Global Optimization of Relay Placement for Seafloor Optical Wireless Networks
GGlobal Optimization of Relay Placement forSeafloor Optical Wireless Networks ∗ Yoshiaki Inoue † , Takahiro Kodama ‡ , and Tomotaka Kimura § June 5, 2020
Abstract
Optical wireless communication is a promising technology for underwater broadband access net-works, which are particularly important for high-resolution environmental monitoring applications.This paper focuses on a deep sea monitoring system, where an underwater optical wireless network isdeployed on the seafloor. We model such an optical wireless network as a general queueing networkand formulate an optimal relay placement problem, whose objective is to maximize the stabilityregion of the whole system, i.e., the supremum of the traffic volume that the network is capableof accommodating. The formulated optimization problem is further shown to be non-convex, sothat its global optimization is non-trivial. In this paper, we develop a global optimization methodfor this problem and we provide an efficient algorithm to compute an optimal solution. Throughnumerical evaluations, we show that a significant performance gain can be obtained by using thederived optimal solution.
Keywords:
Underwater communication, Visible light, Optical network, Queueing network, Globaloptimization, Reverse convex programming.
Real-time monitoring of underwater environments, such as ocean trenches and submarine volcanoes,is of great importance for scientific research toward the prevention and mitigation of natural disasters.In such monitoring applications, underwater wireless communication is a key enabling technology forbringing data from seafloor sensors to terrestrial base stations [1]. Traditionally, acoustic signals havebeen the primary medium for underwater wireless communications due to their ability to propagateover long distances with little energy dissipation. However, the main weakness of the acoustic channelis the quite limited data transmission capacity, which is inherent in the use of kHz-class carrierfrequencies. Therefore, acoustic-based underwater communication networks cannot accommodate thelarge amount of traffic generated by high-specification sensors such as underwater LIDARs and videocameras [2], which will be essential for near-future real-time underwater monitoring systems.Underwater optical wireless communication (UOWC) is a promising solution to this problem, whichcan achieve data rates of several hundred Mbps to about ten Gbps, provided that the transmissionrange is limited to tens to hundreds of meters [3, 4]. Because of this limitation on the propagation ∗ This work was supported in part by JSPS KAKENHI Grant Number 18K18007. † Y. Inoue is with Department of Information and Communications Technology, Graduate School of Engineering,Osaka University, Suita 565-0821, Japan (e-mail: [email protected]). ‡ T. Kodama is with Faculty of Engineering and Design, Kagawa University, Takamatsu 761-0396, Japan (e-mail:[email protected]). § T. Kimura is with Faculty of Science and Engineering, Doshisha University, Kyotanabe 610-0394, Japan (e-mail:[email protected]) a r X i v : . [ c s . N I] J un istance, it is necessary for the practical use of UOWC to construct a networked optical wireless in-frastructure consisting of multiple relay nodes. Such an underwater network is called an underwateroptical wireless network (UOWN), and its optimal design has become a major challenge for realiz-ing underwater real-time monitoring applications. Although a wired link (optical fiber) can also beconsidered as a connection method between relay nodes, this paper focuses on a relay system that is in-terconnected with wireless optical communication, because the ease of relocation provides operationalflexibility desirable for seafloor monitoring systems that are currently under development. Motivation : Most previous works on UOWNs assume vertical network architectures [4–8], wheredata packets generated by seafloor sensors are transferred to a terrestrial base station in multi-hopfashion via vertically deployed optical wireless relay nodes. In such a vertical network architecture,autonomous underwater vehicles (AUVs) hovering in the water are inevitably used as relay nodes inaddition to those anchored to the seafloor.Such architectures with relay AUVs are targeted at relatively shallow marine environments withdepths not exceeding 1000 meters, and their use for deep-sea monitoring is impractical due to thefollowing two reasons. Firstly, the monitoring of deep-sea environments with a vertical network requiresa very large number of AUV relay nodes to connect seafloor sensors to nodes at the sea surface, resultingin enormous costs. Secondly, the AUV relay nodes must be controlled to keep hovering in the turbulentwater, making it difficult to keep all the links stable. To the best of our knowledge, there has notbeen sufficient attention paid to investigating network architectures that can solve these problems ofdeep-sea environment monitoring.
Contributions : This paper proposes a seafloor optical wireless network (SOWN) , which enablesefficient data acquisition from deep-seafloor environments without employing hovering AUV relaynodes. The main components of the proposed SOWN are (i) a terrestrial base station, (ii) a sinknode on the seafloor connected to the terrestrial base station with an optical fiber, and (iii) anchoredrelay nodes horizontally deployed on the seafloor; see Fig. 1 for an illustration. The SOWN serves asan infrastructure to accommodate data traffic originating from a variety of sensors on the seafloor.The sensing data generated by each sensor is first collected at the nearest relay node, then deliveredto the sink node by optical wireless multi-hop transmission, and then transferred to the terrestrialbase station via the optical fiber. Its main advantage being constructed without hovering AUV nodes,the SOWN is a suitable network architecture for deep sea monitoring systems in terms of the cost-effectiveness and stability.The main focus of this paper is on the development of an optimal relay placement method , whichis the most fundamental challenge toward the optimal design of the SOWN. Underwater relay-nodeplacement problems have traditionally been discussed for acoustic-based networks [9–12], where itis known to be optimal to use a constant relay spacing [9], provided that the carrier frequency isappropriately selected. The key observation in this paper, however, is that such a constant-spacedrelay placement cannot fully extract the transmission capacity of the whole network in the SOWN,but rather a placement with optimally determined non-constant node spacing significantly improvesthe network performance. Such an improvement basically stems from the fact that the capacity of anunderwater optical channel is significantly affected by the node distance, due to the rapid attenuationof the optical signal with propagation distance [13–15]. In order to efficiently utilize the resources ofthe whole network, it is then necessary to arrange relay nodes in such a way that the distance betweenthe nodes gradually increases from the sink node to the end (leaf) node, because optical wireless linksclose to the sink node have to relay a large amount of sensing data transferred from upstream nodesand require a larger channel capacity than those away from the sink node.In this paper, we make this idea concrete by modeling the relay placement in the SOWN mathe-matically and performing its detailed analysis. More specifically, we first introduce a queueing-networkmodel whose input process differs depending on the relay-node placement, under a mild assumptionthat the packet generation follows a general stationary point process. Using this model, we then for-2 ptical fiberOptical wireless node
Sink nodeCore NW
Optical fiff bii er
Sink nodeAggregate link • • • •
Figure 1: A seafloor optical wireless network.mulate an optimal relay placement problem that aims to maximize the stability region of the wholesystem. The stability region is defined as the range of total traffic load that the network can accom-modate without exceeding the capacity of any communication links, which is of primary importancein designing communication networks because it determines the fundamental performance limit of thesystem as well known in the queueing theory.The main technical challenge we have to address in this relay placement problem is that theformulated optimization problem is inherently non-convex , as will be shown later. Therefore its globaloptimization is non-trivial, and general-purpose off-the-shelf algorithms can basically yield only localoptimal values. In this paper, we perform a detailed theoretical analysis on the optimal relay placementproblem and develop a global optimization algorithm that can be executed quite efficiently.As an initial study of the relay placement problem in the SOWN, this paper mainly focuses on aone-dimensional network, i.e., the case where relay nodes are placed along a straight line. Althoughthis assumption may restrict the direct applicability of the results to be obtained, this simplification al-lows us to reveal the exact structural properties of the global optimal solution, as will be shown in thispaper. Since the one-dimensional network is a fundamental building block of a more general two- orthree-dimensional UOWNs, the mathematical analysis developed in this paper also provides theoreticalinsights into the network design of such general UOWNs; we shall later demonstrate how the mathe-matical results obtained for the one-dimensional network can be extended to the two-dimensional case.It is also worth noting that the one-dimensional SOWN itself has important practical applications formitigating natural disasters (particularly earthquakes), such as high-resolution real-time monitoringof ocean trenches.
Organization : The rest of this paper is organized as follows. In Section 2, we provide a brief reviewof previous studies related to UOWNs and relay placement problems. In Section 3, we introduce aqueueing network model representing the SOWN and formulate an optimal relay placement problembased on it. In Section 4, we develop a global optimization method for the relay placement problemand investigate the mathematical structure of the obtained optimal solution. In Section 5, we firstexamine the performance of the obtained optimal solution through extensive numerical experiments.In particular, we show that the optimal placement with non-constant spacing significantly improvesthe system performance compared to the constant spacing case. We then demonstrate an extension ofthe obtained results to a two-dimensional seafloor network. Finally, this paper is concluded in Section3.
Gbps-class transmission capacity in UOWC has been achieved by using visible light bands, where theeffects of absorption and scattering losses are relatively small. UOWC is still in the early stages ofdevelopment, and several demonstration experiments have been carried out in recent years [16–20].On the other hand, theoretical investigations on UOWC channel characteristics have been carried outfrom earlier years, and various channel models have been proposed. Giles and Bankman [13] derived abasic signal-to-noise ratio (SNR) formula for UOWC channels, which was further extended to an end-to-end signal strength model by Doniec et al. [14], where its validity was confirmed in a real system.Elamassie et al. [15] have also extended this SNR formula and proposed a correction that takes intoaccount the contribution of the scattered light that partially reaches the detector. For a more detailedcharacterization of the UOWC channel, Tang et al. [21] have proposed a channel impulse responsemodel with a double gamma function. Jaruwatanadilok [22] has developed a channel model basedon radiative transfer theory as well and Zhang et al. [23] have presented a stochastic channel modelrepresenting the spatiotemporal probability distribution of propagating photons, taking into accountthe non-scattering, single-scattering, and multiple-scattering components.From the perspective of UOWC networking, Akhoundi et al. [24] have introduced an optical code-division multiple access (CDMA) underwater cellular network and evaluated its performance in severalwater types. Optical CDMA underwater networks have been further studied by Jamali et al. [25],reflecting the turbulent behavior of underwater channels. Jamali et al. [26] have also presented thebenefits of serial relayed multi-hop transmission using a bit detection and transfer (BDF) strategy,showing that multi-hop transmission can significantly improve system performance by mitigating ad-verse effects on all channels. Vavoulas et al. [27] have studied an effective path loss model in UOWCand characterized the connectivity of long-distance underwater communications. Saeed et al. [8] havediscussed network localization performances in UOWNs, through an analysis on network connectivity.In [28], they have also discussed an optimal placement of seaface anchor nodes in terms of the local-ization accuracy. To evaluate the performance of a video streaming under the sea, Al-Halafi et al. [29]have modeled UOWC channels with M/G/1 queues, assuming that there are multiple laser diodes inthe transmitter and multiple avalanche photodiodes in the receiver. Celik et al. [5] have analyzed theend-to-end bit error rates for the decode and forward (DF) and amplify and forward (AF) relaying in avertical UOWN. Furthermore, in [6], a sector-based opportunistic routing protocol have been devisedwhere packets are transmitted simultaneously to multiple relay nodes that fall within the range of adirectional beam. Xing et al. [30] have investigated problems of minimizing energy consumption andmaximizing SNR by performing relay node selection and power allocation simultaneously in the AFscheme.As mentioned earlier, relay-node placement under water has been studied in the context of acousticcommunication systems. Kam et al. [9] considered a problem of optimizing the frequency and nodelocation to minimize the energy consumption. Souza et al. [10] considered the minimization of energyconsumption taking into account the optimal number of hops, retransmission, coding rate, and SNR,where the distance between nodes of each hop is assumed to be constant. Liu et al. [11] have developedflow assignment and relay node placement methods in a vertical UOWN to maximize network lifetime,where it is assumed that relay nodes are fixed in horizontal coordinates and can be changed only invertical coordinates. Prasad et al. [12] have discussed a problem for a two-hop network that minimizesthe probability of receiving power falling below an outage-data-rate threshold by properly controllingthe locations of relay nodes and the transmission power.4 x =0 x x x x = La a a a a b b b b Figure 2: An illustration for the system model( N = 4). Node N Node N -1 Node λB|C N |λB|C N -1 |λB|C | Sink node
LLL
Figure 3: The SOWN modeled as a network ofG/G/1 queues.
Throughout the paper we follow the convention that for any k -dimensional ( k = 1 , , . . . ) vector y ∈ R k , its i th element is denoted by y i . We further define empty sum terms as zero.Let N = { , , . . . , N } ( N = 1 , , . . . ) denote the set of relay nodes. Relay nodes are aligned ona subset L := [0 , L ] of the real half-line R + , and the sink node is placed at the origin x = 0. Let x n ( n = 1 , , . . . , N ) denote the position of the n th node. We assume 0 ≤ x n ≤ x n +1 ( n = 1 , , . . . , N − x := 0 is defined accordingly.We assume x N = L holds, which implies that the one-dimensional region L is completely covered bythe sink node and N relay nodes.We assume that generation times of data packets follow a general stationary point process and thatthe generation points of those packets are uniformly distributed on L . Each packet is first collectedby the nearest node from its generation point and then transferred to the sink node with multi-hoptransmissions. More formally, we define the coverage area C n ⊆ R + ( n = 0 , , . . . , N ) of the n th nodeas its Voronoi cell, which is given by a half-open interval C n = [ a n , b n ) with a = 0 , a n = x n − + x n , n = 1 , , . . . , N, b n = a n +1 , n = 0 , , . . . , N − , b N = x N . (1)See Fig. 2 for an illustration. Clearly we have ∪ Nn =0 C n = [0 , x N ) and C i ∩ C j = ∅ for i (cid:54) = j . We furtherassume that packet transmissions are performed in the store-and-forward manner (DF relaying, inother words). The system is then represented as a network of N G/G/1 queues depicted in Fig. 3,where λ denotes the mean number of generated packets per unit time (within the whole covered area L ) and B denotes the mean data size.We define ρ n ( n = 1 , , . . . , N ) as the traffic intensity of external arrivals to the n th node, i.e., ρ n = λ |C n | L · B = q |C n | , q := λBL . (2)Observe that q represents the amount of data brought into the system per unit time, normalized bythe area length. Owing to [31, Page 142], the stability condition of this system is given by that foreach node i , the total traffic intensity of relayed packets does not exceed the link capacity: N (cid:88) n = i ρ n < R ( d i ) , i ∈ N , (3)where d n := x n − x n − ( n = 1 , , . . . ) denotes the distance between the ( n − n th nodes, and R ( d ) ( d ≥
0) denotes the effective transmission rate between two nodes with distance d , which isformulated as follows.Let SNR( d ) ( d ≥
0) denote the electrical SNR at distance d . A widely used model [13–15,30,32] forrepresenting the SNR of a UOWC channel is that the SNR( d ) takes a form proportional to d − α e − Kd for some coefficients α > K >
0. In this expression, d − α represents the signal attenuation dueto the geometric spreading of the light beam and α = 2 is usually used to represent the spherical5preading. On the other hand, e − Kd represents the contribution of absorption and scattering losses,and K is given by the sum of the absorption and scattering coefficients, which vary depending onthe type of water and the light wavelength. Readers are referred to [3, 7, 14, 33] for more detailedexplanations on such a theoretical characterization and its validation in a real system.In this paper, to avoid the singularity of d − α at the origin, we consider the following boundedexpression for SNR( d ), with a small (cid:15) > d ) = Ae − Kd ( (cid:15) + d ) − α , (4)where A denotes a constant that depends on physical parameters (an example will be given later inSection 5). It should be noted here that (cid:15) does not have a specific physical meaning: it is a parameterintended to correct the singular behavior that d − α diverges near the origin, and the value of (cid:15) has littleeffect on SNR( d ) unless d is very small (such a correction term is often used in the radio communicationliterature [34]). Owing to the Shannon-Hartley theorem, with W denoting the bandwidth, the effectivetransmission rate R ( d ) ( d ≥
0) is then expressed as R ( d ) = W log(1 + SNR( d )) . (5)In order not to restrict the applicability of our theoretical results, however, we do not assume anyspecific expression for the function R ( d ) in performing mathematical analysis below. Instead, we makeonly the following assumption on R ( d ), which is clearly satisfied by (4) and (5): Assumption 1.
The effective transmission rate R : [0 , ∞ ) → [0 , ∞ ) is a strictly decreasing, continu-ously differentiable convex function of the node distance, and lim d →∞ R ( d ) = 0 . Remark 2.
Another example of an expression for R ( d ) (other than the Shannon capacity (5)) isgiven as follows. Suppose that sensing information is coded and modulated with (i) the modulationlevel M [bits/symbol] and (ii) a forward-error-correction (FEC) code with code rate η ( < η < ). Alsosuppose that the FEC code enables the receiver to decode the signal with a negligible error-rate, providedthat the SNR does not exceed a threshold ζ . This is an abstraction of UOWC channels implementedwith standard modulation techniques, such as the on-off keying (OOK) and the quadrature amplitudemodulation (QAM) [35].In this setting, it is reasonable that the transmitter uses the maximum symbol rate (which equalsto the bandwidth W ) such that the constraint SNR( d ) ≤ ζ for error-free transmission is satisfied.Assuming that the noise spectral density is constant (i.e., white noise) over the operation frequencyrange, the expression (4) is rewritten as SNR( d ) = A (cid:48) e − Kd ( (cid:15) + d ) − α W − , where A (cid:48) does not dependon the symbol rate W . SNR( d ) then decreases with W , so that the maximum symbol rate is achievedif (and only if ) SNR( d ) = ζ , i.e., W = A (cid:48) e − Kd ( (cid:15) + d ) − α ζ − . As the effective transmission rate equals η · M · W , we then conclude R ( d ) = ηM A (cid:48) e − Kd ( (cid:15) + d ) − α ζ − , which clearly satisfies Assumption 1. Remark 3.
A refinement of the SNR equation correcting the exponential term as e − Kd β ( β ∈ (0 , )is proposed in [15]; the discussion above is still valid under such an extension. We see from (2) and (3) that the stability region of the system varies depending on the nodeplacement x := ( x , x , . . . , x N ) (cid:62) . Let ρ n ( q, x ) ( n = 1 , , . . . , N ) denote the traffic intensity ρ n ofexternal arrivals to the n th node, represented as a function of the normalized traffic intensity q andthe placement of relay nodes x (cf. (2)). The size of the stability region is characterized by thenormalized throughput limit q sup ( x ), which is defined as the least upper bound of the normalizedthroughput q for which the system is stable: q sup ( x ) = sup (cid:26) q ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = i ρ n ( q, x ) < R ( x i − x i − ) , i ∈ N (cid:27) (cid:26) q ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = i ρ n ( q, x ) ≤ R ( x i − x i − ) , i ∈ N (cid:27) . (6)The size of the stability region (i.e., the value of the normalized throughput limit q sup ( x )) is themost fundamental performance metric in designing the communication network. In this paper, we thusemploy q sup ( x ) as the objective function of our optimal placement problem. Specifically, we developa solution method to the following optimization problem:maximize x ∈ R N q sup ( x ) s . t . x N = L, x i +1 ≥ x i ≥ , i ∈ { , , . . . , N − } , which is rewritten as (cf. (6))maximize q ∈ R , x ∈ R N q s . t . N (cid:88) n = i ρ n ( q, x ) ≤ R ( x i − x i − ) , i ∈ { , , . . . , N } ,x N = L, x i +1 ≥ x i , i ∈ { , , . . . , N − } ,q ≥ , x i ≥ , i ∈ { , , . . . , N } . (U )Note that an optimal solution of (U ) provides not only an optimal placement x ∗ of relay nodes butalso the achievable maximum value of the normalized throughput limit q ∗ sup := q sup ( x ∗ ). In this section, we develop a global optimization method for (U ). We start with rewriting (U ) intoa more comprehensive form. We have from (1) and (2), ρ n ( q, x ) = q ( x n +1 − x n − )2 , n = 1 , , . . . , N − , ρ N ( q, x ) = q ( x N − x N − )2 , (7)so that we can rewrite (U ) in terms of the distance d n = x n − x n − ( n = 1 , , . . . , N ) of nodes:maximize q ∈ R , d ∈ R N q s . t . R ( d i ) − qd i − N (cid:88) n = i +1 qd n ≥ , i ∈ { , , . . . , N } ,q ≥ , N (cid:88) i =1 d i = L, d i ≥ , i ∈ { , , . . . , N } . (U)It is readily verified that (U) does not have a convex feasible region: each inequality constrainttakes the form that a convex function is not less than 0 (cf. Assumption 1), so that its feasible regionis the complement of a convex set. Such constraints are known as reverse convex constraints, and (U)belongs to a class of optimization problems called the reverse convex programming (RCP) [36]. Ingeneral, an optimization problemmaximize y ∈ R K u ( y ) s . t . f i ( y ) ≥ , i = 1 , , . . . , M, (R)with K variables and M constraints ( M ≥ K ) is called RCP if u and f i ( i = 1 , , . . . , M ) are quasi-convex. Note here that any equality constraints of the form (cid:80) Kk =1 w k y k = c ( c ∈ R , w k ∈ R ) can betranslated into the double number of linear (thus reverse convex) inequality constraints (cid:80) Kk =1 w k y k ≤ c and (cid:80) Kk =1 w k y k ≥ c . Due to the non-convexity of its feasible region, global optimization for RCP is notan easy task in general, and various algorithms to find a globally optimal solution have been developedin the literature (see e.g., [36–39] and references therein). Here we introduce a known theoreticalproperty of RCP, which will be used in our analysis. Let A := { y ; f i ( y ) ≥ i = 1 , , . . . , M ) } denotethe feasible region of (R) and let I ( y ) := { i ∈ { , , . . . , M } ; f i ( y ) = 0 } ( y ∈ A ).7 efinition 4 ([36, Def. 1]) . ¯ y ∈ A is called a basic solution of (R) if the matrix with row vectors {∇ f i ( ¯ y ); i ∈ I ( ¯ y ) } has rank K . Lemma 5 ([36, Th. 9]) . If u and f i ( i = 1 , , . . . , M ) are quasi-convex, (R) has an optimal solutionwhich is also basic. Remark 6.
A basic solution must satisfy at least K constraints with equality. Lemma 5 thus impliesthat there exists an optimal solution that satisfies at least K constraints with equality. In what follows, we develop a global optimization method (U) utilizing its special structure. Tothat end, we first introduce the following subproblem for each q > d ∈ R N N (cid:88) n =1 d n s . t . R ( d i ) − qd i − N (cid:88) n = i +1 qd n ≥ , i ∈ { , , . . . , N } ,d i ≥ , i ∈ { , , . . . , N } . (S q )The main difference between (U) and (S q ) is that q is not variable but fixed in (S q ). Also, the coverage (cid:80) Ni =1 d i of relay nodes is to be maximized in (S q ), while it is fixed to be L in (U).It is readily verified that the subproblem (S q ) still belongs to RCP. Owing to its special structure,however, a globally optimal solution of (S q ) is explicitly obtained. For a fixed q >
0, we define afunction g q : [0 , ∞ ) → ( −∞ , R (0) /q ] as g q ( x ) = R ( x ) q − x , x ≥ . (8)From Assumption 1, g q is a strictly decreasing continuous function, so that it has a unique inversefunction g − q : ( −∞ , R (0) /q ] → [0 , ∞ ). The following results show that an optimal solution of (S q ) isexplicitly constructed in terms of g q and g − q : Theorem 7. (i) If g − q (0) ≥ g q (0) , then the following d ∗ is an optimal solution of (S q ): d ∗ = (0 , , . . . , , g − q (0)) (cid:62) ∈ R k . (9) (ii) If g − q (0) < g q (0) , a backward recursion d ∗ N = g − q (0) , d ∗ i = g − q (cid:32) N (cid:88) n = i +1 d ∗ n (cid:33) , i = 1 , , . . . , N − well-defines d ∗ , d ∗ , . . . , d ∗ N such that < d ∗ i < d ∗ i +1 , i = 1 , , . . . , N − , (11) and the following d ∗ is an optimal solution of (S q ): d ∗ = ( d ∗ , d ∗ , . . . , d ∗ N ) (cid:62) ∈ R N . (12) Remark 8.
The proof of Theorem 7 is somewhat complicated because a careful treatment is necessaryto differentiate between the two cases (i) and (ii). Basically, our proof is based on the fact mentionedin Remark 6 that there exists an optimal solution satisfying at least N constraints with equality. Recallthat (S q ) has N constraints: N out of these are of the form g q ( d i ) ≥ (cid:80) Nn = i +1 d n and the others are ofthe form d i ≥ . The solution (12) satisfies the former with equality for i = 1 , , . . . , N and it has allnon-zero elements. On the other hand, the solution (9) has only one non-zero element, and it satisfies g q ( d i ) > (cid:80) Nn = i +1 d n for i = 2 , , . . . , N .
8e provide the proof of Theorem 7 in Appendix A.The optimal solution given in Theorem 7 takes a different form depending on whether or not g − q (0) ≥ g q (0) holds. While it is easy to check if this inequality holds for given q , we also have asimple criterion shown in the following lemma, which is useful in theoretical analysis below: Lemma 9.
Let q denote the unique solution of R (cid:18) R (0) q (cid:19) − R (0)2 = 0 , q > . (13) We then have g − q (0) ≥ g q (0) , ∀ q ≥ q , g − q (0) < g q (0) , ∀ q < q . (14) Proof.
Because g q ( x ) is strictly decreasing with respect to x , g − q (0) ≥ g q (0) ⇔ g q ( g q (0)) ≤ ⇔ R (cid:18) R (0) q (cid:19) − R (0)2 ≤ , (15)where we used (8) and q > R ( R (0) /q ) − R (0) / q andlim q → (cid:20) R (cid:18) R (0) q (cid:19) − R (0)2 (cid:21) = − R (0)2 < , lim q →∞ (cid:20) R (cid:18) R (0) q (cid:19) − R (0)2 (cid:21) = R (0)2 > , (16)so that (13) has the unique solution. Lemma 9 now follows immediately from (15).We then relate the subproblem (S q ) with the original problem (U). For q >
0, let d ∗ q := ( d ∗ q, , d ∗ q, , . . . , d ∗ q,N ) (cid:62) denote the optimal solution of (S q ) given in Theorem 7, and let x ∗ q,N := (cid:80) Ni =1 d ∗ q,i denote the corre-sponding optimal value. The following theorem shows that we can obtain a globally optimal solutionof (U) by iteratively solving (S q ): Theorem 10. (a) The optimal value x ∗ q,N of (S q ) is a continuous, strictly decreasing function of q with lim q → x ∗ q,N = ∞ and lim q →∞ x ∗ q,N = 0 .(b) The optimal value q ∗ sup of (U) is characterized as follows: x ∗ q,N > L ⇔ q < q ∗ sup , x ∗ q,N < L ⇔ q > q ∗ sup , x ∗ q,N = L ⇔ q = q ∗ sup . (17) Furthermore, ( q ∗ sup , d ∗ q ∗ sup ) is an optimal solution of (U). The proof of Theorem 10 is provided in Appendix B.An important consequence of Theorem 10 is that a globally optimal solution of (U) is obtained byiteratively solving the subproblem (S q ) with Theorem 7. Theorem 10 (b) indicates that we can judgeif a given value of q > q ∗ sup of the originalproblem (U), by comparing the optimal value x ∗ q,N of the subproblem (S q ) with the area length L .It also indicates that an optimal placement d ∗ for (U) is equal to that for the subproblem (S q ) with q = q ∗ sup , which is explicitly calculated from Theorem 7 once q ∗ sup is obtained. Theorem 10 (a) ensures(i) the existence of q such that x ∗ q,N = L (equivalently q = q ∗ sup ), and (ii) the monotonicity of x ∗ q,N withrespect to q . Therefore, a standard bisection method enables us to numerically find the value of q ∗ sup ,so that we can effectively compute the optimal placement d ∗ for the original problem (U); Algorithm1 summarizes such a procedure.Before closing this section, we conduct further investigations on mathematical structures of theobtained optimal solution. Let q ∗ sup ( L, N ) (
L > N = 1 , , . . . ) denote the optimal value q ∗ sup of thenormalized throughput limit, represented as a function of the area length L and the number of relaynodes N for a fixed transmission rate function R . 9 lgorithm 1 A global optimization algorithm for (U), in which the optimal solution d ∗ q and the optimal value x ∗ q,N of (S q ) is computed with Theorem 7. Input:
Number of nodes N , area length L , transmission rate function R ( d ) ( d ≥ (cid:15) . Output:
An optimal solution ( q ∗ sup , d ∗ ) of (U). Find q low and q up satisfying 0 < q low < q up , x ∗ q low ,N ≥ L , and x ∗ q up ,N < L . while q up − q low ≥ (cid:15) do q ← ( q low + q up ) / if x ∗ q,N ≥ L then q low ← q else q up ← q . q ∗ sup ← ( q low + q up ) / d ∗ ← d ∗ q sup . Lemma 11.
For a fixed N ( N = 1 , , . . . ), q ∗ sup ( L, N ) is a strictly decreasing function of L .Proof. Lemma 11 immediately follows from Theorem 10 (a) and (b): q ∗ sup ( N, L ) for fixed N is deter-mined by the unique solution of x ∗ q,N = L, q >
0, which is strictly decreasing with L . Theorem 12.
Let L := g − q (0) , where q is defined in Lemma 9. For any N = 1 , , . . . , we have g − q ∗ sup ( L,N ) (0) ≥ g q ∗ sup ( L,N ) (0) , ∀ L ≤ L , g − q ∗ sup ( L,N ) (0) < g q ∗ sup ( L,N ) (0) , ∀ L > L , (18) i.e., if L ≤ L , an optimal node placement for (U) is given by the form (9), and otherwise it is givenby the form (12), regardless of the number of nodes N .Proof. We readily obtain Theorem 12 from Theorem 7 (i), Lemma 9, Theorem 10, and Lemma 11,noting that q ∗ sup ( L, N ) = q ⇔ L = x ∗ q ,N = g − q (0).Theorem 12 shows that if the area length L is smaller than or equal to L , the normalized through-put limit q ∗ sup ( L, N ) is not affected by the number of nodes N , i.e., no performance gain can be obtainedby increasing the number of relay nodes in that case. On the other hand, if L ≥ L , we can verifyfrom Theorem 10 (a) and Theorem 7 (ii) that q ∗ sup ( L, N ) strictly increases with the number of nodes N . Finally, we provide a further characterization of the sequence d ∗ , d ∗ , . . . , d ∗ N defined by (10), re-stricting our attention to the case g − q (0) > g q (0) (i.e., L > L in view of Theorem 12). As shown inTheorem 7, it is optimal to place relay nodes with ascending node intervals in this case. In other words,if one determines node intervals in the reversed order (i.e., the interval between N th and ( N − d ∗ N , d ∗ N − , . . . , d ∗ is decreasing. Thefollowing theorem shows that in the optimal placement, the decrease in node intervals is at leastexponentially fast: Theorem 13.
Let γ denote a real number given by γ = 1 + ( g − q ) (cid:48) ( g (0)) = 1 + 1 g (cid:48) q (0) , (19) where f (cid:48) denotes the derivative of function f . If g − q (0) > g q (0) , then < γ < and(i) if ( g − q ) (cid:48) (0) > − , we have d ∗ i ≤ γ N − i d ∗ N ( i = 1 , , . . . , N − ), and(ii) if ( g − ) (cid:48) (0) ≤ − , we have d ∗ i ≤ γ N − i − d ∗ N − , ( i = 1 , , . . . , N − ). The proof of Theorem 13 is provided in Appendix C.10able 1: Default parameter values in numerical experiments.Symbol Unit Value P t Watt 0 . P n Watt 2 × − D Meter 0 . ϕ Degree 10 θ Degree 10 W Hz 5 × K × − Green light (550 [nm]):7 × − Blue light (450 [nm]): 2 × − (cid:15) Meter 1
In this section, we evaluate the performance of the obtained optimal solution. We first present extensivenumerical experiments to illustrate the effectiveness of the optimal solution, focusing on the one-dimensional SOWN discussed so far. We then provide an example of an optimal relay placementproblem for a two-dimensional
SOWN, which demonstrates how our result can be extended to a moregeneral situation.Throughout this section, we employ R ( d ) given in (5) and the following SNR equation [13]:SNR( d ) = P t D cos ϕ θ ) P n · e − Kr ( (cid:15) + r ) , (20)where P t denotes the transmitter power, P n denotes the noise power, D denotes the receiver aperturediameter, ϕ denotes the angle between the optical axis of the receiver and the line-of-sight betweenthe transmitter and the receiver, θ denotes the half angle transmitter beam width, K denotes thebeam attenuation coefficient, and (cid:15) denotes the constant introduced in (4). For the noise power P n ,we employ a constant value representing thermal noise, assuming that the contribution of shot noiseto P n is negligible due to the small power of received optical signals.Unless otherwise mentioned, we use parameter values summarized in Table 1 as the default values.We consider three different values for the beam attenuation coefficient K , reflecting its dependenceon the light wavelength [40] (we restrict our attention to the case of pure water, based on empiricalevidence [41] in the deep sea). We also set the area length L = 500 [m] unless otherwise mentioned. We start with providing an example of the optimal, non-constant node intervals we have obtained. Fig.4 illustrate the optimal node placement for the case of green light, and Fig. 5 shows the correspondingsequence of optimal node intervals ( d ∗ i ) i =1 , ,...,N . We observe that the optimal node interval d ∗ i isincreasing with i and that there is large difference between the values of d ∗ and d ∗ N . Fig. 6 shows themaximum normalized throughput limit q ∗ sup (achieved by the optimal relay placement) as a functionthe number N of relay nodes for the three wavelengths. We observe that adding a few relay nodesto the network drastically expands the stability region of the system. Fig. 7 shows the maximumnormalized throughput limit q ∗ sup as a function of the area length L for the blue light. For large valuesof L , we observe that q ∗ sup exponentially decreases as L increases. Furthermore, for relatively smallvalues of L , only little difference can be seen between the values of q ∗ sup for the number of nodes N = 5,11 from sink node [m] N=5 N=10 N=20
Figure 4: An illustration of the optimal placement of relay nodes for the green light, where eachsymbol represents a relay node. D i s t an c e d i be t w een node s i - and i Node number iN=5N=10N=20
Figure 5: Optimal distances { d i } i =1 , ,...,N be-tween relay nodes for the green light. N o r m a li z ed T h r oughpu t li m i t q * s up [ bp s / m ] Number of relay nodes NBlueGreenRed
Figure 6: Impact of the number of relay nodes N on the normalized throughput limit q ∗ sup . N = 10, and N = 20, and they coincide each other for quite small L , i.e., no performance gain can beobtained by using additional relay nodes (cf. Theorem 12).As shown in Fig. 6, q ∗ sup is a concave function of N : the impact of adding a relay node onimprovement in q ∗ sup decreases with an increase in the number N of relay nodes. The saturation of q ∗ sup with an increase in N is due to the fact that link capacity is inherently bounded above by R (0),so that q ∗ sup cannot exceed R (0) /L ( (cid:39) . × in the settings of Fig. 6). Therefore, a reasonablenumber of relay nodes can be determined by taking the cost-performance tradeoff into consideration.To discuss the tradeoff between the number of nodes and the system performance, we introduce atradeoff metric δ := q sup /N . By definition δ represents the (normalized) throughput of the system per relay node. Therefore, the number of relay nodes N maximizing δ is optimal in the sense that itmaximizes the cost-performance ratio. See Fig. 8, where the tradeoff metric δ is plotted as a functionof N for the three types of wavelengths. We observe that the optimal number of relay node dependson the light-wavelength and that the use of blue light is far more effective than that of green and redlights.Fig. 9 compares the optimal placement with the constant-interval placement d i = L/N ( i =1 , , . . . , N ), in terms of the tradeoff metric δ ; note here that between two different placements withthe same N , the ratio of δ is equal to that of the normalized throughput limit q sup itself. From Fig. 9we observe that a significant performance improvement is gained by using the optimized, non-constantnode intervals.We next discuss the effect of several practical aspects of the UOWC channel on the system perfor-12 N o r m a li z ed t h r oughpu t li m i t q * s up [ bp s / m ] Area
Length L [m]N=20N=10N=5 Figure 7: The normalized throughput limit q ∗ sup as a function of coverage length L . T r adeo ff m e t r i c δ Number of relay nodes NBlueGreenRed
Figure 8: The tradeoff metric δ as a function ofthe number N of relay nodes. T r adeo ff m e t r i c δ Number of relay nodes NOptimalConstant
Figure 9: Performance improvement gained bythe optimal solution, compared to the node place-ment with constant intervals. T r adeo ff m e t r i c δ Number of relay nodes N θ =10 θ =15 θ =20 θ =25 θ =30 Figure 10: Effect of the beam width θ on thetradeoff metric δ . T r adeo ff m e t r i c δ Number of relay nodes N φ =10 φ =20 φ =30 φ =40 φ =50 Figure 11: Effect of the misalignment φ on thetradeoff metric δ . T r adeo ff m e t r i c δ Number of relay nodes NFOV=25 [deg]FOV=20 [deg]FOV=15 [deg]FOV=10 [deg]FOV= [deg] Figure 12: Effect of the receiver FOV on thetradeoff metric δ .13ance. For brevity we present the results focusing on the case of blue light. Fig. 10, Fig. 11, and Fig.12 respectively show the effects of the beam width θ , the misalignment φ , and receivers’ field-of-view(FOV) on the tradeoff metric δ , where we assume the focal length F = 0 . φ has a less impact onit. This result suggests that (i) improving the receiver FOV is particularly of great importance indeveloping optical devices for SOWNs and (ii) narrowing the beam width (as long as the line-of-sight(LOS) link is maintained) can effectively increase the system performance.In general, underwater nodes may have uncertainty in their positioning due to the localizationerror. Fig. 13 shows the effect of such uncertainty on the system performance δ , where the positions x , x , . . . , x N − of intermediate relay nodes are perturbed by independent Gaussian noise with meanzero and standard deviation σ . We observe that the optimal placement still attains a good performanceeven with the localization error. Also, we see that the optimal number of nodes in terms of the cost-performance ratio is invariant regardless of δ .Finally, we compare the proposed SOWN with a conventional UOWN with vertical relays. Supposethat the seafloor is covered by N L relay nodes each of which collects data packets from an intervalof length L/N L and relays the packets to a tandem network of N V vertically aligned relays withinterval V /N V , where V denotes the depth of the seafloor from the surface of the sea. In this verticalnetwork, the normalized throughput limit is given by q sup = max { q >
0; (
L/N L ) q ≤ R ( V /N V ) } = N L R ( V /N V ) /L . In Fig. 14, q sup of the vertical UOWN is plotted as a function of total number of relaynodes N = N L ( N V + 1) for a case with V = 3000 [m] and N L = 5, where q sup in our proposed SOWNwith N = 10 is also plotted as a reference. As shown in the figure, to achieve a similar performance tothe proposed SOWN only with N = 10, the vertical UOWN requires at least N = 150 relay nodes forthe blue light and more than N = 200 relay nodes for the green light, which highlights the efficiencyof the proposed scheme in collecting data from deep sea. The optimization procedure we have developed can be extended to a two-dimensional SOWN asfollows. Suppose that a two-dimensional area
L × H ⊂ R of a seafloor is covered by N relay nodes,where L = [0 , L ] as before and H := [0 , H ] ( H > ,
0) and relay nodes are to be placed in a grid with non-constant spacings; the i th spacingalong the x -axis is denoted by (cid:96) i and the j th spacing along the y -axis is denoted by h j . See Fig. 15for an illustration.Let N L (resp. N L ) denote the number of nodes placed along the x -axis (resp. y -axis) for each row(resp. column). Note that the total number of relay nodes in the network (excluding the sink node) isgiven by N = N L N H −
1. Hereafter we refer to the node placed at the i th column ( i = 0 , , . . . , N L − j th row ( j = 0 , , . . . , N H −
1) as the ( i, j )th node, where the (0 , x i , y i ) of the ( i, j )th node is written as ( x i , y i ) = ( (cid:80) ik =1 (cid:96) k , (cid:80) jk (cid:48) =1 h k (cid:48) ).Similarly to the one-dimensional case, we assume that generation times of data packets followa general point process with intensity λ , and the generation points of those packets are distributeduniformly on L × H . The normalized traffic load q is then defined as (cf. (2)) q = λBLH , (21)where B denotes the mean data size as before. Each packet is collected by the node nearest from itsgeneration point and delivered to the sink node with multi-hop transmission. The cover area C i,j of the( i, j )th node is then its two-dimensional Voronoi cell, which is a rectangle as depicted in Fig. 15. Also,the traffic intensity ρ i,j ( q, x , y ) of external arrivals to the ( i, j )th node is given by ρ i,j ( q, x , y ) = q |C i,j | ;note here that cover areas C i,j are, by definition, determined by the node placement ( x , y ).14 T r adeo ff M e t r i c δ Number of relay nodes N σ =0.0 [m] σ =1.0 [m] σ =3.0 [m] σ =5.0 [m] Figure 13: Effect of localization uncertainty onthe tradeoff metric δ . (N=10)BlueGreen N o r m a li z ed T h r oughpu t li m i t q s up [ bp s / m ] Total number of relay nodes NVertical (Green)Vertical (Blue)
Figure 14: Comparison of the proposed SOWNwith a conventional vertical UOWN ( N L = 5). LH l l l h h Packet Transmission
Figure 15: A two-dimensional SOWN with N L =4 and N H = 3. T r adeo ff m e t r i c δ Number of relay nodes N H along the y-axisOptimalConstant Figure 16: Comparison of the optimal and con-stant relay intervals in the two-dimensional case( N L = 6).To formulate an optimal relay placement problem in the two-dimensional case, we have to specifythe routing paths. Here we concentrate on the basic routing policy shown in Fig. 15: each packet isfirst transmitted along the x -axis to the left-most node and then transmitted along the y -axis to thesink node. In this setting, the one-dimensional optimal relay placement problem (U) maximizing thenormalized throughput limit q sup readily extends to the two-dimensional case:maximize q ∈ R , (cid:96) ∈ R N L − , h ∈ R N H − , q s . t . R ( (cid:96) i ) − max j ∈N H (cid:18) h j + h j +1 (cid:19) (cid:32) q(cid:96) i − N (cid:88) n = i +1 q(cid:96) n (cid:33) ≥ , i ∈ N L ,R ( h j ) − L qh j − N (cid:88) n = j +1 qh n ≥ , j ∈ N L ,q ≥ , N L (cid:88) i =1 (cid:96) i = L, N H (cid:88) j =1 h j = H, (cid:96) i ≥ , i ∈ N L , h j ≥ , j ∈ N H , (U )15here N L := { , , . . . , N L − } , N H := { , , . . . , N H − } , and h N H := 0.Observe that for a fixed spacings (cid:96) (resp. h ) along the x -axis (resp. y -axis), the optimizationproblem (U ) reduces to the one-dimensional problem (U) by replacing R ( · ) with R ( · ) /L (resp. R ( · ) / max j ∈ N H (( h j + h j +1 ) / y -axis becomebottlenecks of the system (we shall shortly come back to this point), we can obtain optimal spacings( (cid:96) , h ) by first solving (U) with R ( · ) replaced by R ( · ) /L to obtain an optimal spacing h ∗ along the y -axis, and then again solving (U) with R ( · ) replaced by R ( · ) / max j ∈ N H (( h ∗ j + h ∗ j +1 ) /
2) to obtain anoptimal spacing l ∗ along the x -axis. Write the optimal value obtained at the first step as q ∗ y · L andthat obtained at the second step as q ∗ x · max j ∈ N H ( h ∗ j + h ∗ j +1 ) /
2. The achieved objective value is thengiven by min( q ∗ x , q ∗ y ), as it is the maximum q satisfying all the constraints. That is, we can ensurethat the above-mentioned assumption (communication links along the y -axis are bottlenecks) is indeedsatisfied by checking if q ∗ y < q ∗ x is satisfied. Because q ∗ x increases with N L , we can find the minimum N L such that q ∗ y < q ∗ x holds, and in that case, the maximum normalized throughput limit q ∗ sup (i.e.,the optimal value of (U )) is given by q ∗ sup = q ∗ y .Similarly to the one-dimensional case, we define the tradeoff metric δ := q sup /N . Fig. 16 com-pares the optimal relay placement with the constant placement l = ( L/N L , L/N L , . . . , L/N L ), h =( H/N H , H/N H , . . . , H/N H ) for N L = 6 and the blue light with the parameter values in Table 1. Weobserve that the optimal relay placement developed in this paper provides a significant performanceimprovement also in the two-dimensional network. In this paper, we considered an optimal relay placement problem for a one-dimensional SOWN. Wemodeled such a network as a queueing network with a general input process, and formulated therelay placement problem whose objective is to maximize the stability region of the whole system.We showed that this problem belongs to RCP, whose global optimization is generally a difficult task.We then developed an algorithm (presented in Algorithm 1) to efficiently compute a globally optimalsolution, and investigated mathematical structure of the obtained optimal solution. Through numericalevaluations, we showed that the obtained optimal solution provides a significant performance gain,compared to the conventional constant-interval relay placement. We further proposed a method todetermine a reasonable number of relay nodes by introducing the tradeoff metric δ , defined as theachieved system performance per relay node. We also presented extensive numerical experiments,where the proposed method is compared with a conventional vertical relay, and discussions on severalpractical aspects of UOWC channels, such as the misalignment, the FOV, and the uncertainty in nodeplacement, are given.We finally demonstrated how to extend the developed optimal placement into a two-dimensionalSOWN. While we focused on the case of regular-grid topology and a simple routing policy depicted inFig. 15, there would be various possible other directions for extensions. For example, the mathematicalresult shown in Theorem 13 suggests that it is efficient to employ relay intervals d N , d N − , . . . , d which decrease at least exponentially fast. Future works include an application of this insight totwo-dimensional networks with more flexible topologies and sophisticated routing mechanisms thatenables us to deal with occurrences of node and link failure, which is the important aspect for thereliability of underwater networks as an infrastructure. A Proof of Theorem 7
For ease of presentation, we introduce a slightly generalized problem. Let g : [0 , ∞ ) → [ −∞ , g (0))denote a convex function with g (0) > k = 1 , , . . . , we define f ( k ) i ( y ) ( y ∈ R k , y ≥ ) as f ( k ) i ( y ) = g ( y i ) − i − (cid:88) n =1 y n , i = 1 , , . . . , k. (22)Let f ( k ) ( y ) := ( f ( k )1 ( y ) , f ( k )2 ( y ) , . . . , f ( k ) k ( y )) (cid:62) and u ( k ) ( y ) := (cid:80) ki =1 y i . We consider the followingoptimization problem for k = 1 , , . . . :maximize y ∈ R k u ( k ) ( y ) s . t . f ( k ) ( y ) ≥ , y ≥ . (P ( k ) )Since f ( k ) i and u ( k ) are both convex, (P ( k ) ) belongs to RCP. We can readily verify that (P ( N ) ) reducesto (S q ), letting d n = y N − n +1 ( n = 1 , , . . . , N ) and g ( x ) = g q ( x ) ( x ≥ g − : ( −∞ , g (0)] → [0 , ∞ ) denote the inverse function of g . Because g is assumed to be convexand strictly decreasing, g − is also convex and strictly decreasing. Note that g − ( g (0)) = 0 , g − ( z ) > , −∞ < z < g (0) . (23)Below we provide a proof of the following lemma, which readily implies Theorem 7: Lemma 14. (i) If g − (0) ≥ g (0) , then the following y ∗ is an optimal solution of (P ( k ) ) y ∗ = ( g − (0) , , , . . . , (cid:62) ∈ R k , (24) (ii) If g − (0) < g (0) , a recursion y ∗ = g − (0) , y ∗ i = g − i − (cid:88) j =1 y ∗ j , i = 2 , , . . . , (25) well defines a sequence { y ∗ i } ∗ i =1 , ,... such that < y ∗ i +1 < y ∗ i , i = 1 , , . . . , (26) and for k = 1 , , . . . , the following y ∗ is an optimal solution of (P ( k ) ). y ∗ = ( y ∗ , y ∗ , . . . , y ∗ k ) (cid:62) ∈ R k . (27)We start with considering the number of zeros that an optimal solution can have (see Remark 8).Let A ( k ) ⊆ R k denote the set of feasible solutions of (P ( k ) ), and let Y ( k ) ⊆ A ( k ) denote the set ofoptimal solutions. For y ∈ R k , we define κ ( y ) as the number of elements of y which are equal to zero.We define φ ( k ) as the maximum number of zeros in an optimal solution of (P ( k ) ): φ ( k ) = max { κ ( y ); y ∈ Y ( k ) } . (28) Lemma 15.
For k = 1 , , . . . , the optimal value of (P ( k ) ) is equal to that of (P ( k − φ ( k ) ) ): max { u ( k ) ( y ); y ∈ A ( k ) } = max { u ( k − φ ( k ) ) ( y ); y ∈ A ( k − φ ( k ) ) } . (29) Proof.
Since the case of φ ( k ) = 0 is trivial, we assume φ ( k ) >
0. For any y ∈ R k , let y + ,i denotethe i th non-zero element of y . Let Y ( k )+ := { ( y + , , y + , , . . . , y + ,k − φ ( k ) ) (cid:62) ; κ ( y ) = φ ( k ) , y ∈ Y ( k ) } . It isreadily verified that Y ( k )+ ⊆ A ( k − φ ( k ) ) , and therefore max { u ( k ) ( y ); y ∈ A ( k ) } = max { u ( k − φ ( k ) ) ( y ); y ∈Y ( k )+ } ≤ max { u ( k − φ ( k ) ) ( y ); y ∈ A ( k − φ ( k ) ) } . We then obtain Lemma 15 because max { u ( k − φ ( k ) ) ( y ); y ∈A ( k − φ ( k ) ) } ≤ max { u ( k ) ( y ); y ∈ A ( k ) } also follows from that ( ¯ y , , , . . . , (cid:62) ∈ R k is a feasible solutionof (P ( k ) ) for any ¯ y ∈ Y ( k − φ ( k ) ) . 17 orollary 16. φ ( k − φ ( k ) ) = 0 ( k = 1 , , . . . ).Proof. Because the case of φ ( k ) = 0 is trivial, we assume φ ( k ) >
0. If φ ( k − φ ( k ) ) > ( k − φ ( k ) ) )has an optimal solution ˆ y ∈ Y ( k − φ ( k ) ) such that κ ( ˆ y ) >
0. It then follows from Lemma 15 thatˆ y e := ( ˆ y , , , . . . , (cid:62) ∈ R k is an optimal solution of (P ( k ) ). This implies κ ( ˆ y e ) = κ ( ˆ y ) + φ ( k ) > φ ( k ) ,which contradicts the definition (28) of φ ( k ) .We can verify that ∇ f ( k ) ( y ) is a lower-triangular matrix with non-zero (negative) diagonal ele-ments. We thus have det( ∇ f ( k ) ( y )) (cid:54) = 0, so that rank( ∇ f ( k ) ( y )) = k . Therefore, if ¯ y ∈ R k satisfies f ( k ) ( ¯ y ) = and ¯ y ≥ , then it is a basic solution of (P ( k ) ). Furthermore, the following Lemma 17immediately follows from Remark 6: Lemma 17. If ¯ y ∈ R k is a basic solution of (P ( k ) ) satisfying ¯ y > , then f ( k ) ( ¯ y ) = holds. Lemma 18.
For fixed k ( k = 1 , , . . . ), the followings hold:(i) If there exists no vector ¯ y ∈ R k satisfying ¯ y > and f ( k ) ( ¯ y ) = , then φ ( k ) > .(ii) If φ ( k ) = 0 , then (P ( k ) ) has an optimal solution ¯ y ∈ R ( k ) satisfying ¯ y > and f ( k ) ( ¯ y ) = .Proof. We first consider (ii). When φ ( k ) = 0, the elements of each optimal solution of (P ( k ) ) are allpositive. It then follows from Lemma 5 that (P ( k ) ) has an optimal solution ¯ y > which is also basic.Therefore, we have f ( ¯ y ) = from Lemma 17, which proves (ii).We next consider (i). The contraposition of (ii) is that if there exists no optimal solution ¯ y of(P ( k ) ) satisfying ¯ y > and f ( k ) ( ¯ y ) = , then φ ( k ) >
0. We thus have (i) from Y ( k ) ⊆ R k .We can readily verify that if { y ∗ i } i =1 , ,... in (25) is well-defined, y = ( y ∗ , y ∗ , . . . , y ∗ k ) (cid:62) is the uniquesolution of f ( k ) ( y ) = 0. Note that y ∗ is always well defined, while y ∗ i ( i = 1 , , . . . ) is not well-definedif (cid:80) i − j =1 y ∗ i > g (0) because the domain of g − is ( −∞ , g (0)]. We then define N ∗ ∈ { , , . . . } ∪ {∞} as N ∗ = sup i ∈ { , , . . . } ; i − (cid:88) j =1 y ∗ j < g (0) . (30)By definition y ∗ i is well-defined at least for 1 ≤ i < N ∗ . In addition, if N ∗ < ∞ , then (cid:80) N ∗ − j =1 y ∗ j ≥ g (0),so that g − ( (cid:80) N ∗ − j =1 y ∗ j ) is either equal to zero or not well-defined. We thus have y ∗ i > , ≤ i < N ∗ . (31)Furthermore, because g − is a strictly decreasing function, y ∗ i < y ∗ i − , ≤ i < N ∗ . (32)Let { z ∗ i } ≤ i
Either N ∗ = 2 or N ∗ = ∞ holds. Specifically, if g − (0) ≥ g (0) , then N ∗ = 2 , andotherwise N ∗ = ∞ .Proof. Because g − (0) ≥ g (0) ⇒ N ∗ = 2 immediately follows from the definitions of y ∗ i and N ∗ , weconsider the case of g − (0) < g (0) below. We first show that g − (0) < g (0) ⇒ g (cid:48) (0) < − . (42)Since g is assumed to be a convex function, its derivative g (cid:48) is a non-decreasing function. It thenfollows that if g (cid:48) (0) ≥ −
1, then g (cid:48) ( y ) ≥ − y ≥
0, so that y ∗ = (cid:90) g − (0)0 d y ≥ (cid:90) g − (0)0 ( − g (cid:48) ( y ))d y = g (0) , (43)i.e., g (cid:48) (0) ≥ − ⇒ y ∗ ≥ g (0). We thus obtain (42), taking the contraposition.Below, we proceed by considering two exclusive cases, under the assumption g − (0) < g (0). Case 1: ( g − ) (cid:48) (0) > − g − : ( −∞ , g (0)] → [0 , ∞ ) is a convex function as noted above, its derivative ( g − ) (cid:48) is anon-decreasing function, so that we have from ( g − ) (cid:48) (0) > − g − ) (cid:48) ( z ) > − , ≤ z ≤ g (0) . (44)It then follows from (38) that h is a strictly increasing function. Furthermore, we obtain from g − ( z ) > −∞ ≤ z < g (0)) and (41), 0 < h ( z ) < g (0) , ≤ z < g (0) . (45)From (39) and (45), we can readily show by induction that0 < z ∗ i < g (0) , i = 1 , , . . . , (46)which implies N ∗ = ∞ . Case 2: ( g − ) (cid:48) (0) ≤ − g − ) (cid:48) ( g (0)) = 1 /g (cid:48) (0), we have (cf. (44))( g − ) (cid:48) ( g (0)) > − . (47)19ecall that g is assumed to be continuously differentiable, so that ( g − ) (cid:48) ( y ) is a continuous function.We can then verify that there exists β ∈ [0 , g (0)) satisfying ( g − ) (cid:48) ( β ) = −
1, using ( g − ) (cid:48) (0) ≤ − β is not necessarily unique, because ( g − ) (cid:48) is notnecessarily strictly increasing. Instead, the set of such β is bounded above, so that its maximum value β ∗ is uniquely obtained: β ∗ = max { β ∈ [0 , g (0)); ( g − ) (cid:48) ( β ) = − } . (48)It is then readily verified that ( g − ) (cid:48) ( z ) ≤ − , ≤ z ≤ β ∗ , (49)( g − ) (cid:48) ( z ) > − , β ∗ < z ≤ g (0) . (50)We have from (49), β ∗ = (cid:90) β ∗ d z ≤ (cid:90) β ∗ ( − · ( g − ) (cid:48) ( z )d z = g − (0) − g − ( β ∗ ) < g − (0) , (51)so that (37) and g − (0) < g (0) imply β ∗ < z ∗ = g − (0) < g (0) . (52)It follows from (38) and (50) that h ( z ) is strictly increasing for β ∗ < z ≤ g (0). Furthermore, (40)implies h ( β ∗ ) > β ∗ . We can then verify that (cf. (45)): β ∗ < h ( z ) < g (0) , β ∗ ≤ z < g (0) . (53)Therefore, we can readily show by induction using (52) and (53) that β ∗ < z ∗ i < g (0) , i = 1 , , . . . , (54)which implies N ∗ = ∞ . Lemma 20.
For k = 2 , , . . . , the followings hold:(i) If g − (0) ≥ g (0) , there exists no vector ¯ y ∈ R k such that f ( k ) ( ¯ y ) = and ¯ y > .(ii) If g − (0) < g (0) , ¯ y = ( y ∗ , y ∗ , . . . , y ∗ k ) (cid:62) is the unique solution of f ( k ) ( ¯ y ) = and ¯ y > .Proof. Lemma 20 immediately follows from (22), (25), Lemma 19, and the definition of N ∗ .We are now in a position to prove Lemma 14. Proof of Lemma 14.
We first consider the case of g − (0) ≥ g (0). In this case, we have φ ( k ) > k = 2 , , . . . from Lemma 18 (i) and Lemma 20 (i). Note that y = g − (0) is the optimal solution of(P (1) ), and its optimal value is also equal to g − (0). Obviously, we have φ (2) = 1, so that the optimalvalue of (P (2) ) equals to g − (0). Owing to Lemma 15, the optimal value of (P (3) ) is then equal to g − (0), which implies φ (3) = 2. Therefore, proceeding in the same way, we can readily show that φ ( k ) = k − ( k ) ) is equal to g − (0) for k = 2 , , . . . . Because (24) achievesthe optimal value g − (0) of (P ( k ) ), we obtain Lemma 14 (i).We next consider the case of g − (0) < g (0). Note first that the well-definedness of { y i } ∗ i =1 , ,... and (26) have been proved in (31), (32), and Lemma 19. It then follows from Lemma 20 (ii) that( y ∗ , y ∗ , . . . , y ∗ k − φ ( k ) ) (cid:62) ∈ R k is the unique solution of f ( k − φ ( k ) ) ( ¯ y ) = and ¯ y > . Therefore, from Corol-lary 16 and Lemma 18 (ii), we can verify that ( y ∗ , y ∗ , . . . , y ∗ k − φ ( k ) ) (cid:62) is an optimal solution of (P ( k − φ ( k ) ) )and that from Lemma 15, max { u ( k ) ( y ); y ∈ A ( k ) } = (cid:80) k − φ ( k ) i =1 y ∗ i . Because ( y ∗ , y ∗ , . . . , y ∗ k ) (cid:62) ∈ A ( k ) , thisequation implies (cid:80) ki =1 y ∗ i ≤ (cid:80) k − φ ( k ) i =1 y ∗ i . Therefore, from (31) we have φ ( k ) = 0, so that ( y ∗ , y ∗ , . . . , y ∗ k ) (cid:62) is an optimal solution of (P ( k ) ). 20 Proof of Theorem 10
B.1 Proof of Theorem 10 (a)
Note first that Theorem 7 and Lemma 9 imply x ∗ q,N ≥ g − q (0) , < q < q , x ∗ q,N = g − q (0) , q ≥ q , (55)so that we obtain lim q → x ∗ q,N ≥ lim q →∞ g − q (0) = ∞ , and lim q →∞ x ∗ q,N = lim q →∞ g − q (0) = 0 . For q ≥ q , we have from (55) that x ∗ q,N is continuous and strictly decreasing in q (cf. (8)). Wethen assume 0 < q < q . Let h q : ( −∞ , g q (0)] → R be defined as (cf. (38)) h q ( s ) = s + g − q ( s ) , s > . (56)We define s q,i ( i = 1 , , . . . , N ) as (cf. (33) and (39)) s q, = g − q (0) , s q,i = h q ( s i − ) , i = 2 , , . . . , N. (57)It is then readily verified from Theorem 7 that x ∗ q,N = s q,N , < q < q . (58) x ∗ q,N is thus a continuous function of q . By definition, s q, and h q ( s ) (for a fixed s ) are strictlydecreasing with respect to q (cf. (8)). Furthermore, as shown in the proof of Lemma 19, for a fixed q , h q ( s ) is strictly increasing with respect to s for s q, = g − q (0) < s ≤ g q (0). We can then show byinduction that s q,i > s q (cid:48) ,i , ( i = 1 , , . . . , N ) for any 0 < q < q (cid:48) < q , which and (58) prove that x ∗ q,N iscontinuous and strictly decreasing for 0 < q < q .What remains is to prove that x ∗ q,N is continuous at q = q . By definition of q , we havelim q → q − g − q (0) = g q (0), so that lim q → q − s q,i = g q (0) ( i = 1 , , . . . , N ). x ∗ q,N is thus continuousat q = q because lim q → q + x ∗ q,N = x ∗ q = g − q (0) = g q (0) . B.2 Proof of Theorem 10 (b)
We first show that x ∗ q,N > L ⇒ q < q ∗ sup , x ∗ q,N < L ⇒ q > q ∗ sup . (59)Suppose x ∗ q,N > L and define ˆ q := qx ∗ q,N /L and ˆ d i := d ∗ q,i L/x ∗ q,N ( i = 1 , , . . . , N ), where we have ˆ q > q and ˆ d i ≤ d ∗ q,i . It is then verified that (ˆ q, ˆ d , ˆ d , . . . , ˆ d N ) (cid:62) is a feasible solution of (U), as (cid:80) Ni =1 ˆ d i = L and R ( ˆ d i ) ≥ R ( d ∗ q,i ). We thus obtain q ∗ sup ≥ ˆ q > q from the optimality of q ∗ sup , which proves the firstrelation in (59). On the other hand, the second relation follows from that x ∗ q,N < L ⇒ For any d ∈ R N , ( q, d ) is an infeasible solution of (U), (60)which is proved by contradiction: if there exists d ∈ R N such that ( q, d ) is a feasible solution of (U), d is also a feasible solution of (S q ) satisfying (cid:80) Ni =1 d i = L , contradicting x ∗ q,N < L .Taking the contrapositions of (59), we have q ≥ q ∗ sup ⇒ x ∗ q,N ≤ L and q ≤ q ∗ sup ⇒ x ∗ q,N ≥ L , whichimplies q = q ∗ sup ⇒ x ∗ q,N = L . Owing to Theorem 10 (a), this also implies x ∗ q,N = L ⇒ q = q ∗ sup , sothat we obtain the last equivalence relation in (17). Furthermore, ( q ∗ sup , d ∗ q ∗ sup ) is an optimal solutionof (U) because it is a feasible solution with the optimal objective value q ∗ sup .Finally, the first and second equivalence relations in (17) are immediately obtained, noting thatTheorem 10 (a) and the last relation in (17) imply q < q ∗ sup ⇔ x ∗ q,N > x ∗ q ∗ sup ,N = L and q > q ∗ sup ⇔ x ∗ q,N < x ∗ q ∗ sup ,N = L . 21 Proof of Theorem 13
We consider the slightly generalized problem (P ( k ) ) considered in Appendix A, assuming g − (0) < g (0).It is sufficient to show that under this assumption, γ := 1 + 1 g (cid:48) (0) ∈ (0 , , (61)and y ∗ i defined in (25) satisfies the followings: if ( g − ) (cid:48) (0) > −
1, then0 < y ∗ i ≤ γ i − g − (0) , i = 1 , , . . . , (62)and otherwise 0 < y ∗ i ≤ γ i − g − ( g − (0)) , i = 2 , , . . . . (63)Because (61) immediately follows from (42), we show (62) and (63) below.We first consider the case ( g − ) (cid:48) (0) > −
1. As shown in the proof of Lemma 19, h ( z ) (0 ≤ z ≤ g (0))is a strictly increasing function in this case. In addition, we have ( g − ) (cid:48) ( z ) ≤ ( g − ) (cid:48) ( g (0)) (0 ≤ z ≤ g (0)) because g − ( z ) is a convex function. We then have for any 0 ≤ t ≤ t ≤ g (0), | h ( t ) − h ( t ) | = t − t + g − ( t ) − g − ( t )= t − t + (cid:90) t t ( g − ) (cid:48) ( t )d t ≤ t − t + ( g − ) (cid:48) ( g (0)) (cid:90) t t d t = γ | t − t | . (64)Furthermore, it follows from (41) and (45) that 0 ≤ h ( z ) ≤ g (0) (0 ≤ h ( z ) ≤ g (0)). Therefore, we canverify from (39) and Banach fixed point theorem that { z ∗ i } i =0 , ,... converges to the unique fixed pointof h ( z ) (0 ≤ z ≤ g (0)) given by z = h (0). In addition, we have from (34) and (64), y ∗ i +1 = z ∗ i +1 − z ∗ i = h ( z ∗ i ) − h ( z ∗ i − ) ≤ γ ( z ∗ i − z ∗ i − ) = γy ∗ i , i = 1 , , . . . (65)so that (62) is obtained by induction using y ∗ = g − (0).We next consider the case ( g − ) (cid:48) (0) ≤ −
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