weights in the adaptive PSP (see Examplewhere ho E K, P9k E [0, 2M(1)] and 1). Based on this weighting technique, an efficient stereo-

phonic echo canceling scheme, which can be readily extended( Pk) (hk)hk 2 to multi-microphone systems, has also been proposed [18].

,'Ekht' (k)S hk)hk 2,if hk SFLIk 5(k) The scheme in [18] is compared to NLMS and APA in Fig. I

k = P(k)ps (hk)-hk under SNR = 25 dB to find h* E R2000 with speech input1, otherwise. signals. For all methods, we employ a nonlinear preprocessing

technique to decorrelate the input signals. We observe that theObviously this is a generalization of Algorithm 1 in [5], hence proposed scheme is approximately twice faster than APA.it includes NLMS [1], APA [2,3] and the adaptive parallel B. MAL Suppression for DS/CDMA systems

4Recently the metric projection PK in (1) was extended to arbitrary stronglyattracting nonexpansive mapping T [24]. By this generalization, we can In DS/CDMA systems, as users share the same frequencyminimize asymptotically the sequence of nonnegative convex functions ekZ band at the same time, the signal originated from a single user(Vk: E N) over the fixed point set Fit(T) :={x E 7-( T(x) - X}. iS affected by the interference originated from other users atFor example, if we define T := E WiPK (wi > 0, i 1, 2,..,m

i= wi = 1) or Hli= PK. for closed convex sets Ki (i =1, 2,..,m) the receiver side [27, 28]. Conventional receivers, which ares.t. ni= 1.7 7& 0, we have Fix(T) =O7-m= Ki. essentially matched filters, completely ignores the structure of

270

G 15RLS

7g.,.I..\

-~~~~ ~NL M,S 10-10 ~~~APAI

5 NLMS (step size=0.6)

-20 E

Proposed:-30 8 Proposed

0 20 40 60 (number of symbols used at each iteration=128,Time (sec.) step size=0.2M

forgetting factor for amplitude estimation=0.01)Fig. 1. System mismatch curve: Proposed versus NLMS and APA. -10

Gold sequences of length 31, SNR=15 dB, 6 users

Interfering users have 20 dB power advantage0 500 1000 1500 2000

this multiple-access interference (MAI) and thus yields a bit- Iteration numbererror-rate (BER) that is usually much worse than the minimumachievable probability of error [28]. Therefore, a great deal Fig. 2. Output signal to interference-noise ratio (SINR) curves for differentlinear filters. The proposed (blind) method based on the APSM [19] has onlyof effort has been devoted to developing linear detectors knowledge about the desired user's signature. The Generalized Projection (GP)that present good performance and manageable complexity algorithm [31] knows the desired user's amplitude and signature. The RLS

[29, 30]. Basically, these detectors are adaptive linear filters and NLMS algorithms use training sequences.that minimize either the mean-square-error (MSE) betweenthe desired output and the filter output, or the filter outputenergy subject to a constraint that is determined by the desired denotes expectation, * stands for complex conjugate trans-user's signature. In order to track the optimal solution of position) is the correlation matrix of the random processthese two minimization problems, set-theoretic receivers can (y(k))k C CN received by an array of N elements. Givenbe applied [19-21,31,32] (Excellent performance realized by k C N, the minimizer that 'rejects' interference and noisethe set-membership filtering approach was also confirmed in by the mapping y(k) H-* w*y(k) under the assumption thatits applications [33,34] to the multiuser detection and the the Signal Of Interest (SOI) suffers no distortion is the arrayadaptive beamforming). Ry(k) s The SQl steeringThe APSM is a set-theoretic approach that gives the de- weighting vector wcB(k) = Ry,n(k) hSO

signer the option to use as much information as available vector s0 gathers all the spatial characteristics of the array asat each iteration. Therefore, fast convergence rate and good well as the SOI Direction Of Arrival (DOA).steady-state performance can be achieved at the same time Unfortunately, CB is sensitive to errors like DOA mis-[19-21] (see also Fig. 2). The main idea of the receivers based matches, poor array calibration, unknown sensor mutual cou-on this method is as follows: for each received symbol, sets pling [35], which cause erroneous estimates of s0. Unlike thethat describe desired characteristics of an optimal filter are classical Diagonal Loading (DL) method where for a fixedbuilt (NOTE: explicit knowledge of the transmitted symbol empirical value of EDL > 0 the matrix EDLIN (IN is themay not be necessary), and then the current filter is projected identity matrix in CN XN) is added to Ry(k) prior its inversiononto the available sets, which may include sets based on Vk C N, very recent research [35] devises iterative methodspast transmitted symbols. The additional complexity gained by to calculate the optimal (in some sense) DL parameter bydealing with more data at each iteration can be alleviated by explicitly using the uncertainty about the SOI steering vector.using parallel computing and/or by calculating low-complexity In [22], we follow a different path from DL techniques andapproximations of the true projections. By exploiting geomet- we design a robust CB by the APSM. First, a characterizationrical properties of the projections onto closed convex sets, we of a data-independent (independent from (y(k))kcN) robusthave shown [20] that it is possible to achieve remarkably good beamformer w C CN is given as a time-varying convexconvergence speed even when the receiver uses a moderate feasibility problem in the IR2N+3 space: the set of all robustnumber of symbols at the same time. beamformers against SOI steering vector errors is the intersec-More recently, the results in [19, 20] have been extended to tion of a finite number of time-independent icecream cones and

cope with a variety of assumptions about the channel model a finite number of time-dependent hyperplanes. Furthermore,and the available information at the receiver side [21]. The in order to employ the received data in the design, we generateblind receivers described in this work have lower complexity a sequence of halfspaces that most likely contain the desiredand better BER for fast time-varying flat fading channels as beamformer. Finally, we use Example 1 where instead of thecompared to some algorithms that require a matrix inversion. mapping PK we have used the composition of the metricIt is also shown that many receivers based on the MSE projection mappings onto the icecream cones; a stronglycost function or the output energy cost function are simple attracting nonexpansive mapping. This algorithmic solutionexamples of the adaptive projected subgradient method. becomes a special case of the extension [24] (See the footnote

C. Robust Capon Beamforming gvnwt hoe )* ~~~~~~~~~~~~~~~Fiveinterferences arrive at a Uniform Linear Array (ULA),The Capon Beamformer (CR) is a celebrated solution to N =20, each one of power o2, j=1, ... ,5. In Fig.

the problem of finding w C argminzCN) sOz=1 z*Ry(k)z, 3, 'Ideal' stands for CB with no uncertainty, 'CB-DL' forVlk C N, where Ry(k) := E{y(k)y*(k)}, k C N, (E{.} its DL version, and 'SOCP' for the approach in [35]. The

271

(Contemporary Mathematics 313), Z. Nashed and 0. Scherzer, Eds., pp.269-305. American Mathematical Society, 2002.

Ideal [12] I. Yamada and N. Ogura, "Hybrid steepest descent method for variational22 inequality problem over the fixed point set of certain quasi-nonexpansive

mappings," Numerical Functional Analysis and Optimization, vol. 7/8,20 APSM pp. 619-655, 2004.

[13] I. Yamada, "Adaptive projected subgradient method - a unified viewfor projection based adaptive algorithms," J. IEICE, vol. 86, no. 8, pp.4 18 SOCP 654-658, Aug. 2003, in Japanese.

[14] I. Yamada and N. Ogura, "Adaptive projected subgradient method andz; < \ X its applications to set theoric adaptive filtering," in Proc. 37th Asilomarx 16 - CB-DL Conference on Signals, Systems and Computers, Nov. 2003.

[15] I. Yamada and N. Ogura, "Adaptive projected subgradient method for14 asymptotic minimization of sequence of nonnegative convex functions,"

Numerical Functional Analysis and Optimization, vol. 25, no. 7/8, pp.593-617, 2004.

12 [16] B. T. Polyak, "Minimization of unsmooth functionals," USSR Comput.Math. Phys., vol. 9, pp. 14-29, 1969.

l0 [17] M. Yukawa and I. Yamada, "Efficient adaptive stereo echo cancelingB10 15 20 25 30 35 40 45 50 schemes based on simultaneous use of multiple state data," IEICEINR (dB) Trans. Fundamentals, vol. E87-A, no. 8, pp. 1949-1957, Aug. 2004.

[18] M. Yukawa, N. Murakoshi, and I. Yamada, "Efficient fast stereoFig. 3. Define SINR = (aoaS1w2)/(Z> Wl ,2 + a,-2 .W12), acoustic echo cancellation based on pairwise optimal weight realization

wr ihsegj jm 1 2 i technique," EURASIP J. Appl. Signal Processing, 2006, to appear.where s iS the steering vector of the j-th jammer, j = 1, ,5, CJo is [19] R. L. G. Cavalcante, I. Yamada, and K. Sakaniwa, "A fast blind MAIthe SOI power, and a2 is the noise power. We set SNR = 10dB. Let reduction based on adaptive projected subgradient method," JEICEINR : 1 0r /07 n. Trans. on Fundamentals, vol. E87-A, no. 8, Aug. 2004.

[20] M. Yukawa, R. L. G. Cavalcante, and I. Yamada, "Efficient blindMAI suppression in DS/CDMA systems by embedded constraint parallelprojection techniques," IEICE Trans. on Fundamentals, vol. E88-A, no.8, pp. 2062-2071, Aug. 2005.results exhibit the excellent performance of the APSM in [21] R. L. G. Cavalcante, M. Yukawa, and 1. Yamada, "Set-theoretic

cases where the DL techniques face difficulties, i.e. where DS/CDMA receivers for fading channels by adaptive projected subgra-the INR is moderately larger than SNR or in other words in dient method," in Proc. GLOBECOM, 2005.SOIcontaminated situations. 7[22] K. Slavakis, M. Yukawa, and I. Yamada, "Efficient robust adaptiveSQL-contaminated situations. beamforming by the Adaptive Projected Subgradient Method: A set

Acknowledgement: The authors would like to thank Prof. Ko- theoretic time-varying approach over multiple a-priori constraints," Tech.hichi Sakaniwa of Tokyo Institute of Technology for helpful Rep. SIP2005-52, WBS2005-10 (2005-07), IEICE, July 2005.

[23] B. Hassibi, A. H. Sayed, and T. Kailath, "h° optimality of the lmsdiscussions and comments. The 1st author also would like to algorithm," IEEE Trans. Signal Processing, vol. 44, no. 2, pp. 267-280,thank Prof. Yih-Fang Huang of University of Notre Dame for Feb. 1996.hiskininitaiontoISCAS2006. This work was supported [24] K. Slavakis, I. Yamada, N. Ogura, and M. Yukawa, "Adaptive Projected

hs kd i o tSubgradient Method and set theoretic adaptive filtering with multiplein part by JSPS Grants-in-Aid (C-17500139). convex constraints," in Proceedings of the 38th Asilomar Conference on

Signals, Systems and Computers, Nov. 2004, pp. 960-964.[25] J. Benesty, D. R. Morgan, and M. M. Sondhi, "A better understanding

REFERENCES and an improved solution to the specific problems of stereophonicacoustic echo cancellation," IEEE Trans. Speech Audio Processing, vol.

[1] J. Nagumo and J. Noda, "A learning method for system identification, [26] M. Yukawa and I Yamada, "Pairwise optimal weight realizationIEEE Trans. Autom. Control, vol. 12, no. 3, pp. 282-287, March 1967. Acceleration technique for set-theoretic adaptive parallel subgradient

[2] T. Hinamoto and S. Maekawa, "Extended theory of learning identifica- projection algorithm," IEEE Trans. Signal Processing, 2006, to appear.tion," Trans. IEE Japan (in Japanese), vol. 95C, no. 10, pp. 227-234, [27] U. Madhow and M. L. Honig, "MMSE interference suppression for1975. direct-sequence spread-spectrum CDMA," IEEE Trans. Commun., vol.

[3] K. Ozeki and T. Umeda, "An adaptive filtering algorithm using an 42, no. 12, pp. 3178-3188, Dec. 1994.Orthogonal projection to an affine subspace and its properties," JEICE [28] S. Verdu, "Minimum probability of error for asynchronous GaussianTrans., vol. 67A, no. 5, pp. 126-132, 1984. multiple-access channels," IEEE Trans. Inf. Theory, vol. 32, no. 1, pp.

[4] S. Gollamudi, S. Kapoor, and Y. H. Huang, "Set-membership filtering 85-96, Jan. 1986.and a set-membership normalized LMS algorithm with an adaptive step [29] M. Honig, U. Madhow, and S. Verdu, "Blind adaptive multiusersize," IEEE Signal Processing Lett., vol. 5, no. 5, pp. 111-114, Feb. detection," IEEE Trans. Inf. Theory, vol. 41, no. 4, pp. 944-960, July1998. 1995.

[5] I. Yamada, K. Slavakis, and K. Yamada, "An efficient robust adaptive [30] M. L. Honig, S. L. Miller, M. J. Shensa, and L. B. Milstein, "Per-filtering algorithm based on parallel subgradient projection techniques," formance of adaptive linear interference suppression in the presence ofIEEE Trans. Signal Process., vol. 50, no. 5, pp. 1091-1101, May 2002. dynamic fading," IEEE Trans. Commun., vol. 49, no. 4, April 2001.

[6] M-L. R. Campos, S. Werner, and J. A. Apolinario, "Constrained adaptive [31] S. C. Park and J. F. Doherty, "Generalized projection algorithm for blindalgorithm employing householder transformation," IEEE Trans. on interference suppression in DS/CDMA communications," IEEE Trans.Signal Processing, vol. 50, no. 9, pp. 2187-2195, Sep. 2002. Circuits Syst. II, vol. 44, no. 6, pp. 453-460, June 1997.

[7] L. Guo, A. Ekpenyong, and Y. H. Huang, "Frequency-domain adaptive [32] J.A. Apolinario, S. Werner, P.S.R. Diniz, and T.I. Laakso, "Constrainedfiltering a set-membership approach," in Proc. 37th Asilomar normalized adaptive filters for CDMA mobile communications," in Proc.Conference on Signals, Systems and Computers, Nov. 2003. EUSIPCO, 1998, vol. IV, pp. 2053-2056, Greece.

[8] H. H. Bauschke and J. M. Borwein, "On projection algorithms for [33] S. Kapoor, S. Gollamudi, S. Nagaraj, and Y. F. Huang, "Adaptivesolving convex feasibility problems," SIAM Review, vol. 38, pp. 367- multiuser detection and beamforming for interference suppression in426, 1996. cdma mobile radio systems," IEEE Trans. Vehicular Technology, vol.

[9] P. L. Combettes, "Foundation of set theoretic estimation," Proc. IEEE, 48, no. 5, pp. 1341-1355, Sep. 1999.vol. 81, pp. 182-208, 1993. [34] S. Nagaraj, S. Gollamudi, S. Kapoor, Y. F. Huang, and J. R. Dellar

[10] I. Yamada, "The hybrid steepest descent method for the variational Jr., "Adaptive interference suppression for cdma systems with a worst-inequality problem over the intersection of fixed point sets of non- case error criterion," IEEE Trans. Signal Processing, vol. 48, no. 1, pp.expansive mappings," in Inherently Parallel Algorithm for Feasibility 284-289, Jan. 2000.and Optimization and Their Applications, Y. Censor D. Butnariu and [35] 5. A. Vorobyov, A. B. Gershman, and Z.-Q. Luo, "Robust adaptiveS. Reich, Eds., pp. 473-504. Elsevier, 2001. beamforming using worst-case performance optimization: A solution to

[11] I. Yamada, N. Ogura, and N. Shirakawa, "A numerically robust hybrid the signal mismatch problem," IEEE Transactions on Signal Processing,steepest descent method for the convexly constrained generalized inverse vol. 51, no. 2, pp. 313-324, Feb. 2003.problems," in Inverse Problems, Image Analysis, and Medical Imaging

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