Queue-Aware Joint Dynamic Interference Coordination and Heterogeneous QoS Provisioning in OFDMA Networks
We propose algorithms for cloud radio access networks that not only provide heterogeneous quality of-service (QoS) for rate- and, importantly, delay-sensitive applications, but also jointly optimize the frequency reuse pattern. Importantly, unlike related works, we account for random arrivals, through queue awareness and, unlike majority of works focusing on a single frame only, we consider QoS measures averaged over multiple frames involving a set of closed loop controls. We model this problem as multi-cell optimization to maximize a sum utility subject to the QoS constraints, expressed as minimum mean-rate or maximum mean-delay. Since we consider dynamic interference coordination jointly with dynamic user association, the problem is not convex, even after integer relaxation. We translate the problem into an optimization of frame rates, amenable to a decomposition into intertwined primal and dual problems. The solution to this optimization problem provides joint decisions on scheduling, dynamic interference coordination, and, importantly, unlike most works in this area, on dynamic user association. Additionally, we propose a novel method to manage infeasible loads. Extensive simulations confirm that the design responds to instantaneous loads, heterogeneous user and AP locations, channel conditions, and QoS constraints while, if required, keeping outage low when dealing with infeasible loads. Comparisons to the baseline proportional fair scheme illustrate the gains achieved.
Heterogeneous QoS, finite backlog, dynamic interference coordination, dynamic user association.
[findent=1pt]U biquitous connectivity is a key goal in designing wireless networks enabling broad ranges of reliable services to users. Cellular networks are evolving toward a distributed access point (AP) architecture controlled remotely over a cloud radio access network (C-RAN). By coordinating transmissions across APs, the C-RAN approach provides many benefits including cost, coverage, and capacity improvements. The resulting architecture, with edgeless virtual cells, meeting heterogeneous quality-of-service (QoS) metrics, is crucial to future wireless networks .
Designing for heterogeneity in user demands is relatively new; historically, traffic has been assumed homogeneous in time and space, therefore, interference coordination was performed by static frequency planning. More recently, reuse-1 (reusing frequency resources potentially everywhere) enhances throughput, but largely ignores users’ QoS demands. At best, LTE schedulers are allowed to identify applications as guaranteed bit rate (GBR) or non-GBR. However, networks now deal with a broad range of applications, some that are delay-sensitive (DS), some rate-sensitive (RS), and others that just require best effort (BE). These complex heterogeneous demands cannot be served effectively without advanced adaptation of dynamic interference coordination, dynamic user association, and fine-grained scheduling.
The growing heterogeneity in applications, and in traffic distributions in space and time, motivates changing the network architecture from assigning resources a-priori to APs, toward assigning resources dynamically to the users. In such a design, a user may be associated with multiple APs and the association may change over time, based on both channel and AP load conditions. This flexibility helps meet QoS constraints and allows for offloading to under-utilized cells making the frequency resources to follow the traffic loads and be reused adaptively.
In this paper, we consider radio resource management (RRM), in a multiuser orthogonal frequency division multiple access (OFDMA) network. Unlike other works, we consider dynamic interference and dynamic user association (also called short-term user association), while jointly addressing heterogeneous multiple QoS. Crucially, our finite backlog, queue-aware formulation, with random arrivals addresses delay sensitive flows. Importantly, unlike related works, this allows us to avoid treating a delay constraint as equivalent to a constant rate guarantee. Moreover, unlike many other works [2, 3, 4, 5], we no not use time-sharing and will provide explicit scheduling. We also provide solutions to manage infeasible load conditions, when the core optimization problem becomes infeasible (due to the high input load) making the setup robust to the input load.
Delay is a measurement across frames, i.e., inherently we have a multi-frame problem. This is in contrast to many other works focusing on single frames. Therefore, we first translate the multi frame problem (through a set of closed loop controls) into an optimization of frame rates. Because of our assumptions of dynamic interference and dynamic user association, our formulation results in a complex problem, preventing the use of conventional methods. We devise several techniques to develop an effective iterative QoS aware interference coordination (QoSaIC) algorithm for this challenging problem. We then propose a systematic approach for infeasible load conditions, combining the QoSaIC algorithm with an infeasible load management (ILM) algorithm.
2 Related Works, Research Gap, Approach, and Contributions
2.1 Related Works and Identifying the Research Gap
In this section, to identify the research gap addressed in this paper, we place the related literature into four categories. As a summary, the first category focuses on the QoS without interference awareness; the second one considers exclusively static interference coordination (with or without QoS awareness); the third set, addresses dynamic interference coordination but without QoS awareness; finally, the fourth group studies dynamic interference coordination without delay guarantees. As clearly evident, the categories show that there is a research gap on RRM decision making jointly considering heterogeneous QoS, including queue-aware delay sensitive flows, with dynamic interference coordination, and dynamic user association.
1 - QoS without interference awareness: This set of works studies the sub-problem of queue scheduling and resource allocation addressing only QoS and fairness without adequate attention to interference. Examples include maximizing average utilities balancing efficiency and fairness , analysis of generalized proportional fairness , scheduling for elastic traffic using convex optimization , single cell throughput maximization with rate guarantees , utility maximization with rate constraints through a token counter , minimum rate guarantees using a Lagrangian approach , joint channel- and queue-aware scheduling for mean-delay utility maximization , mean-delay fairness via gradient method , joint real-time and non-real-time packet scheduling and resource allocation , mean-delay guarantees through time-coupling constraints and Lagrange dual-based solutions , maximizing goodput for multihop networks through dual solutions , utility maximization and routing with probabilistic delay requirements , adapting rates with delay constraints to increase network video capacity , QoS-aware routing and subchannel allocation in time-slotted realy networks, without interference coordination, and dynamic user association , and, in single AP, single frequency networks, optimizing secondary users’ delay based on interference .
2 - Static interference coordination with or without QoS awareness: This set of works addresses a-priori static interference coordination, such as soft frequency reuse (SFR) [20, 21], two-phase coarse interference management and fine-scale resource-allocation based on graph-theoretic approaches , joint optimization of user association and use of almost blank subframes (ABS) , single frame constant rate guarantees with interference threshold and a-priori user association , and outage guarantees on constant rate requirements with RRM in cognitive small cells using cooperative Nash bargaining . We note that [2, 3, 4, 5] consider delay constraints as constant rate guarantees; this approach is unsuitable for random arrivals with finite backlog (particularly relevant for delay sensitive flows).
3 - Dynamic interference coordination without QoS awareness: This set of works aims at dynamic interference coordination for BE flows. Examples include throughput and fairness oriented interference coordination by blanking and based on dominant interferers , dynamic interference avoidance for cell edge users , a range of weighted sum signal-to-interference-plus-noise ratio (SINR) maximizations through Perron-Frobenius theory , and uplink clustering scheme decreasing both the intra- and inter-cluster interference without increasing the size of clusters . These interference coordination schemes improve cell-edge rates, but, importantly, do not address QoS.
4 - Dynamic interference coordination without delay guarantees: This set focuses on rate QoS and interference coordination, without delay guarantees. Examples include energy efficiency maximization while guaranteeing minimum rates [28, 29], interference management accounting for minimum throughputs with heterogeneous APs , hierarchic interference coordination with rate constraints , load balancing and interference coordination with infinite backlog  (note that infinite backlog assumption prevents the control of queue lengths), hybrid coordinated multipoint transmission, based on Markov decision process (MDP), improving the overall delay performances, but without delay guarantees , and heuristics for related sub-problems of interference coordination and queue equalization, based on static delays . Among the studies considering interference and QoS, while rate metrics are useful with infinite backlog, they do not account for metrics such as delay, especially relevant for real-time flows.
2.2 Our Approach and Contributions
Our goal in this paper is to develop an algorithm enabling resources to follow users’ traffic in a QoS- and interference-aware manner. Specifically, our goal is to associate users with APs and allocate time/frequency resources to maximize network utility while meeting rate and, crucially, delay constraints in a queue-aware manner, with random arrival. Importantly, unlike the works reviewed in Section 2.1, we require joint dynamic interference coordination, dynamic user association, and finite backlog random arrivals (particularly relevant for delay sensitive flows).
This requirement implies that our algorithm(s) must meet five criteria: (1) We design for finite backlogs (queue-aware), because without queue-awareness, the algorithm cannot adapt frequency reuse to traffic situations. (2) We design for QoS- and load-awareness, accounting for (and exploiting) the heterogeneity of QoS classes, QoS requirements, and load conditions. (3) We design for network-wide interference awareness caused by frequency reuse. (4) We design for an opportunistic setup in order to exploit user, time, and frequency diversities. (5) Finally, if faced with an infeasible load condition, due to high mean input rates, or a spike in input rates, our design should allow for graceful degradation of the QoS satisfaction.
Meeting all these criteria is a complex problem: The problem is coupled across flows and across APs. Moreover, it is a nonlinear combinatorial program with a non-convex relaxed version. Unlike other works resulting in convex problems after integer relaxation (such as [2, 3, 4]), in which a time-sharing approach can be used, our problem, even for a single AP, does not lead to a convex problem, rather a convex maximization, similar to . This is a strong indicator of an NP-hard problem . We reemphasize that modeling of joint dynamic user association, dynamic interference, and finite backlog (unlike other works) inevitably lead to this challenging problem. With the global optimum essentially impossible to find, our QoSaIC algorithm allows us to meet the first above-mentioned four criteria while our ILM algorithm meets the fifth.
Having clarified the research gap and our approach, the contributions of this paper are:
We formulate a systematic multi-cell utility maximization problem, with heterogeneous QoS guarantees, including importantly queue-aware (finite backlog with random arrival) delay sensitive flows. This enables matching (and relocating) of available time and frequency resources based on both the spatial dimension (user locations) and the temporal dimension (traffic arrivals). Unlike the related works (which use a constant rate constraint to serve delay sensitive flows), we consider the more realistic finite backlog with random arrivals. Furthermore, again unlike related works, we do not use a static unilateral interference threshold or static user association, in order to not limit the network efficiency.
Unlike other works, our design allows for control of both instantaneous and mean QoS metrics. We derive a set of closed loop controls that observe the mean QoS (rates and delays) measurements, compare them with the given QoS requirements, and adapt the allocation, in each frame. These controls are derived based on translating the high-level requirements to frame level requirements through Function QoSiFT, in Section 4.
Given the channel and load information, our RRM makes decision jointly on time/frequency scheduling (QoS provisioning), short-term user association (including load balancing), and frequency reuse patterns (interference coordination), per frame. As extensively discussed in Section 2.1, our work is the first to address the above-mentioned RRM decisions jointly, to the best of our knowledge. This is done based on our several original techniques we develop in this paper
1. Without these subtle techniques, the solution was not possible.
Unlike the related works, we also develop an effective strategy for infeasible load management to account for scenarios with infeasible demands. This part of our solution enables us to have an RRM robust to the input load (not becoming infeasible due to input load) with graceful degradation.
The paper is organized as follows: Section 3 presents our system model, our novel formulation, and our novel translation of the high-level RRM to the frame level optimizations. Section 5, then, discusses the solution, based on several original techniques, developed in this paper, setting up the QoSaIC and ILM algorithms in Section 6. Section 7 presents the simulations to illustrate the effectiveness of our algorithms. Finally, Section 8 make the conclusions.
We use the conventional notation system: Boldface lower-case letters, e.g., , represent vectors, while boldface upper-case letters, e.g., , represent matrices. Calligraphy style letters are exclusively used for sets, e.g., with superscripts as required. Subscripts usually represent flow (user) and AP indices, while superscripts represent the frequency subchannel index, such as in
||Dimensions of the problem, corresponding to indices .|
||Size of an RB, in sec., and in Hertz.|
||Algorithm fixed constants.|
||Optimization main variable and its compact matrix representation.|
||SINR on the link from user , to port , on RB , in frame .|
||Dual variables and the matrix representation of them.|
||Set of flows in QoS classes: DS class, RS class, BE class.|
|Sets representing integer constraint, PHY-1 constraint, PHY-2 constraint, MAC constraint for RS class, MAC constraint for DS class, and MAC constraint translated into frame rates.|
||Target QoS demands controlling .|
, , , ,
|Fairness weight, frame rate, queue length, mean queue length, and instant arrivals.|
||Translated minimum and maximum frame rates.|
||Intermediate variables translating the mean-delay and mean-rate constraints to frame level optimizations.|
||Channel coefficient, in frame , noise level.|
|Individual flow utility, Sigmoid function, overall translation, Lagrangian, overall primal update, overall dual update, dual update for variable , dual update for variable , dual update for variable , AMC function.|
||Overall interference on link from user to port on RB , in frame , comprising of intercell interference and intracell interference .|
||Intermediate variables in the fixed point method algorithm.|
||Error tolerances for inner/outer loop and the primal dual gap.|
||Constants controlling the shrinkage of the inner loop error tolerance.|
||Satisfaction/violation margins for different constraints corresponding to the superscripted dual variables.|
||Constants associated with the novel dual update design.|
||Single frame outage for rates.|
||Outage from mean QoS requirements corresponding to the superscripted QoS item (all for flow , in frame ).|
||Outer and inner loops counters and their corresponding allowed maximums.|
3 System Model and High-Level Problem Formulation
3.1 System Model
We consider the downlink of a multi-cell OFDMA network comprising APs, serving flows (users), without a-priori user association. The available bandwidth is divided into resource blocks (RBs), each spanning seconds and Hertz. The system serves three classes of flows: a BE class, denoted by , comprising flows without rate or delay requirements, a DS class, , with a maximum mean-delay constraint for each flow (), and a RS class, , with minimum mean-rate constraint for each flow (); and without loss of generality, a maximum mean-rate constraint (). The APs are connected to a C-RAN. The server also knows the data of all users and the channel state between all APs and all users, similar to other works in the fields, such as [2, 3, 4, 5]. In this paper, we provide solution for the air access. Backhaul scheduling remains as a future item extending this work.
At the server, in frame , each flow is associated with a queue of length bits. The number of bits for flow , that arrive in frame , is denoted by . The product of the transmission power (with uniform transmit power allocation), antenna gain, and channel power, from AP to user , on subchannel , in frame , is denoted by , and is assumed known.
3.2 Components of High-Level Problem Formulation
Our network objective is to maximize sum flow utilities, subject to the QoS constraints. The overall optimization problem is given in (5) on page 5; we first develop optimization components of (5), namely, optimization objective, optimization variable, interference metrics, flow rates, queuing delays, and optimization constraints.
The optimization objective (5a) is the network utility, , where is the mean-rates vector. The objective is sum of the individual flow utilities, , a function of individual mean-rate, .
The main optimization variable is the binary : , if flow is scheduled to be served by AP , on subchannel , in frame , else . Therefore, for each frame, the optimization variables form a 3D array denoted by .
We now define the interference metrics in order to first calculate the signal-to-interference-plus-noise ratios (SINRs), and then find the rates on RBs. We denote the total interference impacting the link from AP , supporting flow , on subchannel , by comprising inter-cell and intra-cell interference.
The inter-cell interference to flow , associated with AP , on subchannel , is due to undesired APs () communicating on the same sub-channel:
The intra-cell interference, on the same link, is given by
Intra-cell interference occurs when an AP serves more than one flow on a single RB. Later, we eliminate this totally undesirable situation via an explicit constraint
Summing up the intra-cell and inter-cell interference metrics, the total interference is given by
Based on the interference, the SINR, and the corresponding achievable spectral efficiency, are given by
explaining (5i). Notations, denotes the noise power and the capacity of the corresponding RB. Since any differentiable is allowed, an SINR gap to capacity can be added to (4). Having calculated the rate on RBs, the rate of a flow is given by (5h) summing up the RBs it is assigned. We emphasize that, unlike e.g., [2, 3, 4, 5], we do not use a static interference threshold, but dynamic interference coordination. Furthermore, we do not use a-priori user association as in [2, 3, 4]; our user association is part of and changes from frame to frame.
We now describe the required constraints. The first constraint is on RB scheduling - (5b) below. Furthermore, the physical layer imposes two constraints on any subchannel: first, frequency reuse is not allowed inside a cell - (5c); and second, a single flow cannot pass through two APs simultaneously over a single RB - (5d). Note that while a flow cannot be connected to more than one AP, on a single RB, it can be connected to multiple APs, across different RBs, allowing for data aggregation and load balancing. As such, we emphasize that our formulation importantly allows for frequency reuse across APs. Since we use the joint approach, the frequency reuse adapts to channels and QoS requirements.
The QoS requirements, in (5e) and (5f) below, represent the MAC constraints imposed as explicit mean-rate and mean-delay constraints. BE flows do not impose QoS constraints. RS flows impose the constraints in (5e) while DS flows impose the constraints in (5f), where and denote the mean-delay and mean-rate achieved by flow , respectively. The relation in (5g) relates the mean-rate to the instantaneous rate, in frame , , using forgetting factor . We discuss the connection of mean and frame quantities, in Section 4.
Having explained the optimization objective, variables, and the constraints, our core proposed optimization problem is given in (5).
To the best of our knowledge (and as extensively reviewed in Section 2.1), the formulation in (5) is the first one incorporating delay and rate QoS as explicit constraints, on a multi-frame problem, while also accounting for dynamic interference and dynamic user association. Unlike other works, we consider finite backlog resulting in queue awareness and addressing time varying random arrivals, crucial for delay sensitive flows. This formulation makes our first major contribution, summarized in Section 2.2. In the next part, we translate the problem in (5) into a parameterized frame-by-frame rate optimization problems. We highlight that the optimization in (5) is executed for every frame .
4 Translation to Frame-level
4.1 Translating MAC Constraints to Frame Rate Constraints
We begin with the RS flows. The instantaneous rate, in frame , is given by (5h), aggregating all the RBs given to a link. Mean-rate is calculated using exponential averaging (with averaging coefficients ) in (5g). We use . After simple manipulation of the inequalities (substituting (5h) into (5g) and solving for frame rate ), the constraints on the minimum and maximum mean-rates translate to single frame constraints as in
We now translate the mean-delay requirements into frame rates requirements. In contrast to mean-rate, mean-delay requires a more detailed analysis. Using Little’s formula (), queue evolution (conservation on arrivals and departures), and estimating the arrival through its empirical expected value, the mean-delay can be approximated as (see ):
where the approximation is valid when there is enough backlog in the queue:
This frugality constraint ensures that the service rate to be less than the backlog and prevents resources being wasted on a flow without a sufficient backlog. As derived in , the constants are functions of previous queue lengths and service rates:
where is now a second min constraint on frame rates due to the delay constraint (the first minimum rate, , was due to RS flows). We highlight that, unlike other works [2, 3, 4, 5], a constant minimum rate guarantees across frames is not sufficient for DS flows. Instead, as derived here, the intricate function in (8) is needed.
We highlight that in this paper we choose to use the mean-delay for the DS flows, similar to . Using other metrics, particularly head-of-the-line (HOL) delay  makes the problem highly complex in terms of connecting the optimization variable to the delay metric. Nevertheless, interestingly, since we guarantee a bound on mean-delay, we also implicitly guarantee a probabilistic bound on HOL. Based on Markov inequality, to control the outage on HOL-delay bound, we can perform it through a bound on mean-delay outage. In other words, bounding mean-delay to bounds the HOL-outage at most to .
Having translated the MAC constraints, we see that the mean-rate and the mean-delay constraints are equivalent to two independent min frame rate constraints and two independent max frame rate constraints. These four constraints simplify to a single minimum of and a single maximum of . We denote the feasible set of these frame requirements as , while we keep using for the feasible set of mean requirements.
4.2 Linearizing the Objective Function
We now focus on the objective in (5). Following the common practice in resource allocation literature (e.g. see [39, 9.3.2]), we use a Taylor expansion to linearize the objective with respect to the previous frame. Using (5g), the order Taylor series yields
The fairness weights are given by , at . With the Taylor expansion, sum utility is approximated by maximization of a weighted sum of frame rates. We note that utilities are concave increasing functions in order to model diminishing marginal utility.
4.3 Approximating the Frugality Constraint with a Soft Constraint
Here, we particularly consider the feasibility of the frugality constraint. In low load conditions, the frugality constraint (7) often makes the requirements on frame rates infeasible; this is especially a problem when we have a low backlog and/or when RB granularity does not match the needed frame rates. We therefore approximate the frugality constraint with a soft constraint inside the optimization objective. In , the authors substitute the weighted sum objective with . However, since we want to use a fixed point method for the primal problem, we must resolve the discontinuity in the derivative of . Therefore, we approximate this soft constraint with a Sigmoid function as
where the parameter controls the sharpness of the Sigmoid function (we use ).
Note that we cannot keep the frugality constraint as an explicit constraint, because there are backlog scenarios, where including the constraint explicit makes the problem infeasible. Having an infeasible problem, especially only due to the frugality constraint is not acceptable, in terms of robustness of the formulation to the input loads. We selected using Sigmoid approach after implementing the explicit frugality constraints, understanding its limitations, comparing alternative approaches for the soft constraints, and finally selecting the most effective one.
We emphasize that approximating the frugality constraint inside the objective does not impact the heterogeneous QoS guarantees nor the dynamic interference coordination capabilities. Moreover, this constraint is active only in low load conditions, and is inactive in moderate and high load conditions, anyways. Our optimization, even without frugality constraint is valid. The reason is that when the allocated data from a flow exceeds the actual backlog, one can always cut the surplus. However, we include the frugality constraint in order to open up space more efficiently for BE flows, and push BE flows further in service (whenever possible).
4.4 Summary of Translation
The function QoS inter-frame translator (QoSiFT) described below summarizes the discussed translations so far. It represents a set of closed loop controls adjusting the frame requirements, based on comparing the measurements with the targets, in order to maintain the RS and DS users’ satisfaction while serving as many BE flows as possible.
Here, represents (5h) and (5i). Symbol denotes the intersection of PHY layer constraints: . The other notations used here are as in (5). The translation in this section forms our second major contribution, discussed in Section 2.2. We note that, unlike many of the related works, our problem in (12) is not convex (due to dynamic interference coordination and dynamic user association). As such, in the next section, we combine several techniques in order to devise an effective solution.
5 Solution Approach
Having formulated the RRM problem, we now use a tailored primal-dual approach, coupled with a number subtle novel techniques, to devise an effective solution. Properly revised versions of the primal-dual method have proven to be powerful to devise approximation algorithms for combinatorial optimization, e.g., Hungarian algorithm [40, 41]. We first relax the integer constraint and decompose the problem into primal and dual domains. We then use the Karush Kuhn Tucker (KKT) conditions to form a system of equations for the primal domain. Next, we solve the primal problem using fixed point iterations (in an inner loop) while also using a novel approach to update the dual values (in an outer loop). The integer constraint is imposed iteratively, with a primal dual interface projection, for each outer iteration.
5.1 Decomposing into Primal-Dual Domains
The Lagrangian of the relaxed constrained problem in (12) is given by
where are the Lagrangian multipliers associated with the minimum rate requirement, , and constraints, respectively. Vector versions of the Lagrangian multipliers are denoted by , and . For simplicity, in the derivations, we drop the frame index . The constrained optimization now becomes an unconstrained problem , as in
where is element-wise. Forming the dual problem suggests an iterative solution between the primal and dual domains: iterate between solving in order to find the primal variables and solving in order to find the dual variables.
With explicit QoS constraints, an insightful interpretation is that the Lagrangian is equivalent to solving a multi-objective optimization, where, in addition to the conventional objective, the other objectives satisfy the QoS constraints. The Lagrangian jointly finds the appropriate scale factors to this multi-objective optimization. This is in contrast to including the constraints within the objective function, where the scale factors have to be adjusted manually. Works based on QoS constraints inside the objective have reported difficulty adjusting these scale factors .
5.2 Dealing with Primal Variables
In this section, we solve for the primal variables, assuming fixed dual variables. Vanishing the derivative with respect to the primal variables yields to
We denote the derivative of the Sigmoid function in (10) as leading to . The derivative of the Lagrangian has three components, , arising from the derivative of . Considering the fact that the derivative of rate can be written as in (16), the conditions in (15) dictates , where the components, , are as below.
The first component () is a single term depending on as
The second component () is due to inter-cell interference. The derivative of has non-zero elements only when , and is equal to . This yields terms:
The third component () is based on intra-cell interference. The derivative of has non-zero term only when , and is equal to . This yields terms:
It is insightful to note that the terms and are the aggregate rate gain sensitivity (for the inter- and intra-cell interference), as functions of . Dynamic interference coordination occurs when is satisfied. Intuitively, this is when the aggregate rate gain in using an RB, minus its dual cost, is equal to the aggregate rate loss in using that RB.
Having calculated the derivative of the Lagrangian, vanishing the derivative yields to . This results in our novel primal update of
which updates the primal variable . Here, . Importantly, the allocations depend not only on interference but also on the relative priorities due to QoS requirements.
The system of equations in (20) represents equations, with primal unknowns, and dual unknowns. We note that these primal equations have a special structure that each primal unknown can be written explicitly in terms of other primal unknowns. This special structure makes the system amenable to fixed point iterations. For this inner loop, the dual variables are constants (to be determined in an outer loop, in Section 5.3).
We use and to denote the outer and inner loop counters, respectively. In the fixed point method, solving for the primal variables involves iterations (on ), for a fixed , as
The inner loop is terminated if , where is the inner loop error tolerance. A convergence analysis of the associated fixed point method is beyond the scope of this paper. Having solved the primal equations, the outer loop index is increased passing the results of the primal variable to the next outer iteration as , where is the smallest index satisfying the error.
5.3 Dealing with Dual Variables
In this section, we solve the outer optimization, , finding the dual variables for fixed primal variables. Since the value of the Lagrangian is an upper bound on the original optimization and any feasible primal solution provides a lower bound, if the gap , then the primal and dual solutions are within of local optimality . The outer iterations aim at , for large . To find the best upper bound, the Lagrange multipliers are updated in the direction opposite to the gradient of the Lagrangian (with respect to the multipliers), in the outer loop.
The bisection method  is the standard approach to dual updates. However, in our problem, the several dual variables have very different roles making bisection ineffective. We, instead, design a customized update rule building on the basic idea of increase (decrease) the multiplier, if the corresponding constraint is violated (satisfied). In addition, a computationally efficient update rule should have four features: (i) providing exponential convergence; (ii) accounting for the violation/satisfaction margins; (iii) being able to sweep the whole interval from zero to the multipliers maximum; and finally (iv) shrinking the steps sizes with the outer iteration. The bisection method lacks features (ii) and (iii), making it ineffective in our problem. The classical gradient method lacks feature (i), making it very slow.
We introduce a novel dual update method, based on Lagrangian gradients, combining aforementioned features. Define the Lagrangian gradient with respect to as ; the Lagrangian gradient with respect to as ; and the Lagrangian gradient with respect to as . Negative values of any correspond to a constraint violation, while positive values indicate constraint satisfaction.
We use the multiplicative factors of , if a constraint is violated (and if satisfied). The multiplicative factors provide feature (i). In addition, we amplify them, based on the ratio of satisfaction or violation, based on the mapping , in order to provide feature (ii). The image of this mapping is , for both cases of violation () and satisfaction () of the minimum rate constraint. Nevertheless, using the term makes the amplification aggressive (acting as an increasing convex function), when the constraint is violated, in comparison to when it is satisfied (acting as an increasing concave function). Note the absolute sign in . We also use the outer counter to shrink dual update steps, based on dividing the