Recursive quadratic programming for constrained nonlinear optimization of session throughput in multiple‐flow network topologies

In this article, we propose and formulate a single‐objective nonlinearly constrained programming problem with delay and capacity constraints. Specifically, we consider the utility resource allocation with the goal of maximizing the throughput amongst the multiple heterogeneous sessions in the network. Several numerical experiments for multiple topologies exhibit the performance and global convergence process of the proposed iterative algorithm. These network topologies are considered with distinct source‐destination communication sessions to emphasize the traffic flow analysis and optimal rate allocation vectors. For this, we employ recursive quadratic programming method and Lagrangian multiplier approach to solve the proposed optimization problem. The presented numerical results are obtained using Octave simulation tool. The results obtained from implementation of numerical simulation demonstrate the robustness and convergence performance of the proposed optimization scheme to optimal solution within finite number of iterations. Finally, we employ various global measures of fairness including the entropy‐based index, G's fairness index, linear product‐based fairness index, and Bossaer's fairness index to assess the performance of the proposed resource allocation scheme.


INTRODUCTION
sequential quadratic programming (SQP) method as one of the most effective general algorithms for solving nonlinear recursive programming problems.
For real-time applications to support delay-sensitive traffic, the link transmission delay is constrained to be less than the maximum admissible threshold value. This maximum tolerable delay value is determined by the specific quality of service (QoS) requirements for the interactive multimedia traffic at the higher layer of the traditional network stack architecture. The network delay is comprised of various components such as transmission delay, propagation delay, contention delay, buffer delay, and packet processing delay. In our proposed model, we analyze only transmission delay onto the communication link and ignore other types of delay considering them as either fixed quantities or negligibly small values.
Real-time applications running in wireless communication networks require the packet transmission delay to be bounded by a specific threshold before the packet can reach the intended destination node. In this work, the proposed optimization problem is to find the optimal source rate allocation for static topology network with multiple flows. We study the convergence behavior of nonlinear objective function under linear as well as nonlinear inequality constraints. To illustrate this behavior, we employ a recursive quadratic programming approach. 1 This algorithm exhibits a quadratic rate of convergence for a feasible optimal solution in the search space.
The SQP methods, also referred to as the successive or recursive quadratic programming approaches, are used to solve mathematical optimization problems, in which the objective function and the constraint functions are twice continuously differentiable real-valued functions. In the theory of differential equations, these functions are of class Ω 2 , such that the first and the second derivatives of the functions both exist and are continuous on some open set of En, where En is the n-dimensional Euclidean space.
The SQP algorithm is based on the proceedings of highly sequential iterative design phenomenon. This effective optimization technique does not require any side information to compute the global optimal solution. In this sequential search procedure, the information produced at the previous iteration is used in evaluating the subsequent iteration data. It is an iterative procedure based on the quasi-Newton algorithm to directly solve the Karush-Kuhn-Tucker (KKT) conditions for the original problem. As a consequence, the optimization programming subproblem becomes the minimization of a quadratic approximation to the Lagrangian function optimized over a linear approximation to the feasible constraint functions. 1 Therefore, this approach is also known as the projected Lagrangian approach.
Besides, the concept of fair resource allocation is prevalent in wireless communication systems for quantifying the levels of fairness measures. With rapid increase in the number of connected devices across the wireless network, the allocation of relatively scarce and shared resources such as available wireless bandwidth with the goal of maximizing the throughput has been identified as a crucial challenge in the design of these networks. Resource allocation in terms of achievable throughput should be distributed cautiously and fairly to individual sessions in order to enforce practically feasible resource management. Thus, sharing of limited resources such as utility and throughput among the multiple sessions is interrelated with the fairness issue at the network and node levels.
In this work, we propose and formulate a univariate single-objective nonlinear programming problem subject to linear as well as nonlinear inequality constraints. Moreover, we employ various existing global measures of fairness to assess the performance of the proposed scheme. These measures include the entropy-based index, G's fairness index, linear product-based fairness index, and Bossaer's fairness index. These fairness measures are implicitly tied with other networking objectives of throughput optimization and optimal allocation of constrained source data rates. In our numerical analysis, we have utilized several topology scenarios with multiple network flows to perform simulation of the proposed mathematical optimization model. Our method guarantees convergence to the global and unique optimal solution. The optimal solution entails the evaluation of objective function value for the set of problem variables x while satisfying all the functional constraints. The simulation experiments are performed using the Octave simulation tool. Specifically, Octave is a high-level programming language that comes with a graphical user interface support for performing considerable numerical computations of generalized linear, quadratic, least squares, and nonlinear programming problems.
The rest of this article is organized as follows. Section 2 presents an overview of the existing works that are related to our research field of employing mathematical optimization formulation for assessing the performance of non-linearly constrained wireless networks. The system model is discussed in Section 3. Section 4 formulates the proposed optimization problem, followed by the discussion of numerical simulation results in Section 5. Finally, Section 6 draws the conclusions for the article, succeeded by outlining the broad areas of future research work. The employed symbols and variables in the following sections and their descriptions are summarized and given in Table 1.

LITERATURE SURVEY
It is well known that the SQP algorithm is one of the most simple, fast, and effective procedures with local quadratic convergence properties. Therefore, many research efforts have adopted this method of optimization for solving nonlinear constrained optimization problems. Recently, the authors in Reference 2 formulated SQP method for large-scale nonlinear optimization that employed both positive-definite approximate and exact Hessian method information. Another work in Reference 3 considered distortion minimization problem for wireless communication video system subject to transmission rate and end-to-end delay constraints. An SQP algorithm based on infeasible interior-point method has been proposed in Reference 4 in which step length is computed adaptively at each iteration instance depending on quadratic programming subproblem. A recursive quadratic optimization algorithm for solving nonconvex constrained problem was proposed in Reference 5 that employed the Broyden-Fletcher-Goldfarb-Shanno (BFGS) procedure with no convergence guarantees. S. Deshpande 6 proposed a novel iterative simplex procedure for multi-dimensional nonlinear programming problem with convergence consideration for optimality. A literature survey on algorithms and applications of nonconvex mixed-integer nonlinear optimization problems has been presented in Reference 7. Besides, a distributed optimization algorithm is developed in Reference 8 for mixed constraints associated with a set of agents with local cooperation. In our previous work, 9 we considered a single objective of maximizing the network utility among a set of end-to-end communication sessions as the criterion function of the proposed problem and formulated it as a convex optimization problem. The problem was solved using subgradient projection procedure by deriving a distributed iterative algorithm. Moreover, in Reference 10, we employed three different numerical methods, viz, active-set, interior-point, and SQP methods to solve a bi-objective nonlinear optimization problem subject to the underlying network operating conditions. The parametric SQP-based algorithm employing the exact Hessian and active-set estimates is used for solving the nonconvex nonlinear program in Reference 11. In addition, a recursive quadratic programming algorithm is proposed in Reference 12 to maximize the joint utility of transmission sources over the network, while achieving lower packet loss and lower delays. Furthermore, the authors in Reference 13 developed joint optimization solution for energy-constrained wireless networks using the nonlinear integer SQP programming algorithm. The presented analytical model established a global solution for the optimization problem, together with exploring the tradeoff between energy conservation and throughput maximization. A minimax optimization problem is proposed in Reference 14 for reducing the bit error rate and computing the transmission power of OFDM/A systems based on the application of quadratic programming techniques. Besides, computation of optimized overall throughput for multiuser multiple-input, multiple-output (MIMO) systems with near-optimal performance is implemented in Reference 15. This optimal formulation is considered by employing the nonlinear numerical techniques and minimal mean squared errors approach. Tan et al 16 presented a distributed power control algorithm for throughput optimization in wireless cellular networks through nonlinear geometric programming technique. Likewise, fairness issues are considered in Reference 17 for interference-limited OFDM networks by exploiting SQP method to solve the proposed resource allocation scheme. Recently, this numerical method is employed in Reference 18 for optimal allocation of transmit power resource to establish secure communication in bi-directional relay networks. Also, the authors in Reference 19 considered the problem of joint power allocation and rate optimization over the downlink macrocell and femtocell networks based on the SQP technique. A fundamental work on data delivery systems is proposed by Cheng et al, 20 which deploys Lagrangian multiplier and SQP methods to compute optimal link transmission rate for coded caching architecture. The body of work in Reference 21 employed SQP and Augmented Lagrangian approaches together with artificial neural networks for optimal power assignment in optical code-division multiple access (OCDMA) systems.
Additionally, stochastic subgradient descent method is applied in the dual domain for formulating the resource allocation problem in wireless device networks. 22 It characterized the convergence rate analysis of the developed algorithm in attaining the optimum objective function. Mohindra and Sharma 23 utilized gradient descent algorithm to find the optimal energy-efficient cluster head for data transmission to base stations in static sensor networks. This first-order iterative optimization technique is less complex than the SQP method and guarantees convergence to the local optimal solution. Hijazi et al 24 introduced a location estimation technique in wireless sensor networks implemented with noisy radio propagation model based on nonlinear gradient descent optimization. Moreover, a conglomeration of gradient descent and particle swarm optimization algorithms is applied in Reference 25 to effectively enforce the guaranteed node localization approach in underwater sensor networks. Karkoub et al 26 employed the distributed Newton method for modeling the dynamic network scenario consisting of a group of autonomous underwater vehicles. This approach incorporated the descent direction and line search techniques for fast convergence performance of the node positioning problem. Subsequently, the steepest gradient descent methodology is adopted in Reference 27 to develop an adaptive localization search algorithm in wireless sensor networks.
Despite these previous works, this article investigates the application of SQP optimization algorithm to single-objective nonlinear network programming model with linear inequality link capacity and bound constraints, and nonlinear inequality delay constraint. This feasible optimization technique provides an effective and robust way for solving smooth nonlinear constrained problems with fast asymptotic convergence rates. The end aim of the proposed design is to carry out nonlinear constrained optimization to achieve enhancement in network throughput with the underlying delay and capacity constraints. The criterion function of the proposed optimization problem is the utility function of the source rate variables. We consider this utility function to be additive in nature, so that the aggregated utility of all network flows in the system can be maximized. This facilitates the implementation of limited memory storage requirements in wireless nodes and tractable analysis of the network performance. Besides, the previous works in literature do not consider the fairness management issues concerned with the fair resource allocation among the competing sessions in the wireless network. Unlike existing methods, the contribution of our work is determining the fair allocation of a divisible network resource among heterogeneous sources with predetermined routes. These sources transmit data at non-negative source rates as the distributed network resources.
In addition, no prior work exists in the research literature that has considered several widely popular network topologies for the implementation of nonlinear optimization techniques. Assessing the potential of SQP algorithm for nonlinear optimization, we provide numerical analysis and comparison of the proposed optimization scheme for four different topology scenarios of the same network size. This ensures the general application of our proposed system framework with multiple source-destination communication flows. Generally, the size of a network is determined by the number of nodes in it. This iterative method searches for the global optimal solution of the original problem by searching through the feasible region in the space En defined by the constraints. It has been established through the simulation results that the implementation of the presented nonlinear optimization algorithm is computationally efficient for the proposed system model with improved accuracy. In addition, the analysis of convergence rate behavior is provided for the contemplated network topologies in attaining the optimal objective function value.

SYSTEM MODEL
We adopt a slightly similar system model as the one used in our previous works. 9,10 In particular, we simulate a multiple session ad-hoc network to implement the proposed optimization framework represented by an undirected graph with three parameters N, L, M. Here, N represents the set of static nodes with indices n ∈ N, L represents the set of links exchanging information between the nodes in the network indexed by l ∈ L, and M denotes the set of data transmission source nodes, with indices m ∈ M. Here we assume that the communication system is subjected to additive white Gaussian noise (AWGN) channel with approximately immobilized or slowly varying network topology from source toward their destination via a number of intermediate nodes.

SYSTEM COMMUNICATION MODEL
We consider a variable-rate M-level quadrature amplitude modulation (MQAM) scheme implemented at the physical layer. Furthermore, the achievable transmission rate of communication links is considered consistent over time. Thus, each link l ∈ L transmits data with a constant bit rate of Cl referred to as the finite link capacity. The flow rates across each link in the network are bounded by the associated link capacities. The wireless channels are assumed to be discrete-time Gaussian channels. In this time-discretized channel system, the time is distributed into discrete and identical slots, with the commencement of transmission of information by a node at the start of the allocated time slot. In the proposed communication system, the wireless channels are defined by single-dimensional pseudo-random code words such that each wireless user is assigned a single orthogonal code. 28 As an illustration, consider the optical orthogonal codes (OOCs) employed in OCDMA systems. In such a channel, the transmitter can coordinate the transmission of each code word across the single time coordinate. The total frequency bandwidth available for communication across all links in the network is considered as unity. Furthermore, the wireless communication system is considered as uniform and time-balanced with relatively stable network topology between the source and relay nodes.

NETWORK LAYER MODEL
We use a network layer model with multiple commodities in which there are several disparate source-destination pairs (sessions) that communicate data synchronously along point-to-point links. Let T |L | × | M| denote the binary routing matrix that maps the end-to-end traffic demands into the link-by-link network layer flows. Note that |.| denotes the number of distinct elements in a set. The network topology with respect to the cooperation between network flows and data links for routing of information across the network can be described with this link-session incidence matrix. An entry t lm of this matrix, associated with session m and link l, is defined as:

PROBLEM FORMULATION
In this article, we consider the following mathematical optimization program with the objective of maximizing the net utility function.
The criterion function of the proposed optimization problem designated in Equation (2) is the utility function of the source rate variables x. We assume that each source node m ∈ M attains a nonlinear utility U m (x m ) = log(1 + x m ) by transmitting at a data-rate of x m bits per second (bps). This utility function U m :R + → R is assumed to be smooth, increasing consistently over the interval (0, ∞) and twice differentiable function in x m . The specified logarithmic utility function corresponds to the TCP Vegas protocol implementation at the transport layer. In addition, since 2 U m x m 2 < 0, ∀m ∈ M, the utility function is strictly concave with negative curvature. In the constraint set, the first functional constraint in Equation (3) specifies a bandwidth conservation constraint enforced for each link l. This capacity constraint states that the average amount of network traffic load generated by a source node on a link l should be less than the mean transmission rate of that link. For the ease of exposition, we assume constant and identical nonzero transmission capacity across each link l. The second constraint in Equation (4) limits the link transmission delay to be less than the maximum permissible delay threshold value. The link transmission delay is defined as the ratio of the packet size to the instantaneous link flow rate. Here, Z is the packet size (in bits) and D max is the maximum tolerable delay threshold in the communication system. Furthermore, the feasible region is constrained by upper and lower bounds on the rate allocation variables.
Here, the polyhedral set Γ is used to specify x min m ≥ 0 and x max m < ∞ as the minimum and maximum data-rate requirements of each source m ∈ M.
Let x * be a unique global optimal solution to the above primal optimization problem. The KKT optimality conditions for this problem require a primal feasible solution x * ∈ |M| and a unique Lagrangian multiplier vector * ∈ ℜ |L| + associated with the maximal link capacity constraint in Equation (3), such that * ≤ C l , ∀l ∈ L, x * ∈ Γ, where the first three conditions in Equations (7) to (9) are primal feasibility conditions, the fourth inequality constraint in Equation (10) is the dual feasibility condition, the fifth Equation (11) is the complementarity condition, and the last Equation (12) is the stationarity condition. Together these conditions constitute the first-order KKT necessary conditions to achieve optimality. The symbol h(y) denotes the vector gradient of h at the n-dimensional variable y ∈ R n , that is, For a given primal-dual iterate (x k , l ), the standard quadratic programming (QP) subproblem determining an appropriate search direction s k is formulated as: where the computations of the usual Hessian 2 ( − ∑ U(x k ) ) and 2 ( k − C l ) are consistent with the BFGS approximation that is updated at each iteration. The last term in the objective function of the above QP subproblem incorporating these computationally extensive Hessians represent the curvature of the objective and the capacity constraint functions of the original optimization problem. Furthermore, the constraint in Equation (15) models the linear approximation of the link capacity requirements. Below, we provide an outline of the SQP algorithm to solve the formulated nonlinear optimization problem. This optimization procedure can be perceived as a generalization of the Newton's method to the constrained programming framework for computing the gradient and Hessian approximations with the centralized approach.

Algorithm 1 SQP
1. Set k = 0. Select arbitrary initial iterate vector (x 0 , 0 ). Let tol = exp(−6). 2. Repeat steps 3 to 5 until x k converges to x * or the search direction has become too small, that is, s k < < tol * ||x k || 3. Solve the above QP subproblem using the BFGS scheme to determine (s k , k + 1 ). 4. Set x k + 1 = x k + s k . Set k = k + 1. 5. If the above first-order KKT necessary conditions are not satisfied, return to Step 2.
This algorithm iteratively obtains and refines the approximate subproblem solution until the improved optimal solution (x * , * ) is eventually found. Moreover, it ensures global convergence to the unique optimal iterate starting from the random and feasible initial iterate (x 0 , 0 ) for an infinitesimally small tolerance value.

NUMERICAL RESULTS AND DISCUSSION
In order to empirically validate the proposed mathematical optimization framework, we employ four topology scenarios with multiple network layer flows to compare the algorithm performance and to verify its generality. The algorithm is implemented in Octave simulation tool. In this section, we evaluate the performance of the proposed mathematical optimization model through simulation of four widely used network topologies, namely, linear, ring, star, and bus topologies. In our numerical analysis, the wireless nodes in ad-hoc mode employ the IEEE 802.11 Distributed Coordination Function (DCF) as the MAC sublayer protocol. The antenna for each node is configured as omnidirectional and the physical layer uses the IEEE 802.11b PHY specification model. 29 The initial rate at which data is generated by each source node is taken as 5 Kbps. We consider wireless communication systems such as GSM or CDMA, which have the modest data rate of about tens of Kbps and support basic voice call kind of traffic. 30 The system parameters with their corresponding values for the proposed optimization model are listed in Table 2. This model is implemented using the SQP algorithm intended for solving the constrained optimization problems in Octave-4.2.1 software package. 31 The results obtained from numerical computer implementation indicate that the proposed optimization scheme converges to optimal solution within finite number of iterations. Convergence is achieved in a finite number of iterations, since the number of function evaluations performed to stabilize on an optimal solution is in direct proportion to the number of iterations taken. If k is the number of iterations required to achieve optimal convergence, then the number of function evaluations to determine the cost of optimization is k + 1. These evaluations include the computation of all the system functions comprising the objective function and the constraint functions, as well as the computation of gradients of these functions.
We first execute numerical simulations of the proposed optimization framework in a linear network topology comprising five links and four sessions as shown in Figure 1. Figure 2 demonstrates the evolution of optimal objective function value of −14.8 attained in 15 iterations. Note that each successive marker on the figure denotes the estimated objective function value at each new iteration point. The convergent sequence of points generated converges to the limit point of that sequence, which is the globally constrained optimal value. Figure 3 plots the objective function value against the number of iteration instances for the ring topology shown in Figure 4. It can be observed that SQP converged in 17 iterations with 18 function evaluations with an optimal objective function value of −15.7. Similarly, the SQP method took 20 iterations (21 function evaluations) to extremize the formulated objective function in the case of star topology for the same network size shown in Figure 5. The globally maximal net utility converges to the optimal value of −16.58 as shown in Figure 6. Finally, Figure 7 shows the bus topology consisting of seven links and four source-destination pairs. The extrema found by the SQP method in the case of bus topology is achieved in eight iterations (nine problem function evaluations) as shown in Figure 8. Therefore, the convergence speed in case of bus topology is highest with the maximum optimal objective function value of −12.9 with a least number of function evaluations, compared with the other three topology scenarios.
The expected value of dual variable associated with the capacity constraint for different topologies is illustrated in Figure 9. It can be observed that the shadow price is highest in case of star topology, indicating the maximal improvement in the objective function of the underlying primal optimization problem. Figure 10 demonstrates the optimal rate allocation vector for each session in different topologies. The mean session throughput in case of linear topology is 43.83 Kbps, while that for ring, star and bus topologies, it is 53.75 Kbps, 66.75 Kbps, and 26.5 Kbps, respectively. Furthermore, Figure 11 depicts the evolution of objective function value vs the number of communication links in the network. For a F I G U R E 6 Evolution of objective function value in star topology F I G U R E 7 Bus topology comprising seven links and four sessions F I G U R E 8 Evolution of objective function value in bus topology specific packet size, the utility of network traffic decreases with an increase in the number of hops between the source and the destination nodes. This is because the network congestion and packet error rate increase with the enhanced network size, leading to reduction in network throughput.
In addition, the performance of these investigated network topologies for the proposed optimization algorithm is evaluated and compared in terms of different existing fairness metrics determining the fair division of limited resources such as utility and throughput among the multiple sessions in the network. Often referred to as statistical diversity measures, these fairness metrics are specified in terms of sophisticated functions that map the shared resource allocation into the normalized real number between 0 and 1, with 0 representing the least fairness and 1 representing the maximal fairness corresponding to the identical and ideal resource allocation system. Shannon entropy used in the context of information theory measures the uncertainty of resource allocation and is defined as: where the probability distribution m of each session m satisfies Equations (17) and (18): and is given as: Observe that the entropy increases with the increase in fairness of shared resources among the competing network flows.

F I G U R E 12 Different fairness performance measures for the various network topologies
Directly affected by potentially unfair resource allocation, G's fairness index with -th order ( ℜ + ) powered sine modification and Bossaer's fairness index are, respectively, defined as: The specific instance of Bossaer's fairness index with = 1 corresponds to the linear product-based fairness index. These different fairness metrics are computed and compared for the previously considered four network topologies, as illustrated in Figure 12. It can be observed that ring and bus topologies have approximately similar and maximal values for the various fairness indices, indicating their appropriateness for the effective implementation of the proposed optimization model.

CONCLUSION
This article addresses a nonlinearly constrained optimization in multiple-flow ad-hoc networks using the conventional SQP algorithm. The optimization results are plotted for four different network topology scenarios to indicate wide general applicability of the proposed methodology. The simulation results demonstrate the convergence process of objective function value to its global optimal solution within finite number of iterations. For the investigated topologies of the same network size, we found that the optimal convergence achieved in the bus topology scenario is the most efficient in terms of fastest convergence speed and minimum number of function evaluations as compared with the other topology scenarios. Moreover, if all topologies are operated in the same networking constraints with similar system setup, the highest optimal session throughput is obtained using the star topology with the average value of 66.75 Kbps. Finally, the considered network topologies are evaluated and differentiated in terms of several fairness metrics determining the fair allocation of relatively scarce and shared utility and throughput resources amidst the multiple heterogeneous sessions in the network. The results show that the ring and bus topologies have nearly identical and highest values for the various fairness measures. Testing results in real-time model to implement better practical and adaptive mechanisms can be considered in future. Also, the employed first-order KKT conditions for optimality can be extended to implement second-order optimality conditions for the developed mathematical model to achieve faster convergence and more efficient optimization. Furthermore, the mathematical optimization architecture contemplated in this work makes the assumption that the nodes are nearly immobile with slowly-varying topology. Thus, mobility characteristics with time-varying channel model for realistic network implementations are of particular interest for future work.

PEER REVIEW INFORMATION
Engineering Reports thanks Oscar Danilo Montoya Giraldo and Anton Abdulbasah Kamil for their contribution to the peer review of this work.

CONFLICT OF INTEREST
The author declares no potential conflict of interest.