Hopfield Learning-based and Nonlinear Programming methods for Resource Allocation in OCDMA Networks

Hopfield Learning-based and Nonlinear Programming metho ds for Resource Allo cation in OCDMA Net w orks Cristiane A. P endeza Martinez a , T aufik Abr˜ ao b, , F´ abio Renan Durand a , Alessandro Go edtel a a Universidade T e cnol´ ogic a F e der al do Par an´ a – Campus Corn ´ elio Pr o c´ opio Avenida A lb erto Car azzai, 1640 - CEP 86300-000 Corn ´ elio Pr o c´ opio - PR - Br asil b Dep artment of Ele ctric al Engine ering, L ondrina State University, R o d. Celso Gar cia Cid - PR445, Po.Box 10.011, CEP: 86057-970, L ondrina, PR, Br azil Abstract This pap er proposes the deplo yment of the Hopfield’s artificial neural net w ork (H-NN) approach to optimally assign p o w er in optical co de division m ultiple access (OCDMA) systems. Figures of merit such as feasibilit y of solutions and complexit y are compared with the classical p ow er allo cation metho ds found in the literature, s uc h as Sequential Quadratic Programming (SQP) and Augmented Lagrangian Metho d (ALM). The analyzed metho ds are used to solv e constrained nonlinear opti- mization problems in the con text of resource allo cation for optical netw orks, sp ecially to deal with the energy efficiency (EE) in OCDMA net works. The promising p erformance-complexit y tradeoff of the mo dified H-NN is demonstrated through n umerical results p erformed in comparison with classic metho ds for general problems in nonlinear programming. The ev aluation is carried out considering c hallenging OCDMA net works in which different levels of QoS w ere considered for large n umbers of optical users. Keywor ds: Optical Net works, Po w er Allo cation, Artificial Neural Netw orks, Hopfield Net works, Iterativ e Optimization Metho ds 1. In tro duction Due to the tec hnological adv ancemen t of state-of-the-art media, the researc h is fo cused on finding systems wit h greater efficiency , higher transmission capacities and greater range with few er repeaters. No wada ys, with the evolution of photonic tec hnology , optical transmission media hav e b ecome the most feasible option for large-scale information transmission in a fast and reliable wa y and reac hing high transmission rates in sev eral systems. A fib er optic connection has low loss b et ween its transmitter and receiver, in addition to b eing able to transmit analog or digital signals. The original signal is con v erted from electrical to optical through a media conv erter, in a p oint-to-point transmission or an optical line terminal (OL T) if fib er to the home (FTTH) is deplo yed. Based on the evolution of the co de division m ultiple access (CDMA) tec hnique in wireless systems, Email addr esses: crismartinez@utfpr.edu.br (Cristiane A. Pendeza Martinez), taufik@uel.br (T aufik Abr˜ ao), fabiodurand@utfpr.edu.br (F´ abio Renan Durand), agoedtel@utfpr.edu.br (Alessandro Go edtel) Pr eprint submitte d to ArXiv arXiv.or g Cornel l University Septemb er 6, 2019 the resp ectiv e optical tec hnique (OCDMA) was in tro duced in the mid 1980s [1], [2]. OCDMA systems w ere dev elop ed using the spectral scattering technique, where each user is indicated b y a unique co de. Due to the tremendous growth of OCDMA netw ork, sev eral configurations ha ve b een established whic h can b e classified in to non-coheren t and coherent systems [3]. Moreo ver, OCDMA technology has attracted research in terest b ecause of its many adv antages, such as async hronous op eration, net work flexibilit y , proto col transparency , simplified control and also making the netw ork p oten tially safer [4], [39]. The resources allo cation, such as p o w er and bandwidth, optical netw orks are decisiv e in order to mak e the most efficien t as p ossible the deplo yment of the a v ailable bandwidth in this netw ork mo de, as well as, observing that the transmission o ccurs aiming at minimizing the non-linear effect impacts caused b y the ph ysical and structural limitations of the optical fib ers during transmission. F or this, a robust system is necessary , using enco ders and deco ders that can p erform the transmis- sion safely and efficiently . In this context, a paramoun t resource allo cation problem in OCDMA net works is the pow er allo cation that minimizes the m ultiple access in terference. In solving suc h optimization problem, the optical netw ork is able to accommodate the largest num b er of users shar- ing the same sp ectrum net work, guaranteeing for each user its minim um quality of service (QoS) in terms of minim um data rate and signal to in terference plus noise ratio (SINR) [6]. On the other hand, the p o wer allo cation in optical systems must deal with the optical p ow er budget in the sense that it is paramount guarantee the mitigation of the non-linear optical effects due to ph ysical fib er imp erfections and construction netw ork limitations. W orks as [9] and [10] for CDMA netw ork and for OCDMA net w ork [27] use an analytical and in teractive approach, namely the V erhuslt p opulation model to obtain a new distributed p o w er con trol algorithm (DPCA). The V erhuslt model has demonstrated high sp eed of con v ergence, quality of solutions (pro ximit y of optimal v alue after conv ergence), robustness to estimation errors, among others as p ositiv e asp ects for implementation. The p o w er control (PC) in OCDMA optical netw orks app ears as a nonlinear optimization prob- lem. In this case, the ob jectiv e is to establish p ow er lev els so that the signal-to-noise ratio plus each user’s in terference reac hes a threshold required for acceptable p erformance (maxim um tolerable er- ror rate) and quality of service required (minimum QoS). Sev eral engineering problems are mo deled as optimization problems which require metho ds that provide realistic and pro cessed solutions to digital computers quic kly . In this wa y , traditional metho dologies such as programming metho ds can pro vide inaccurate solutions. Therefore, the search for solutions in such cases b ecomes imp ortant and essen tial, as the structure of certain problems is complex or there are a large num b er of p ossible solutions [19]. The artificial intelligence (AI) area prop oses sev eral techniques and resources in the dev elopment of intelligen t programs, that is, programs capable of making a decision similar to h uman [5]. The area of artificial intelligence began to develop in the sense of mo deling the brain through the creation of artificial neural net w orks (ANN), which hav e the same cognitive and asso ciativ e prop erties of the h uman brain. Problems of difficult treatmen t in conv entional computation, can b e approached by using ANN, so that it elab orates effectiv e solutions [19]. In recen t y ears the theory of ANN has 2 made significant progress that has led to the developmen t of more effective to ols in solving problems across different areas of knowledge. A condition for these adv ances is the more efficient use of the a v ailable computational resources, whic h generates an increase in the manipulation capacity of the information. Since its inception, ANN deploymen t has b een motiv ated by the recognition that the human brain computes in an en tirely different wa y from the conv en tional digital computer highly complex, nonlinear, and parallel computer (information-processing system) [33], [41] and [38]. Sp ecifically , the ANN pro cedures deploy ed to solv e nonlinear optimization problems ha v e b een dev elop ed using p enalt y parameters [20], [21] and [23]. The equilibrium p oin ts of these net works, corresp onding to the solutions of the problem, are obtained through the appropriate choice of p enalt y parameters that m ust b e sufficiently large to guarantee the conv ergence of the netw ork. Th us, the c hoice of a sp ecific ANN structure and it resp ective parameter v alues is a complex task p erformed through empirical tec hniques, which require a v ery excessive computational effort and exhaustive training steps. In addition, the qualit y of the final solution also dep ends on the parameters fitting [24]. Some other difficulties related to the con vergence process for the net work equilibrium p oin ts, whic h represents the solutions of the optimization problem also should b e considered. Numerical results presented in [20], [21] and [23] discuss the infeasible solutions. In this con text, the authors of [19] seek an alternativ e to improv e the efficiency of computer simulations and to provide a new metho dology for mapping constrained nonlinear optimization problems using the modified Hopfield neural net works (mH-NN). The main characteri stics of suc h metho dology deploying mH-NN are: (i) no weigh ting constants; (ii) all structural constraints inv olved in the constrained nonlinear optimiza- tion problem are group ed into a single constraint term; (iii) there is no in terference b et w een the optimization term and the restriction term; (iv) no initialization parameter is required for simulation execution. In this work, an analysis of the dynamic b eha vior of the optical CDMA netw ork is carried out, while the conv ergence pro cess tow ards the optimal p ow er solutions is careful analyzed. By analyzing the numerical and theoretical results, in this work we seek generalizations for the p ow er allo cation problem in OCDMA net works aiming at analyzing the applicability of Hopfield-based ANN as a pro cedure to solve general classes of resource allo cation problem. Herein, the analytical solutions are explored through contin uous optimization metho ds, namely sequen tial quadratic programming (SQP) and augmented Lagrangian metho d (ALM). It is worth noting that in the literature, the optimization approach commonly applied to solv e p o wer assignment problems in optical netw orks has b een heuristic metho ds, for instance [27], [40]. Based on the form ulation of the optimization problem, it is notable that the minim um p o wer allo cation problem presents only one p oin t that satisfies Karush-Kuhn-T uc ker (KKT) conditions. Th us, in this work, the nonlinear programming (NLP) classical metho ds, suc h as SQP and ALM, w ere deploy ed as a baseline to solve the optical p ow er allo cation problem. Con tribution : The contribution of this w ork is threefold. It consists of a ) prop ose the use of mo dified Hopfield-based NN (mH-NN) esp ecially constructed to solve the minim um p ow er allo cation problem in OCDMA with QoS guarantee; b ) a systematic analysis of the minim um p o wer allo cation 3 problem in OCDMA netw orks using the mH-NN; and c ) comparing it with the classical optimization metho ds SQP and ALM which are analytical metho ds of optimization applied sp ecifically to solve the minim um optical pow er allo cation problem taking into accoun t the con v ergence sp eed, feasibilit y , complexit y and optimality , in implementing realistic OCDMA netw orks. The remainder of this do cumen t is divided as follo ws: Section 2 discusses the problem of mini- mizing p o wer allo cation in OCDMA systems based on QoS. The optimization metho ds used in this w ork are presented in Section 3. Also, asp ects related to the implementation of the optimization metho ds applied to the problem of minim um p o wer allo cation and numerical results are analyzed in Section 5. Finally the Section 6 presen ts the final remarks. 2. Definition of Po wer Allo cation Problem The PC is applied to systems users interfere with eac h other. P erforming p o wer control should adjust the transmitted p o wer of all users so that eac h user’s noise ratio plus in terference (SNIR) meets a certain threshold required for acceptable p erformance. In order to ac hiev e a sp ecific QoS, whic h is associated with a bit error rate (BER) that is tolerated b y the i th optical no de, the relation carrier-noise in terference (CIR) required in the netw ork receiver deco der can b e defined as [8]: Γ i = G ii p i P K j =1 ,j 6 = i G ij p j + σ 2 i ≥ Γ ∗ i (1) where p i is the p o wer of the i th no de, K is the dimensional v alue of the column vector of transmitted optical p o w er namely p = [ p 1 , p 2 , ......p K ] | , σ 2 i is the p o wer of additiv e white Gaussian noise (A W GN) inheren t to the comm unication system at the receiving no de i and G ij is the fiber atten uation betw een the j th transmitting no de and i -th receiving no de. Therefore, the BER is a related QoS metric as w ell as the SNIR. The SNIR is asso ciated with the CIR as: γ i = r c r i Γ i , i = 1 , . . . , U (2) where r c is the chip rate, r i is the information rate of the i th user, U is the n umber of users of the system and Γ i is the CIR of i th user. The ob jectiv e of the minimum p ow er allo cation problem is to find the minimum transmission p o w er for eac h system user while satisfying all QoS requirements. Suc h QoS requiremen ts are basically summarized to the minim um information transmission rate and the maximum tolerable BER. This can b e summarized in SNIR using it as a constraint for QoS guarantee. The PC problem can b e mathematically describ ed as: min J 1 ( p ) = 1 | p = K X i =1 p i s . t . Γ i = G ii p i P K j =1 ,j 6 = i G ij p j + σ 2 i ≥ Γ ∗ i (3) p min ≤ p i ≤ p max , ∀ i = 1 , · · · , K , p min > 0 , p max > 0 4 where 1 T = [1 , . . . , 1], Γ ∗ i is the minimum CIR for the i th user reach the desired QoS level, p min is the minim um optical transmission p o w er and p max is the maximum optical transmission p o wer. Using matrix notation one can express the inequalit y as: [ I − Γ ∗ H ] P ≥ ¯ u , (4) where I is the iden tity matrix, H is the normalized interference matrix whose elemen ts are given by: H ij = ( 0 , i = j ; G ij G ii , j 6 = i. , ¯ u i = Γ ∗ i σ 2 i G ii , Λ ∗ =    Γ ∗ 1 ... 0 0 Γ ∗ 2 ... 0 0 ... Γ ∗ k    . (5) Substituting inequalit y by equality , the optimized p o wer vector solution is given by [15] [7]: p ∗ = [ I − Γ ∗ H ] − 1 u . (6) The PC problem in OCDMA net works can b e classified as a non-linear programming problem and not conv ex due to the constrain t of the imp osed CIR. F rom the literature we can cite authors suc h as [15] who use an t colony optimization (A CO) metho d to solve the problem (3). How ever, heuristic metho ds frequen tly do not generate promising results regarding p erformance-complexity tradeoff when compared to deterministic metho ds. Occasionally , one can also find solutions to the problems that are far from optimal v alues when using suc h metaheuristic metho ds. In addition, the input parameters of the heuristic metho ds, m ust b e adjusted b y the exhaustive searc h, whic h often can impact the results of more detailed analyzes if not adjusted prop erly . An alternative to problem solving (3) is the use of analytical approaches [7]. In this approac h, the determination of solution has greater computational complexit y in relation to the heuristic al- ternativ es, but they are able to guarantee the optimality of the solution. In this context, this w ork prop oses an alternativ e solution to the problem (3) with the Hopfield net work. The use of ANNs to solve optimization problems w as first prop osed by Hopfield and T ank [11]. Since then it has b een explored the p ossibilit y of solving problems of optimization with approac hes by neural netw orks. In particular authors such as [19] in tro duce a new metho dology for mapping restricted nonlinear optimization problems called mo dified Hopfield Net w orks in order to b ypass netw ork conv ergence problems and impro ve the efficiency of computer simulations. 3. Approac hed Optimization Metho ds The increase in the demand for transmission rate, related in large part b y the contin uous gro wth of Internet traffic, implies the need to increase the flexibility and the capacity of the net work [29]. Ho wev er, the degradation of the SNIR app ears as a challenge, since the problem of the near-far effect app ears together with the in terference by m ultiple access (MAI). In this wa y , it b ecomes necessary to establish an efficient managemen t of the resources, as example, the optical p ow er control is necessary to ov ercome this problem, increasing p erformance and optimizing netw ork utilization. This solution 5 can b e ac hiev ed b y solving the optimization problem posed b y eq.(3). In this section we revisit the main c haracteristics of the optimization metho ds discussed in this w ork to solve this problem. 3.1. Hopfield Artificial Neur al Network (H-NN) ANNs are computational mo dels inspired b y the central nerv ous system of an animal, in particular the brain, capable of performing machine learning as well as pattern recognition. They are usually presen ted as systems of interconnected neurons, whic h can compute input v alues, simulating the b eha vior of biological neural net works. These mo dels are used to solve v arious engineering problems suc h as function appro ximation, pattern classification and optimization. In this work w e use the mo dified H-NN prop osed by [19] in a nonlinear constrained optimization problem. In 1982, John Hopfield presen ted a net work type differen t from those based on P erceptron [18]. In this mo del the net work presented recurrent connections and was based on the unsup ervised learning with the comp etition among the neurons. This type of artificial neural net w ork arc hitecture has the follo wing c haracteristics: ( i ) typically dynamic b ehavior; ( ii ) ability to memorize relationships; ( iii ) p ossibilit y of storing information; ( iv ) ease analog implemen tation. The deploy ed Hopfield net w ork [18] has the structure as depicted in Figure 1, with a single la yer in whic h all neurons are completely in terconnected, i.e. all neurons of the net work are connected to all the others and themselv es where the outputs feed the inputs. Figure 1: Conv entional Hopfield Netw ork The simplified couple of expressions go verning the con tin uous-time b eha vior of eac h neuron in the Hopfield net work are given by: ˙ u j ( t ) = − η u j ( t ) + N X j =1 W ij v i ( t ) + ι b j , j = 1 , . . . , n (7) v j ( t ) = g ( u j ( t )) (8) where ˙ u j ( t ) is the internal state of the j -th neuron, with ˙ u j ( t ) = du/dt ; v j ( t ) is the output of j -th neuron; W j i is the synaptic weigh t by j th neuron to i th neuron; ι b j is the threshold (bias) applied to j th neuron; g ( . ) is a growing monotonic activ ation function, which limits the output of the neuron; η u j ( t ) is a passive deca y term. Observing the expressions (7) and (8) on can v erified that the b eha vior 6 of the Hopfield net w ork is alwa ys dynamic, that is, a set of inputs is applied; and then the outputs v are calculated and fed back to the inputs. The output is then recalculated and the pro cess rep eats in an iterative manner. Successiv e iteration sequences pro duce (decreasingly) changes in net w ork outputs un til their v alues b ecome constan t (stable). Therefore, given an y set of initial conditions, can b e obtained by second Ly apunov metho d as presen ted in [38] a Lyapuno v function for the Hopfield netw ork whose neurons are changed one at a time is defined b y: E ( t ) = − 1 2 v ( t ) T Wv ( t ) − v ( t ) T ι b (9) where the equilibrium p oin ts of the netw ork corresp ond to the v alues of v ( t ) that minimize the energy function of the netw ork; W is the weigh t matrix; ι b is the input vector asso ciated with the p o w er function of the net work (9). F rom (9) w e obtain the expression for its temp oral drift, that is: ˙ E ( t ) = dE ( t ) dt = ( ∇ v E ( t )) T v ( t ) (10) where ∇ v is the op erator gradient in relation to the vector v . As long as the w eight matrix is symmetric, W = W T , w e hav e: ∇ v E ( t ) = − Wv ( t ) − ι b (11) F rom (7) assuming that the passiv e deca y term is zero, w e conclude the follo wing result with (11): ∇ v E ( t ) = − ˙ u ( t ) (12) Using the ab o v e relations we obtain the expression for the deriv ative of the function E ( t ): ˙ E ( t ) = − n X j =1 ( ˙ u j ( t )) 2 | {z } i . ∂ v j ( t ) ∂ u j ( t ) | {z } ii (13) The p ortion ( i ) is alw ays p ositiv e. F or the p ortion ( ii ) choose an increasing monotonic activ ation function that limits the output of eac h neuron to a predefined in terv al. Th us, the tw o conditions are essential for the dynamic b eha vior of the Hopfield netw ork to b e stable: The matrix W must b e symmetric and the activ ation function g ( · ) m ust b e monotonically increasing. By establishing the ab o ve conditions, then given an y set of initial conditions the net work will con verge to a stable equilibrium p oin t. Then, since the Hopfield netw ork is deterministic, for an y initial p ositions that lie within the region of attraction of a p oin t of equilibrium, the net work will alw ays conv erge asymptotically to this p oin t. 3.2. Mo difie d Hopfield A rtificial Neur al Network (mH-NN) The neural net w orks used to solve constrained nonlinear optimization problems are developed using p enalt y parameters. The equilibrium p oin ts corresp onding to the solutions of the problem are obtained by choosing appropriate p enalt y parameters that m ust b e large enough to guarantee the con vergence of the netw ork [20], [21], [22]. 7 Ho wev er, the choice of these parameters is an arduous task and it is usually done through empirical tec hniques, which may require a v ery excessive computational effort. In addition, the qualit y of the final solution also dep ends on the setting of these parameters. A detailed analysis of the n umerical results in [16] shows that often infeasible results are p oin ted out as solutions to the problem. In order to o vercome problems related to con vergence, the authors [19] use a new metho dology for the mapping of nonlinear optimization problems called the mo dified Hopfield netw ork. The mo dified Hopfield net work w as implemen ted in order that the equilibrium p oin ts corresp ond to the solution of the constrained nonlinear optimization problem. The main c haracteristics of this netw ork are: (i) no w eighting constants; (ii) all structural constrain ts in v olved in the constrained nonlinear optimization problem are group ed into a single constraint term; (iii) there is no in terference b et w een the optimization term and the restriction term and (iv) no initialization parameter is required for sim ulation execution. F or these problems, a t wo-term energy function is used: E ( t ) = E opt ( t ) + E conf ( t ) (14) where E opt ( t ) = − 1 2 v ( t ) T . W opt v ( t ) − v ( t ) T ι opt , (15) E conf ( t ) = − 1 2 v ( t ) T . W conf v ( t ) − v ( t ) T ι conf , (16) the terms E opt and E conf corresp ond to the optimized energy function and the function that confines all constraints in a single term, resp ectiv ely . In (15), the terms W opt and ι opt corresp ond to the optimized w eight matrix and the resp ectiv e bias vector. Finally , the terms W conf and ι conf are the w eight matrix asso ciated with E conf and the resp ectiv e bias v ector. The purp ose of the net w ork is to simultaneously minimize the energy E opt ( t ) asso ciated with the ob jective function of the minimization problem as w ell as minimizing the energy function E conf ( t ) in volving all constraints of the problem. A simple mapping tec hnique enco des the constrain ts as terms in the energy function that are minimized when constrain ts are satisfied [19], that is: E ( t ) = E opt ( t ) + c 1 E rest 1 ( t ) + c 2 E rest 2 ( t ) + . . . + c k E rest k ( t ) (17) where c i are p ositiv e constan ts that give weigh t to constraints. The authors of [16] and [19] used the v alid subspace technique in order to group all constraints in volv ed in a given problem. Thus, the energy function given in (17) defined by: E ( t ) = E opt ( t ) + c 0 E conf ( t ) (18) where E conf confines all restrictions E rest k of (14) in the subspace-v alid. T o ensure that E opt optimized when all constraints con tained in E conf are satisfied, it inv olves assigning a high v alue to the constant c 0 . This condition makes the net work simulation inefficien t, since most of the computational effort is to force constrain t confinement. 8 The architecture of the mo dified Hopfield netw ork is sketc hed in Fig.2, where the pro jection matrix pro jects the vector v in to the v alid subspace, defined by: W conf = I − ∇ h ( v ) T  ∇ h ( v ) ∇ h ( v ) T  − 1 ∇ h ( v ) , (19) where the function h is defined such that the constraints of the optimization problem can b e repre- sen ted as h ( v ) = 0. Fig. 2 represen ts a suitable example of a recurren t net work where the outputs of a neural la yer in step (I I I) are fed bac k to their inputs in step (I). Indeed, it represents the v ariable relationship for conv ergence of the mo dified Hopfield netw ork whose op erating dynamics is implemented through steps (I)-(I II). Figure 2: Hopfield netw ork for solving constraint optimization problems The pseudo-co de depicted in Algorithm 1 illustrates the basic steps of the mH-NN deplo yed to solv e the OCDMA p ow er allo cation problem (3), aiming at finding the minimum transmission p ow er for eac h user sub ject to minim um SINR constraint and maximum p o wer budget. In order to analyze the mH-NN p erformance, this w ork ev aluate numerically the p erformance- complexit y tradeoff of three algorithms in solving the minim um p ow er allo cation problem. In the sequel, we describ e the main characteristics of b oth nonlinear SQP and classical ALM programming metho ds. 3.3. Se quential Quadr atic Pr o gr amming (SQP) The main feature of the sequential quadratic programming metho d is the determination of the NLP solution as the b oundary of the solutions of a quadratic problem sequence. In our case, the J ( p ) function is replaced b y a quadratic appro ximation of the Lagrangian function L , defined b elo w: L ( p , µ ) = J ( p ) + K X i =1 µ i [Γ ∗ i − Γ i ( p )] , (20) where the nonlinear constrain ts are replaced b y linear appro ximations thereof. Thus, eac h iteration of the SQP metho d solves the follo wing quadratic programming problem: max d J ( p k ) + ∇ J ( p k ) | d + 1 2 d | ∇ 2 p k L d (21) s . t . − ∇ Γ i ( p k ) | d + Γ ∗ i − Γ i ( p k ) ≥ 0 , i = 1 . . . K , 9 Algorithm 1 mH-NN for Po w er Assignment in OCDMA 1: Initialize: p with random v alues 2: In tro duce: auxiliary v ariables in the vector p , p ∗ = ( p , q ), such that Γ( p ) ≤ Γ ∗ b ecomes an equalit y constrain t: h ( p ∗ ) = Γ( p ) = Γ ∗ + q = 0. Also denote f ( p ∗ ) = J 1 ( p ). 3: Rep eat: While ( p + do not conv erge) do Get the v alue of h ( p + ) Get the Jacobiana matrix ∇ h ( p + ) Up date the v alue of p + from p + ← ( p + − ∇ h ( p + )) T ( ∇ h ( p + ) . ∇ h ( p + ) T ) − 1 ∇ h ( p + ) Apply the activ ation function End of while Get the vector ι opt giv en by ι opt = − h ∂ f ( p + ) ∂ p + 1 ∂ f ( p + ) ∂ p + 2 . . . ∂ f ( p + ) ∂ p + n i T Up date the v alue of p + ← p + + ∆ t ( W opt p + + ι opt ) . Un til ( p + sta y stationary) 4: End where ∇ 2 p k L = ∇ 2 L ( p k ) = ∇ 2 J ( p k ) − K X i =1 µ i ∇ 2 Γ i ( p k ). A more detailed analysis of the construction of the SQP metho d can b e found at [25]. Properties regarding con v ergence can b e found in [34]. The Algorithm 2 describ es a pseudo-co de for the SQP metho d. Algorithm 2 SQP – Sequential Quadratic Programming 1: Cho ose a starting p oin t ( p 0 , µ 0 ); 2: do k ← 0; 3: Rep eat Ev aluate J ( p k ) , ∇ J ( p k ) , ∇ 2 p k 2 L , Γ i ( p k ) e ∇ Γ i ( p k ); Solv e (21) to get d k and µ k +1 ; 4: p k +1 ← P ( p k + a k d k ); where P is the orthogonal pro jection operator in the b o x p min ≤ p ≤ p max . 5: End(rep eat) 3.4. A ugmente d L agr angian Metho d (ALM) The augmen ted Lagrangian metho d seeks to solve an NLP in an iterative w a y , where at each step an optimization problem with simple constraints is solv ed (in our problem, these constraints define the set Ω = { p ∈ R K p min ≤ p ≤ p max } ), namely the augmen ted Lagrangian function defined b elo w for the problem is minimized (2): 10 A ρ ( p , µ ) = J ( p ) + ρ 2 K X i =1  [Γ ∗ i − Γ i ( p )] + + µ i ρ  2 , (22) where ρ > 0 is the p enalt y parameter and µ i ≥ 0 are appro ximations for the Lagrange multipliers. Generally , in ALM the p enalt y parameter is set small when starting the metho d. So hop ed that the first iterations fa vor the achiev ement of viability . A more detailed analysis of ALM can b e found in [25] and additional results regarding conv ergence. can b e found in [21], [22], while extensiv e results and analysis for this optimization metho d can b e found in [20]. O algor ´ ıtimo 3 apresenta um pseudo c´ odigo para o ALM. Algorithm 3 ALM – Augmented Lagrangian metho d 1: Input Parameters: µ max > 0, µ 1 i ∈ [0 , µ max ], ∀ i = 1 , ..., m, { ε k } ⊂ R + so that lim k →∞ ε k = 0. 2: do k ← 1 3: Rep eat Calculate p k ∈ R n satisfying    P ( p k − ∇ A ( p k , µ k )) − p k    ≤ ε k , (23) where P is the pro jection op erator in the b ox p min ≤ p ≤ p max . 4: Up date the Lagrange m ultipliers and the p enalt y parameter. 5: End(rep eat) 4. Algorithm Implementation, F easibility and Complexit y In this subsection relev an t asp ects on the implementation of the t wo NLP metho ds, as well as the prop osed mo dified Hopfield ANN in solving the p o w er allo cation problem (3) are developed. The implemen tation asp ects of Algorithm 1, 2 and 3, taking into accoun t realistic OCDMA top ologies and considering similar system and channel parameter v alues, as in [28], [29], [25] are discussed in the next section. 4.1. A lgorithm Implementation Asp e cts In this subsection, the implemen tation asp ects of the previously optimization algorithms taking in to consideration realistic optical netw ork top ologies are explored. The adopted OCDMA netw ork arc hitecture is the same as at w ork [35]. The netw ork description and implementation are completely distributed and no training is required. In the follo wing w e discuss relev ant asp ects of implementation for the three optimization metho ds applied to PC OCDMA problem. 4.1.1. SQP Implementation The SQP metho d is initialized deploying a random approximation for the vector p with the elemen t en tries in the range [ p min ; p max ], which is usually adopted in the implementation of the Algo- rithm 2. The gradien t calculus was implemented by finite differences, while the quadratic subproblem w as solved using the interior p oin ts (IP) metho d applied to con vex quadratic programs [34]. 11 4.1.2. ALM Implementation The ALM algorithm is initialized with a random approximation for p 0 ∈ [ p min ; p max ], m ultipliers b eing zeros, and the p enalt y parameter v alue ρ = 10. In addition, if the metho d rep eats the solution of the subproblem with viable points, the algorithm will b e in terrupted and the con vergence attained. Besides, to solv e step 3 we used the BFGS quasi-Newton metho d [34]. 4.1.3. mH-NN Implementation The mH-NN algorithm is initialized by deploying a random appro ximation for p 0 ∈ [ p min ; p max ], follo wing the three steps describ ed in the diagram of Figure 2. ∆ t = 0 . 1 was adopted for distributed implemen tations and no training requirement. 4.2. Stopping Criterion and F e asibility The same stopping criterion has b een considered for all p o w er allo cation algorithms analyzed. Hence, if after the k th external iteration, p k results feasible, and the v alue ξ = k p k − p k − 1 k < 10 − 6 , (24) the algorithm stops execution, reac hing conv ergence. The fe asibility in the context of OCDMA p ow er control problem is considered as: F ( k ) i = [Γ ∗ i − Γ i ] + = max { 0 , | Γ ∗ i − Γ i |} if p min ≤ p k ≤ p max , ∀ k iteration (25) Th us, the null v alue of the feasibilit y F ( k ) = max {F ( k ) 1 , F ( k ) 2 , . . . , F ( k ) K } at the k th iteration indicates that p o w er vector p k satisfying the constrain ts of the problem (3). 4.3. Normalize d Me an Squar e d Err or (NMSE) The quality of the solution reached b y the p o w er allo cation algorithms in an iteration can b e measured b y the degree of proximit y to the optimal solution p ∗ , b eing quantified through the nor- malized mean square error (NMSE) when the equilibrium is reached. The quality of the solution ac hieved by an sp ecific algorithm in solving problem (3) is simply defined b y [15]: nmse = E  || p t [ n ] − p ∗ || 2 || p ∗ || 2  , (26) where || · || 2 denotes the squared Euclidean distance b et ween v ector p t [ n ] to the optimum solution v ector p ∗ at the n -th iteration of the t -th realization and E [ . ] is the exp ectation op erator. This measure will also b e analyzed in the context of this w ork. 4.4. A lgorithm R obustness The algorithm robustness R can be thought as the ratio b etw een the num b er of con vergence success cS to the total n umber of pro cess realizations T : R = cS T . 100 [%] (27) The conv ergence success ev en t is confirmed when the stopping criterion and feasibilit y are achiev ed. 12 5. Numerical Results In this section the ob jectiv e is to analyze through n umerical tests the b eha vior of the mo dified Hopfield neural netw ork for the resolution of the minim um pow er allocation problem in (3), aiming at impro ving the o v erall energy efficiency of the OCDMA system. Also, w e compared it with the non- linear programming metho ds, and taking in to consideration the realistic optical net w ork top ologies based on 2-D multiple-length extende d wavelength hopping prime c o de (MLEWHPC) [35]. The MLEWHPC codes are comp osed by a set of 2-D multiple-length constant-w eight EWHPCs. Suc h co de set is able to support a large v ariety of multimedia services, suc h as data, v oice, image and video, while accommo dating simultaneously all kinds of subscrib ers with v ery different bit-rates and quality-of-service (QoS). These co des hav e ideal correlation prop erties which can b e obtained b y extending wa velength-hopping prime co des with single length. The resulting MLEWHPCs present iden tical auto correlation p eaks and low cross-correlation v alues of at most one [35], [36], [37]. W e apply the three algorithms for different n um b ers of no des of an OCDMA netw ork considering the parameter v alues summarized in T able 1. The adopted target SNIR of 20 dB has b een chosen to achiev e suitable transmission in a single rate net work, resulting in a BER of less than the free limit of error ( ∼ 10 − 12 ). Besides, the c hoice of netw ork size considered for numerical analysis in this section tak es into account tw o scenarios: Scenario A represen ts a medium system loading with K = { 8 , 16 , 32 } users; Scenario B is more c hallenging high loading optical netw ork in considering K = { 48 , 64 , 128 } users and differen t levels of QoS. T able 1: Adopted Parameter V alues P arameter V alue Unit Mo dulation order M = 2 (Binary) T ransp onder Inefficiency ι = 2 . 7 [W/Gbps] White Noise std σ = 0 . 032 [dB] Planc k constant h = 6 . 63 × 10 − 34 [J/Hz] Chip Period T c = 9 [ps] Link length [4; 100] [km] Max. Laser p o wer p max = 20 [dBm] Min. Laser p o wer p min = ( p max − 90) [dBm] Scenario A: single rate Num b er of users K ∈ { 8 , 16 , 32 } [users] Min. user rate r serv min = 30 [Mbps] Sequence length F i = T b T c = 121 Max. BER acceptable ber max ≤ 10 − 12 SNIR target (min) Γ ∗ = 20 [dB] Scenario B: single rate, different QoS Num b er of users K ∈ { 48 , 64 , 128 } [users] Class I: SNIR (I) and Γ (I) ∗ = 17 [dB] Class I I: SNIR (II) Γ (II) ∗ = 20 [dB] Class I II: SNIR (II I) Γ (II I) ∗ = 22 [dB] Min. user rate p er class r serv min = 25 , 30 or 35 [Mbps] A lgorithm Initialization and Conver genc e P ow er vector initialization p 0 ∼ U [ p min ; p max ] Max. n umber of iterations I = 10 or 15 (under p ertubation) Con vergence criterion F ≤ 10 − 4 and ξ < 10 − 6 , p feasible 13 The sim ulation results were p erformed using the soft w are MatLab 8 . 0 running under Windows 10 Home Language, v ersion 1803, In tel(R)Core pro cessor i5-8250U @CPU 1.60GHz, 8.00GB of RAM and 64-bit op erating system. 5.1. Power Assignment Optimization (Sc enario A) F or the first scenario, the three algorithms addressed to ok into accoun t K = 8 , 16 or 32 users of an OCDMA netw ork whose parameters are describ ed in T able 1. Besides, T able 2 reveals the n umer- ical v alues referring to the p erformance of the three p o wer allo cation metho ds, including execution time, minim um v alue of sum-p ow er J 1 ( p ), n umber of iterations required for conv ergence, measure of feasibilit y F , num b er of flo ating-p oint op er ations p er se c ond ( flops ), normalized MSE, eq. (26), and sum-rate P K i =1 r i , with r i ≥ r serv min . The flops were obtained through an adaptation of Hang Qian’s Con tour FLOPS program 1 . T able 2 shows that all metho ds hav e conv erged to feasible p oints and also that the three metho ds reac h the optim um p o wer allo cation v alues for Scenario A. The b est feasibility lev els were obtained b y the SQP and mo dified Hopfield NN metho ds and the b est NMSE v alues were obtained by the mH-NN metho d, with adv antage for mH-NN method when K increases. F or 8 and 16 users, Hopfield and SQP had close amoun ts of FLOPS, while Hopfield w as slightly faster than SQP considering the execution time, sp ecifically for higher problem dimension, i.e., for K ≥ 32, SQP users reac hed lo wer n um b er of FLOPS. How ever, mH-NN metho d has achiev ed con vergence b y requiring shorter pro cessing time. It can b e observed that Hopfield and SQP p erformed v ery closely while ALM consumed a greater num b er of FLOPS and thus consumed more time for con vergence. Finally , the SQP and Hopfield metho ds attain practically the same levels of sum-rate, i.e., P K i =1 r i ≥ K · r serv min , for all user v alues while the ALM metho d presented little difference to the v alues of sum-rate due to its marginal feasibilit y p erformance degradation when compared to the other t wo metho ds. Despite the adopted constrain r serv min = 30 Mbps, the laser p o wer budget is enough to attain an a verage p er user rate of ¯ r i = 34 . 34 Mbps. Fig. 3 depicts the con vergence evolution of the individual p o w er levels for the three p o wer allo cation metho ds in solving (3) in comparison to the in version matrix (T arhuni) solution obtained b y (6). In Figure 3(a) one can observ e that the ALM method starts to approac h the individual pow er lev els of the solution after the fourth iteration and the conv ergence o ccurs in the sixth iteration. While the numerical conv ergence results for the SQP metho d in Fig.3(b) reveals that the optimal p o wer lev els are sim ultaneously attained early in the second iteration, satisfying the con vergence criteria in the third iteration. Finally , Fig.3(c) shows the mH-NN metho d reaching the solution righ t after the first iteration and con vergence criteria in the second iteration. Fig. 4.(a) presents the evolution of sum-p ow er J 1 ( p ) along the iterations. It ma y b e noted that regardless of the num b er of users, mH-NN algorithm reac hes the required minim um p o w er 1 Av ailable for download at www.mathworks.com/matlabcentral/fileexchange/50608- counting- the- floating- point- operations- flops 14 4 10 -6 10 -5 ALM - Iterations 0 1 2 3 4 5 6 Allocated Power [W] 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Power level - K=8 users. -- -- -- Tarhuni Solution a) ALM algorithm 2 10 -5 SQP - Iterations 0 1 2 3 Allocated Power [W] 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Power level - K=8 users. -- -- -- Tarhuni Solution a) SQP algorithm 1 10 -5 mH-NN - Iterations 0 1 2 Allocated Power [W] 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Power level - K=8 users. -- -- -- Tarhuni Solution a) mH-NN algorithm Figure 3: Po wer evolution until the equilibrium in Scenario A, K = 8 users: a) ALM; b) SQP; c) mH-NN algorithms 15 T able 2: Execution time, the num b er of external iterations for con vergence, feasibilit y for the three algorithms, FLOPS, NMSE and sum-rate considering the increase in the n umber of optical no des K . The av erage p er user rate has resulted ¯ r i = 34 . 34 Mbps. Metho d Time [sec.] J 1 ( p ) [W] Iterations F easibility F FLOPS NMSE P K i =1 r i [Mbps] 8 users ALM 0.34078 7.90100e-05 6 3.45931e-05 2.9615e+5 0.49289 2.74721e+2 SQP 0.01577 7.90197e-05 3 2.74802e-14 6.5850e+4 0.25633 2.74751e+2 mH-NN 0.01215 7.90105e-05 2 4.78985e-06 2.8885e+4 2.28502e-12 2.7475e+2 16 users ALM 0.59639 4.70084e-04 6 1.82361e-05 1.4106e+6 0.37642 5.5970e+2 SQP 0.01965 4.70114e-04 3 3.40500e-11 4.1945e+5 0.33332 5.4950e+2 mH-NN 0.01559 4.70094e-04 2 8.66522e-06 2.0925e+5 1.49264e-11 5.4942e+2 32 users ALM 0.98670 0.01869 6 2.7581e-06 2.6283e+7 0.30801 1.0989e+3 SQP 0.34545 0.01869 3 1.1180e-10 2.9736e+6 0.22872 1.0990e+3 mH-NN 0.01508 0.01869 2 9.8134e-18 3.2326e+6 4.44322e-07 1.0990e+3 assignmen t as early as the first iteration. On the other hand, SQP approaches the conv ergence after the second iteration and ALM b egins to approach the optimal p o wer solution after the third iteration. Complemen tary , Fig. 4.(b) depicts the ev olution of the sum of the user rates along with the iterations for the three metho ds discussed. As a result, the b eha vior is similar to the sum-p o w er of the corresp onding graphs (for the same K netw ork loading) in Fig. 4.(a). One can see that the ALM and SQP metho ds, at the b eginning of iterations, get very differen t v alues from the optimal v alue of sum-rate. A deep er analysis of the numerical con v ergence results in Fig.3 and Fig. 4.(a) evidences that nonlinear programming metho ds in the first iterations may preferentially seek b etter v alues of the ob jective function ov er feasibilit y . This b eha vior is describ ed in the literature as the v oracit y in reducing the ob jective function magnitude, whic h has already b een rep orted in [30] for the ALM metho d, but considering the dimension of the net works treated in the Scenario A and suc h c harac- teristic did not affect the conv ergence of the metho d herein. Ho wev er, we can highligh t that ALM demanded a greater num b er of iterations to ac hieve con vergence when compared to the SQP method. On the other hand, the mo dified Hopfield neural netw ork (mH-NN) metho d has demonstrated a dis- tinct b eha vior regarding NLP metho ds, approaching con v ergence v ery so on, typically after the first iteration. 16 Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -10 10 -5 10 0 K=16 users. Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -10 10 -5 10 0 K=8 users. Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -10 10 -5 10 0 K=32 users. mH-NN ALM SQP (a) Sum-P ow er levels Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 500 1000 1500 2000 K=8 users Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 500 1000 1500 2000 K=16 users. Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 500 1000 1500 2000 K=32 users. mH-NN ALM SQP (b) Sum-Rate lev els Figure 4: Sum-Po wer (a) and Sum-Rate (b) allocation across the iterations for K = 8 , 16 and 32 users in Scenario A. 17 5.2. Power Assignment Optimization with Differ ent L evels of QoS (Sc enario B) In this subsection, the minim um p o w er allo cation problem is defined for a single-rate differen t QoS OCDMA system, considering the parameters previously describ ed in T able 1 for Scenario B and larger net works with K = 48, 64 and 128 users. It also has analyzed three differen t levels of QoS determining distinct classes of QoS, which are asso ciated to different attainable single-rate OCDMA systems, namely Class I, I I and I I I, and defined b y the following SNIR: Γ ∗ = 17dB ( r ( i ) min ); 20dB ( r ( ii ) min ); and 22dB ( r ( ii i ) min ) As in the Scenario A, herein the three p o w er assignment algorithms are compared in terms of execu- tion time, minim um pow er solution, n umber of iterations, feasibilit y , FLOPS, NMSE and attainable data rates, as depicted in T able 3. It can b e noted that in Scenario B, the three metho ds maintained similar p erformance for the three classes of service (single-rate), Class I, I I and I I I to that obtained in Scenario A, where SQP and mH-NN metho ds presen t similar execution time v alues while attain the b est levels of feasibility . Moreov er, the mH-NN is able to attain b etter NMSE lev els due to the fact that already in the first iteration the metho d has b een able to ac hiev e suitable appro ximations for the solution even under high loading systems of K = 128 users. In turn, the ALM presen ted the w orst v alues for NMSE b ecause it consumes greater n um b er of iterations to ac hieve the same solution qualit y . Considering the attainable sum-rate, one can observe the b eha vior similar to Scenario A, where SQP and the mo dified Hopfield (mH-NN) presen t similar v alues and ALM a minimum differ- ence, probably due to its inferior p erformance regarding the feasibilit y . Notice that in solving the minim um optimal pow er allocation ( p ∗ ) problem with differen t lev el of QoS, one can observ e that the v alue of Γ i coincides with Γ ∗ i , so the v alue of CIR for eac h user is minimal. On the other hand, after con vergence (using p ∗ ) one can observe that sum-rate is sligh tly greater than K · r i, min . This result rev eals that the minim um p o wer v ector found takes in to consideration all the system impairments while the calculation of the a verage rate p er user do es not tak e suc h factors into account. Figure 5 shows the evolution of sum-p o wer along the external iterations for the three p o wer allo cation algorithms op erating under Class I, I I and I I I OCDMA system, resp ectively . Indeed, considering Fig. 5(a), it can b e noted that regardless of the n umber of users in the high loading Scenario B ( K ≥ 48 users), the SQP and mH-NN algorithms are able to reac h v alues v ery close to the minim um required p ow er in the second iteration, while the ALM metho d again presents difficulties in the first iterations to attain conv ergence due to its v oracit y feature, as already seen in Scenario A. Besides, insp ecting Fig. 5(b) and 5(c), depicting the ev olution of sum-p ow er and sum-rate ov er iterations to Class I I and I I I, resp ectiv ely , one can note that regardless of the num b er of users that SQP and mH-NN p erform v ery closely in terms of num b er of iterations tow ards the con vergence. The same trend is confirmed in terms of sum-rate for this scenario; Fig. 6 exhibits the ev olution of sum-rate with resp ect to the iterations for Class I, I I and I II. 18 T able 3: Class I, I I and I I I – Execution time, sum-p o wer, num b er of iterations for conv ergence, feasibilit y , FLOPS, NMSE and sum-rate for K = 48 , 64 , 128 optical no des. (a) Class I – Γ ∗ = 17dB ( r ( i ) min = 25Mbps) and ¯ r ( i ) i ≈ 29 . 49 Mbps. Method Time [sec.] J 1 ( p ) [W] Iterations F easibility F FLOPS NMSE P K i =1 r i [Mbps] 48 users ALM 1.21379 0.026371 6 1.24862e-06 3.5609e+8 0.36453 1.4155e+3 SQP 0.03135 0.026371 4 7.06701e-13 9.5138e+6 0.14921 1.4155e+3 mH-NN 0.03266 0.026371 3 5.48562e-18 2.2936e+7 2.67154e-09 1.4155e+3 64 users ALM 1.65431 0.035138 6 1.38161e-06 9.4509e+8 0.35719 1.8874e+3 SQP 0.08348 0.035138 4 1.61612e-12 2.2146e+7 0.24360 1.8874e+3 mH-NN 0.05468 0.035138 3 6.49071e-18 5.7321e+7 5.33412e-09 1.8874e+3 128 users ALM 2.02370 0.13220 7 9.88553e-05 3.9875e+9 0.39053 3.7748e+3 SQP 0.15053 0.13200 3 1.53822e-12 1.7241e+8 0.05997 3.7748e+3 mH-NN 0.34171 0.13200 3 6.490744e-18 4.8104e+8 3.19617e-09 3.7748e+3 (b) Class I I – Γ ∗ = 20dB ( r ( ii ) min = 30Mbps) and ¯ r ( ii ) i ≈ 34 . 34 Mbps. Method Time [sec.] J 1 ( p ) [W] Iterations F easibility F FLOPS NMSE P K i =1 r i [Mbps] 48 users ALM 1.73390 0.05262 6 1.27104e-06 4.3685e+8 0.33348 1.6485e+3 SQP 0.03134 0.05262 3 7.39512e-12 9.5230e+6 0.10153 1.6485e+3 mH-NN 0.03260 0.05262 3 1.15068e-17 2.4285e+7 6.49577e-09 1.6485e+3 64 users ALM 1.91301 0.07011 6 6.95347e-04 1.1431e+9 0.33092 2.1980e+3 SQP 0.08343 0.07012 3 1.50302e-11 2.2149e+7 0.14114 2.1980e+3 mH-NN 0.05468 0.07012 3 1.04083e-17 5.7321e+7 1.51221e-08 2.1980e+3 128 users ALM 2.48559 0.26679 7 5.58532e-04 4.9978e+9 0.39589 4.4017e+3 SQP 0.15053 0.26345 3 7.52932e-12 1.7534e+8 0.03146 4.3960e+3 mH-NN 0.17014 0.26345 3 1.09713e-17 4.5572e+8 3.73725e-09 4.3960e+3 (c) Class I I I – Γ ∗ = 22dB ( r ( ii i ) min = 35Mbps) and ¯ r ( ii i ) i ≈ 37 . 60 Mbps. Method Time [sec.] J 1 ( p ) [W] Iterations F easibility F Flops NMSE P K i =1 r i [Mbps] 48 users ALM 1.05472 0.08342 6 3.55791e-07 3.4363e+8 0.31702 1.8046e+3 SQP 0.03901 0.08342 3 2.41023e-11 9.5851e+6 0.05154 1.8046e+3 mH-NN 0.03052 0.08342 3 1.20122e-17 2.2936e+7 7.56132e-09 1.8046e+3 64 users ALM 1.420707 0.11116 6 1.43765e-06 9.7314e+8 0.30859 2.4061e+3 SQP 0.098363 0.11116 3 7.34822e-11 2.1428e+7 0.08301 2.4061e+3 mH-NN 0.105133 0.11116 3 1.551583e-17 5.4137e+7 2.56962e-08 2.4061e+3 128 users ALM 2.734116 0.41761 7 2.36749e-04 4.8290e+9 0.31848 4.8122e+3 SQP 0.215497 0.41761 3 4.23523e-11 1.7381e+8 0.01831 4.8122e+3 mH-NN 0.242005 0.41761 3 1.55134e-17 4.5573e+8 5.34532e-09 4.8122e+3 19 Iterations 0 1 2 3 4 5 6 10 -6 10 -4 10 -2 10 0 K=64 users ALM mH-NN SQP Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -6 10 -4 10 -2 10 0 K=48 users Iterations 0 2 4 6 8 J1= ' P [W] 10 -6 10 -4 10 -2 10 0 K=128 users (a) Γ ∗ = 17dB Iterations 0 1 2 3 4 5 6 10 -6 10 -4 10 -2 10 0 K=64 users ALM mH-NN SQP Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -6 10 -4 10 -2 10 0 K=48 users Iterations 0 2 4 6 8 10 -6 10 -4 10 -2 10 0 K=128 users (b) Γ ∗ = 20dB Iterations 0 1 2 3 4 5 6 10 -6 10 -4 10 -2 10 0 K=64 users mH-NN ALM SQP Iterations 0 1 2 3 4 5 6 J1= ' P [W] 10 -6 10 -4 10 -2 10 0 K=48 users Iterations 0 2 4 6 8 10 -6 10 -4 10 -2 10 0 K=128 users (c) Γ ∗ = 22dB Figure 5: Sum-p ow er levels for K = 48 , 64 and 128 users and differen t Γ ∗ ∈ [17 , 20 , 22]dB. 20 Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=64 users. mH-NN SQP ALM Iterations 0 1 2 3 4 5 6 7 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=128 users. Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=48 users. (a) Γ ∗ = 17dB Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=48 users. Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=64 users. mH-NN SQP ALM Iterations 0 1 2 3 4 5 6 7 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=128 users. (b) Γ ∗ = 20dB Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=48 users. Iterations 0 1 2 3 4 5 6 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=64 users. mH-NN ALM SQP Iterations 0 1 2 3 4 5 6 7 ' rate [Mbps] 0 1000 2000 3000 4000 5000 6000 7000 K=128 users. (c) Γ ∗ = 22dB Figure 6: Sum-rate allo cation across the iterations in for K = 48; 64 and 128 users (Scenario B) with different target SINR: a) Γ ∗ = 17 dB; b) Γ ∗ = 20 dB, and c) Γ ∗ = 22 dB. 21 5.3. Complexity Analysis The quality of the solutions achiev ed b y three algorithm is ev aluated through the NMSE metric presen ted in T ables 2 and 3. Figure ?? also sho ws the analysis of NMSE ev olution as a function of the n umber of in teractions for Scenario A considering systems with K = 8, 16 and 32 users. One can observ e that the b est MSE levels o ccur for SQP and mH-NN. This b eha vior was rep eated in Scenario B. Iterations 1 2 3 4 5 6 10 -10 10 -5 10 0 K=16 users. ALM SQP mH-NN Iterations 1 2 3 4 5 6 NMSE 10 -10 10 -5 10 0 K=8 users. Iterations 1 2 3 4 5 6 10 -10 10 -5 10 0 K=32 users. Figure 7: NMSE evolution for the methods addressed in relation to the p o w er vector p ∗ for U = [8 , 16 , 32] users. Fig. 8 put in p ersp ectiv e the computational time consumption for the mo dified Hopfield netw ork algorithm compared to the SQP for all K = { 8 , 16 , 32 , 48 , 64 , 128 } users configurations, highligh ting its sup eriorit y ov er ALM for the all scenarios ev aluated. W e can observ e that the implementation of the mH-NN Algorithm 1 is simple when compared to the ALM implementation, as well as the SQP metho d. The augmen ted Lagrangian metho d had lo wer p erformance for all users scenarios and it is worth men tioning that, based on results found in the literature, the use of a more sophisticated implemen tation of ALM, for instance, considering an ALM-based solver lik e ALGENCAN [32] would reduce the time burden pro cessing, b ecomes close to those obtained by the mH-NN and SQP metho ds. T o bring more insight on the three algorithms’ complexit y , T able 4 summarizes the algorithm robustness, measured as defined in (27), considering Scenario B with QoS rate 22dB for 100 real- izations. The robustness obtained demonstrated that the SQP and mH-NN metho ds result in full con vergence success under the considered high loading system scenario, while the ALM w as able to attain con vergence to acceptable solutions in the ma jority of the realizations. There are some cases where the ALM is not able to reac h full con v ergence, meaning that the metho d do es not attain a feasibilit y level of F = 10 − 4 , according to the stipulated feasibilit y criterion. Moreo v er, the stopping criterion was maintained in this context, i.e., given b y eq. (24), while the maximum num b er of external iterations I = 10 w as stipulated. In the equilibrium, the system p o wer allo cation solution giv en by eq. (6), the matrix − Γ ∗ H ma y ha ve en tries close to zero, since this is obtained through the gain matrix and target CIR, dep ending 22 Users 8 16 32 48 64 128 Time (seconds) 10 -2 10 -1 10 0 10 1 SQP ALM mH-NN Figure 8: Execution time for the three metho ds considering K = { 8 , 16 , 32 , 64 , 128 } users and QoS level 20 dB. T able 4: Robustness of the ALM, SQP and mH-NN algorithms for the p ow er allo cation problem in eq. (3) ov er T = 100 realizations, considering QoS level of 22dB. Users R -ALM R -SQP R -mH-NN 48 100% 100% 100% 64 98% 100% 100% 128 93% 100% 100% on the scale used. When we p erform the matrix sum ( I − Γ ∗ H ), small loss of information may o ccur. In this sum, one can lose information due to the order of magnitude of the en tries of the matrix ( − Γ ∗ H ). Hence, in the resolution of the system (6), due to the propagation errors phenomenon, esp ecially considering netw ork with large num b er of users, namely K = 48 , 64 and 128 users, the T arh uni solution (6) fails to achiev e feasibilit y levels as obtained b y mH-NN and SQP as depicted in T able 5. T able 5: Class C – Sum-p o wer and attainable feasibility from eq. (6) for increasing num b er of optical no des. # Users J 1 ( p ) [mW] F easibilit y , F T arh uni Σ -Po w er [mW] T arh uni F easibility F 48 83.42 1.20122e-17 83.38 2.51414e-04 64 111.16 1.55158e-17 111.01 3.901331e-04 128 417.61 1.55134e-17 417.36 6.007099e-04 5.3.1. Dynamic al Performanc e A nalysis The p ow er v ariations in the net work are related to the linear and non-linear effects asso ciated with the optical fib er, as w ell as to the coupling effects of c hannel p o wer, whic h are influenced b y the netw ork top ology , traffic v ariation and physics of optical amplifiers, as well as dynamic addition 23 and remov al of c hannels. In addition, there are the effects of the unpredictabilit y of time-v arying p enalties, such as p olarization effects [26]. In this subsection, we extend the PC algorithms robustness analysis by analyzing in a more chal- lenging p o w er allocation scenario considered a larger n umber of users and different QoS requiremen ts. Also, on fly mo difications were in tro duced in the op erational configurations of the optical net work whic h parameters are presented in the T able 1, but no w taking in to account the dynamic addition of c hannels. F or this analysis, after con vergence of the three metho ds considered 32 users and QoS lev el of 22dB, the n um b er of users w as increased b y 300% reac hing 128 activ e users in the OCDMA net work. SQP - Iterations 0 1 2 3 4 5 J1= ' P [W] 10 -3 10 -2 10 -1 10 0 10 1 32 users 128 users mH-NN - Iterations 0 1 2 3 4 5 J1= ' P [W] 10 -2 10 -1 10 0 10 1 32 users 128 users ALM - Iterations 0 2 4 6 8 10 12 J1= ' P [W] 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 32 users 128 users a) SQP b) Hopfield c) ALM Figure 9: Behavior of the a) SQP and b) Hopfield and c) ALM p o wer allo cation metho ds under the on fly increasing n umber of users in 300%. In Fig. 9, it can b e observ ed that after the restart with 128 users, the SQP , mH-NN and ALM metho ds resp ectively consumed 2, 2 and 6 iterations to reach full con vergence in the new 128 user’s p o w er allo cation equilibrium. Besides, the ALM b eha vior rev eals that as it w as taking into accoun t a restart of the metho d not taking adv an tage of the previous p o wer allo cation solution for 32 users, th us consuming 6 a high num b er of iterations to reac h the conv ergence. On the other hand SQP and Hopfield needed just only 2 iterations to fully ac hieve the new equilibrium. In order to ev aluate the capability of p o w er allo cation algorithms to re-establish the p o wer equi- librium after a strong p erturbation, it w as considered an optical p o wer p erturbation in the p o w er in the i th user, mo deled as: p i [ n ] = | α n · sin (1 . 5 π · n ) | + p ◦ i , n ≥ 0 (28) where α = 0 . 65, p ◦ i is the nominal transmitted p o w er for the i th ligh tpath and n represen ts the curren t iteration. F or illustration purp ose, Fig. 10 depicts the p erturbation function (28) considering p ◦ i = 0 . 1 W. It can b e seen that the effect of the disturbance tends to disapp ear as we increase the v alue of 24 n . This p erturbation w as considered in the first half of the iterations, i.e., 2 ≤ n ≤ 7, that w ould b e sp en t by the three metho ds to obtain the conv ergence, for K = 128 users and QoS of 22dB. Notice that under external p erturbation regime, I = 15 iterations has b een adopted. F or the SQP and Hopfield metho ds, the p erturbation was included starting from the second iteration, while for the ALM metho d it was included starting from the third iteration. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 p[n] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 10: Perturbation function p i [ n ] illustration × n umber of iteration n with p ◦ i = 0 . 1 W. The p o wer con trol resp onse to the disturbance is ev aluated along the 15 iterations, as depicted in Figure 11. One can notice that the three OCDMA p o w er allo cation metho ds are able to reco ver to the p erturbation introduced by eq. (28), re-establishing the con vergence equilibrium to the optimum p o w er allo cation p ∗ , but with p erturbation effects on the conv ergence pro cess, mainly noted at the b eginning iteration instants where p erturbation is inserted. Indeed, mo dified Hopfield-based metho d meets the conv ergence criterion v ery early , after 10 iterations, while ALM attains con vergence after 14 iterations, and SQP after 12 iterations. Thus, w e can see that Hopfield needed 1 iteration to r each equilibrium and 3 more to con verge following the F and ξ v alue criteria defined in T able 1. ALM - Iterations 0 5 10 15 J1= ' P [W] 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 mH-NN - Iterations 0 5 10 15 J1= ' P [W] 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 SQP - Iterations 0 5 10 15 J1= ' P [W] 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 Figure 11: Represen tative p ow er allo cation conv ergence p er ONU considering a p erturbation of eq. (10) inserted at 2 ≤ n ≤ 7 iteration for the Hopfield, SQP and ALM metho ds. 25 6. Conclusion In this w ork it was demonstrated that the metho dologies studied are adequate for the problem of minimum p o w er allo cation in OCDMA optical netw orks op erating under different system loading scenarios. W e highlight that the prop osed mo dified Hopfield netw ork prov ed to b e an effective alternativ e to solve the p o w er allo cation problem in OCDMA net works when compared to the classic programming metho ds due to its low computational cost, while its simplicity of implemen tation do es not require previous training. It is also w orth noting that the con v entional Hopfield netw ork usually consumes man y iterations to achiev e the conv ergence, motiv ating us to use the direction of the gradient to optimize the ob jective function in Step I I I of the Algorithm 1. Moreov er, in the problem (3) the gradient is the only direction of descent since the function is linear whic h justifies the fact that the metho d has consumed few iterations to obtain con vergence. While SQP method tak es adv antage of the simplicit y of the ob jective function for the construction of simpler subproblems, on the other hand the ALM minimizes the Lagrangian function in the b ox p min ≤ p ≤ p max , which is not simplified due to the c haracteristics of the ob jectiv e function. In this w ay , among the represen tative NLP metho ds ev aluated, one can alwa ys exp ect a b etter p erformance of the SQP metho d with resp ect to the ALM in solving the p o wer allo cation problem in OCDMA net works. 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