- Research Article
- Open Access

# An Active Constraint Method for Distributed Routing, and Power Control in Wireless Networks

- Alban Ferizi
^{1}Email author, - Armin Dekorsy
^{2}, - Joerg Fliege
^{3}, - Larissa Popova
^{4}, - Wolfgang Koch
^{4}and - Michael Söllner
^{5}

**2009**:656832

https://doi.org/10.1155/2009/656832

© Alban Ferizi et al. 2009

**Received:**17 April 2009**Accepted:**15 December 2009**Published:**8 February 2010

## Abstract

Efficiently transmitting data in wireless networks requires joint optimization of routing, scheduling, and power control. As opposed to the universal dual decomposition we present a method that solves this optimization problem by fully exploiting our knowledge of active constraints. The method still maintains main requirements such as optimality, distributed implementation, multiple path routing and per-hop error performance. To reduce the complexity of the whole problem, we separate scheduling from routing and power control, including it instead in the constraint set of the joint optimization problem. Apart from the mathematical framework we introduce a routing and power control decomposition algorithm that uses the active constraint method, and we give further details on its distributed application. For verification, we apply the distributed RPCD algorithm to examples of wireless mesh backhaul networks with fixed nodes. Impressive convergence results indicate that the distributed RPCD algorithm calculates the optimum solution in one decomposition step only.

## Keywords

- Time Slot
- Source Node
- Power Control
- Wireless Mesh Network
- Subgradient Method

## 1. Introduction

Nowadays, there is an increased interest in communication via wireless mesh networks such as ad-hoc, sensor, or wireless mesh backhauling networks [1, 2]. In wireless networks the link capacities are variable quantities and can be adjusted by the resource allocation such as scheduling, and power allocation to fully exploit network performance. Hence, for efficient data transmission an integrated routing, time scheduling and power control optimization strategy are required. This strategy has to take different transmission constraints into account, for example, maximum available power level or limited buffer size at nodes. The inherent decentralized nature of wireless mesh networks mandates that distributed algorithms should be developed to implement the joint routing, scheduling, and power control optimization. The first step towards a distributed implementation is to break up this problem into manageable subproblems and solve these subproblems by iterative algorithms. Cruz and Santhanam [3] have addressed the problem of finding an optimal link scheduling and power control policy while minimizing total average power consumption. Their algorithm is designed for single-path routing only, does not consider buffer limitations and has a worst case exponential complexity. In [4], Li and Ephremedis solve at first power control and scheduling jointly. They use the obtained power values to calculate a routing distance that in turn is used by Bellman-Ford routing. However, the proposed separation is performed by not considering the combinational structure of the entire routing, scheduling and power control problem. Although less computationally intensive, the algorithm ends up in a suboptimal solution. It further fully neglects multiple path routing as well as buffer restrictions. Xiao et al. proposed in [5] the dual decomposition as a promising decomposition approach. By dual decomposition the overall problem is split into two subproblems while the master dual problem coordinates them. In this paper we consider joint routing, time-scheduling, and power control for single frequency wireless mesh networks. The wireless transmissions are arranged in time-slots. However, we take into account that simultaneously active transmissions suffer from multiple access interference. Dual decomposition is a universal approach to solve such optimization problems [5, 6], but it does not consider the specific combinational structures of optimization problems. By contrast, we propose a novel method that explicitly exploits the combinational structure of a joint routing, time-scheduling, and power control problem by means of an active constraint method. The formulation of the optimization problem is yet generally valid, so that the method proposed here is applicable to a plurality of wireless networks. The proposed approach meets the following requirements: ( ) less iterations to an optimum solution, ( ) distributed implementation, ( ) multiple path routing, and ( ) per hop error performance. In particular, the approach is as follows. We separate scheduling from routing and power allocation by including it in the constraint set of a S imultaneous R outing and P ower C ontrol (SRPC) problem. For scheduling, several well known approximations such as Greedy-based approaches exist [7, Section ], that we can leverage on. The constraints we use in the SRPC problem are induced by a precalculated colored graph of the network that, in turn, reflects the scheduling decisions of any arbitrary scheduler. Consequently, the main contribution is to introduce a R outing and P ower C ontrol D ecomposition (RPCD) method to solve the simultaneous routing and power control problem while meeting the above-mentioned requirements. The clever bits of the RPCD are manifold. ( ) We rewrite the SRPC problem to an equivalent problem by applying the active constraint method. ( ) We decouple the equivalent problem by solving a (convex) network and a (convex) power assignment problem separately. ( ) Iterations are performed by switching between the two subproblems for which network and power variables act as interchanging variables. Apart from the mathematical framework we introduce the RPCD algorithm and prove its convergence to a KKT-point of the joint routing and power control problem. We compare the RPCD algorithm with dual decomposition as state-of-art approach with respect to the number of iterations needed to calculate the KKT-point. This verification is performed by applying both algorithms to a wireless cellular mesh backhauling network [1, 2]. The backhauling network describes a "regular" cellular network. This models the situation where, in order to save infrastructure expenses of laying cable or fiber to each node (base station), we try to extend the range of a given source node with wired backhaul connection by using several other nodes. These intermediate nodes have no wired connection and can only communicate with the backhaul via the source node by wireless mesh communications. The simulation set-up correctly models mobile radio channel characteristics such as path-loss and slow fading. The comparison indicates that the RPCD approach requires only one decomposition step to calculate the optimum solution as opposed to dual decomposition. This paper is organized as follows. In Section 2 we describe the network model used for the wireless data network. In Section 3 we formulate the optimization problem and define the standard interference function. The RPCD algorithm for solving the joint routing and power control problem is presented in Section 4. We extend the RPCD algorithm in Section 5 by introducing distributed algorithms for solving the routing and power assignment problem. Finally, in Section 6 we apply the algorithm to a wireless backhaul network and present the simulation results. We conclude the paper in Section 7.

## 2. Network Model

*multiple path routing*each message is transmitted via several paths from its source node to its destination node. Thus, nodes can send parts of messages to many receivers and receive parts of messages from many transmitters. We denote the nodes by with be a finite set of nodes. At any time, each of these nodes can map any parts of messages onto a single link for transmission. The set of all links is denoted by . A wireless communication link corresponds to an edge between two nodes and is described by the ordered pair such that transmits information directly to . Moreover, we assume that for all . We have that is a directed graph with node set and edge set . For an arbitrary node , denote by and the set of outgoing and incoming edges within at the node , respectively. A link represents a wireless resource characterized by a given bandwidth, time duration, space fraction, or by a given code assignment. We assume a time-slotted single frequency network for which the time is divided into equal slots of length while all nodes occupy the same frequency band of bandwidth . Time slots are indexed by , with as an index set. We take time scheduling into account by assuming that there is a given coloring of the nodes such that adjacent nodes do not have the same color (half-duplex constraint) [4]. That is, we are given a number and a function such that for all nodes with . Here, is at least as large as the chromatic number of . Computing such a coloring can be done by a Greedy approach [7]. To take delay constraints into account we introduce as the maximum number of time slots, a message is allowed to use for transmission from its source to its destination, that is, . The interference model we consider includes multiple access interference caused by simultaneously active transmissions that can not be perfectly separated by, for example, code- or space division multiple access (CDMA/SDMA) techniques. Thus, let be the set of edges interfering edge at time . The signal attenuation from node to node is and it remains unchanged within the duration of a time slot . We further assume perfect knowledge of at the corresponding senders. Let be the transmitting node and let be the receiving node of edge . Hence, denotes the attenuation a signal suffers that is transmitted from but received by node . For link such a signal represents multiple-access interference that is caused by link . Furthermore, with as the (transmit) power to be allocated to link at time slot , the received signal power at node from the transmitter is given by . We define the signal-to-interference-plus-noise ratio (SINR) of edge at time slot as

with as an additive noise power of edge . If we only assume thermal noise to be the same for all edges, we have with noise spectral density . For the optimization problems to be introduced later we have the following design variables. As network flow variables we have as the part of the message sent along edge in time slot (in bits), and as the part of the message stored in a buffer at node directly before the start of time slot (in bits). Communication variable is the transmit power allocated to edge at time slot to transmit the total traffic on edge (in Watt). If we stack the different variables to vectors we obtain , , and . We further use the following parameters. Let be the size of message (in bits) and be the maximum total buffer size at node (in bits). Power constraints are as the maximum transmission power of a node (in Watt) assumed to be the same for all nodes and as the maximum transmission power per edge (in Watt).

## 3. Optimization Problem

### 3.1. Problem Description

Let us consider an operation of a wireless data network with the objective to minimize a convex cost function
(or to maximize a concave utility function). The design variables
,
**,** and
are subject to some constraints. For instance, with
we require the power constraints

forming the polyhedral set

Since we isolated coloring from the joint routing and power control, we have to take the precalculated colored network graph in the flow constraints into account. Similar to power constraints, we require that flow constraints form a polyhedral set . For example, if we assume that given source nodes have to transmit messages of sizes to destinations in a given time , the polyhedral set is defined by the equalities and inequalities

Equations (7) and (8) avoid buffer overload while (9) and (10) initialize buffer values. To account for delay constraints (11) and (12) ensure that messages reach their destinations completely at at last. Coloring is ensured by (13) and (14) is a modified Kirchhoff's Law [6]. The SRPC problem under consideration is now as follows

By the last constraints, we assume that at any time each node can map all part of messages onto a single link for transmission [5]. Furthermore, we assume that the amount of information (in bits) we can transmit on a single wireless link at time slot is bounded from above by a maximum mutual information bound that itself depends on the power setting. The last constraints of (15) are the only constraints coupling network flow variables with communication variables . Thus, we call them coupling constraints [5], and they represent the most challenging constraints of the SRPC problem. All the other constraints are either constraints for the network flow variables or for the communication variables only. Assuming time-invariant channel conditions within the duration of a single time slot , function describes the amount of information of edge and can be expressed with (1) by the well-known Shannon formula

A comprehensive overview on commonly used cost functions for wireless data networks is given in [5].

### 3.2. Standard Interference Function

In the following we give an interpretation of the coupling constraints of the SRPC problem (15). We define for and

to be a *standard interference function* in
[9]. To clarify the meaning of a standard interference function, we restate the definition given in [9]. Here,
and
mean componentwise inequality.

If we insert (1) into (16) and solve the coupling constraints in (15) for power values, we obtain rewritten coupling constraints as

Definition 1.

*standard interference function*if for all the following properties are satisfied.

- (i)
- (ii)
- (iii)

The positivity property ensures positive power values of the joint routing and power control problem (15). If any transmit power level is decreased, the monotonicity guarantees the decrease of the interference on the other links in the network, ensuring the maintenance of the same or even the achievement of a lower interference level for all links. The scalability property implies that if then for .

Interestingly, (20) represents a Quality-of-Service (QoS) constraint, that is, a lower bound on the (implicitly defined) SINR. By reusing the coupling constraints again (15) and solving (16) for SINR we require

SINR is the main indicator for the transmission quality. Hence, given a modulation and coding scheme a specific per-hop error performance implies a respective . In turn, by varying we vary the transmission quality.

Note that for given values , we can use a fixed point iteration algorithm to find a unique power vector with

This power iteration represents a standard power control algorithm as introduced in [9]. The power iteration used herein to solve (15) will be described in detail in Section 5.1. With coupling constraints (20) of problem (15) we can make use of the properties of the standard interference function [9], arriving at the following theorem.

Theorem 1.

Suppose there exists a feasible point of this optimization problem, then there exists a feasible point with the same or better objective function value for which all the constraints (20) are active, that is, equality holds in all of them. Especially, for every optimum objective function value there exists an optimum variable setting such that all constraints in (20) are active. If is strictly monotone in , then all constraints (20) are active at each optimal solution of this problem.

Proof.

See Appendix A.

Theorem 1 is an extension of the results found in [9]. In contrast to [9] we do not assume that is just a sum of powers, instead it can be an arbitrary function being monotone in power values. Moreover, the objective as well as the coupling constraints depend on the flow variables and buffers , a case not considered in [9].

## 4. RPCD-Algorithm

In this section we present the RPCD-Algorithm for solving the SRPC problem (15). In contrast to universal approaches, like the dual decomposition method, we fully exploit our knowledge of active constraints of the joint optimization problem.

Based on Theorem 1 we can formulate an equivalent optimization problem but we avoid the extension of the utility function as usually done by applying dual or penalty approaches. We further keep the constraints and we only have to exchange the common network and power variables.

### 4.1. RPCD-Principle

Let us consider again problem (15). Due to Theorem 1 we know that all coupling constraints (20) of the SRPC problem are active at least at one optimum solution. By means of this observation, we can rewrite the SRPC problem to an equivalent problem as follows. Activity (equality) means that

We now substitute (24) into the objective of the SRPC problem (15) and obtain an equivalent problem with the rewritten cost function as

In the following we use (25) and decompose the SRPC problem into two convex subproblems.

In particular, by assuming feasible power variables, a routing problem with fixed link capacities is formulated and the optimum flow variables for the routing problem are calculated. Equivalently, we can assume fixed routing variables and formulate a power control problem to calculate optimum power values [10].

The two subproblems are as follows.

#### 4.1.1. Network Flow (Routing) Subproblem

We assume feasible power variables . With (25) we need to solve the optimization problem

where are the optimization variables.

We have the following lemma.

Lemma 1.

( ) If is a continuously differentiable and monotone function in and in , then the objective of (26) is a continuously differentiable and monotone function in .

- (a)
- (b)
- (c)

Then, the objective of (26) is strictly convex in and the solution to (26) is unique and continuous on

Proof.

See Appendix B

#### 4.1.2. Power Control Subproblem

We assume feasible network variables . We need to solve the optimization problem

where are the optimization variables.

We have the following lemma.

Lemma 2.

- (1)
problem (27) has a unique solution,

- (2)

Proof.

See Appendix C

### 4.2. RPCD Algorithm

As a consequence of the discussion above, we can replace the SRPC problem (15) by two simple subproblems, coupled to each other via fixed variables (power and network variables). The algorithmic scheme used is exemplified as RPCD algorithm and described in Algorithm 1.

**Algorithm 1:** RPCD

**Input:** All parameters for problem (15).

( ) Choose sufficiently large so that problem (26)

is feasible;

(
) **while** *stopping criterion for*
*not fulfilled*

**do**

( ) Set and solve problem (26).

( ) Set and solve problem (27).

Convergence of the RPCD algorithm is given by the following theorem.

Theorem 2.

Let us consider Lemmas 1 and 2. Under these assumptions and under the assumption that (15) is convex, the RPCD algorithm is well defined and provides a sequence of iterates such that each subsequence of this sequence converges to an optimal point of (15). Moreover, there exists at least one converging subsequence. Additionally, the sequence converges monotonically decreasing.

Proof.

See Appendix D

Note that both subproblems, (26) and (27), are *convex* and represent standard problems for which many efficient (distributed) algorithms exist. Particularly, we have to solve a flow problem with fixed capacities (fixed power values) [11] while computing optimum power values can be done by means of standard power control algorithms [9].

## 5. Distributed RPCD

Generally, we can apply centralized as well as distributed implementation for the RPCD algorithm. In this paper we concentrate on distributed algorithm exclusively. For the interested reader, a detailed survey about the centralized and distributed algorithms and their advantages and disadvantages can be found in [12].

Herein, for the distributed approach locally available information is required and we restrict the internode communication between neighbor nodes only.

In the following, as introduced in Section 4.1, we consider again the two subproblems, routing (26) and power control (27), and present distributed algorithms for solving them.

### 5.1. Distributed Power Control

Let us consider again the power control subproblem (27), which is part of the RPCD algorithm.

Assume given network variables and that the standard interference function is feasible, that is, if the power vector satisfies the coupling constraints (20), then we can use the following fixed point iteration to compute the optimum power settings

However, in practical systems we have taking power limitations into account. Given the original interference function and considering the maximum power vector in (3) we define

It has been proven in [9] that is a standard interference function fulfilling Definition 1.

To include the constraint on the output power of a node, we define

and denote by a projection operator that maps computed power values into the polyhedral set at each iteration step of the power iteration (29).

This projection allows us to consider only feasible power values during the course of the iteration.

By coupling the constrained interference function with this projection on a polyhedral set, we define a new interference function for given network variables

It can be easily shown that for all the interference function satisfies all properties given by Definition 1 and, hence, is also a standard interference function.

For each time step we can now write the standard constrained power iteration as

The power iteration (33) we call *distributed power control algorithm*.

Obviously, (33) is defined in terms of (32), (31), and (19). Due to (19), the information required to update the power values at starting node for a link is the interference caused by the interfering transmissions measured at the end node for a link . Moreover, the projections introduced to consider the power constraints are local only. Hence, (33) represents a distributed power control algorithm [9].

We use (33) to find a unique vector with

If is feasible, then for any initial vector , the iteration (33) converges to a unique fixed point .

Due to Theorems 1 and 2, this unique fixed point of (34) is a solution of the SRPC problem (15). If the SRPC problem is (strictly) convex, the fixed point is the global (unique) solution for the power setting of the joint routing and power control problem.

### 5.2. Distributed Routing

In the following we present a distributed algorithm for solving the routing subproblem (26) with given power values

The key to a distributed algorithm is to apply a decomposition method by means of formulating the dual problem of the optimization problem (26). Therefore we exploit the separable structure of the routing problem (26) via the dual decomposition method (see, e.g., [5, 13]). For solving the dual problem, we propose to apply the common approach of using the subgradient method [14].

To form the dual routing problem we rewrite the original routing problem (26) using the Lagrange function [6]. We introduce the Lagrange multipliers for the most involving constraints, which are the coupling constraints of the SRPC problem (15)

and the flow conservation constraints, that is, modified Kirchhoff's Law (14)

This results in the partial Lagrangian of (26) given as

and the Lagrange multipliers are denoted by and .

The Lagrangian dual function is

Given the Lagrange dual function we can formulate the *dual problem* by [6]

We need to solve the dual problem (40) in order to obtain the best lower bound on from the Lagrange dual function (39). Since the Lagrangian dual function is convex, the dual problem is a convex optimization problem [5]. Moreover, Slater's condition (see, e.g., [6, 13]) holds and thus, strong duality holds. This means, the optimal value of the original routing problem (26) and the dual optimal value from (39) are equal and we can solve the primal problem (26) by its dual (40).

The algorithm to solve (39) and (40) is a two stage optimization algorithm. It solves (39) and (40) separately by using the subgradient method [6, 14] and toggling between the two subproblems until a convergence criterion is met.

For the computation of the dual function (39) we use the projected subgradient method [6, 14], which is an algorithm for minimizing a nondifferentiable convex function with the main feature of enabling distributed implementation.

As a first step we have to calculate the subgradients with the respect to the variables and , for variables and . These subgradients are given by

where denotes the node that represents the starting point for one or more links . Analog to , we denote with the node that represents the end point for one or more links .

The subgradient updates on the variables and are

Note that the projection on the nonnegative orthant by results due to the network constraints and .

Furthermore, and represent the subgradient step sizes and have to satisfy (shown for )

to ensure convergence. By we denote the iteration step.

Finally, we have to solve the dual problem. For this, we compute the subgradients of the Lagrangian dual function due to the dual optimization variables and that are given by

Applying subgradient update we obtain for variables and

Note that the projection on the nonnegative orthant by results due to the constraints .

Furthermore, and represent the subgradient step sizes, both satisfying the conditions in (44) with denoting the iteration step.

As one can see by considering (33), (41), (42), (45), (39), (46), and (41) two types of information are necessary. First, that the information required for the computation to take place at each and every node is the interference caused by the interfering transmissions measured at the receiving node. Second, by (42) Lagrange multipliers from neighbor nodes, for example, and are required.

The distributed routing algorithm tries to achieve an optimum coordination between the network variables and on the one hand and the dual variables and on the other hand. For the considered wireless network, this means that the distributed routing algorithm tries to achieve an optimum coordination between node buffers and capacities allocated to the links, subject to the network constraints as defined in (26).

## 6. Simulation Results

Each intermediate node can transmit to and receive along multiple links from nodes, neither multicast nor broadcast is considered. The network is a single frequency network. For the sake of simplicity, we assume that the scheduler does not take in-band signaling users into account, rather we might interpret in-band users as additive noise. We further require in the simulations that simultaneously active links do not interfere. Hence, the SRPC problem under consideration is convex, therefore, the optimum solution is global. This means, we assume orthogonal transmission between links, possibly performed by Space Division Multiple Access (SDMA) schemes such as sending/receiving beamforming [15, 16]. Due to the setup of the wireless backhaul links, the nodes we consider are cellular base stations with high processing capability. Without loss of generality, the objective function we assume is to minimize total transmitted power with .

The scenario shown in Figure 4 is denoted as [ , 3, 5, 1] scenario. So, we have 10 nodes forming a wireless mesh backhauling network with 23 edges. The simulation parameter set up is as follows. The wireless network has to transmit data of = 10 Mbit size from the source node to the destination, but due to the delay constraint the transmission has to be completed within a maximum number of time slots, that is, . The bandwidth per link is = 5 MHz, the length of an time-slot is ms and the radius per hexagonal cell is m. We assume an exponential path-loss model with factor 3, but no shadow-fading. The thermal spectral noise density is dBm/Hz. The buffer size per node is restricted to Mbit. To account for power constraints we upper bound the power per node by Watt, whereas for each specific link we assume no explicit power restriction. Regarding the distributed RPCD algorithm, we use a stopping criterion based on the variables to check for convergence. We proceed with the iteration as long as the maximum norm of two consecutive iteration steps is greater than . To show convergence, we apply the algorithm for a huge number of different starting points as well as for several networks, that is, [ , 3, 1], [ , 3, 5, 1], [ , 3, 5, 7, 1].

We observe the following result.

*For a given feasible starting point*
*the distributed RPCD algorithm converges to an optimum solution within one step only.*

Since the algorithm converges globally the zero vector is always a feasible starting point.

*multiple path routing*has to be performed. The transmission of the data expands over more routes in the wireless mesh backhaul network. Nevertheless the data traffic is generally concentrated in the inner links, which can be explained with the geometry of the network.

## 7. Conclusion

In this paper we have considered the joint routing, time scheduling and power control problem for single frequency, time-slotted wireless mesh networks. We presented an approach for optimally solving this crosslayer optimization problem while meeting the requirements, such as distributed implementation, multiple path routing, and per-hop error performance. The main contribution is the distributed Routing and Power Control Decomposition (RPCD) Algorithm, which is based on the idea of decoupling the SRPC problem into two subproblems, power control and routing, and including scheduling in the constraint set of the SRPC problem. Moreover, we presented distributed algorithms for solving both, the power control and the routing subproblem. For illustration purpose we applied the distributed RPCD algorithm to a wireless mesh backhaul network. The observed convergence results are impressive: only *one* decomposition step is needed to achieve the optimal solution.

## Appendices

### A. Proof of Theorem 1

Proof.

Choosing feasible variables ( ) for the problem above, we immediately see that the interference function defined by (29) is a standard interference function. Therefore, the power iteration (29) started from, say, converges to a point with (22). Clearly, for this point the constraints (20) are active. Moreover, in [9] it was shown that has the smallest objective function value for all possible choices of the variables ( ) with prespecified and fixed variables ( ). Therefore, the constraints (20) have to be active in all solutions of the optimization problem above.

### B. Proof of Lemma 1

Proof.

( ) This can be seen by differentiating the objective under consideration with respect to .

The objective is strict convex in if and only if this Hessian is positive definite. Now, the first summand above is clearly positive semidefinite, since is convex in . Likewise, the third summand is positive semidefinite, since is convex in . Finally, the diagonal matrix has nonnegative entries on the main diagonal, since is strictly convex in and is monotone in . Accordingly, the Hessian above is positive definite as long as one of the summands is positive definite. Assumption (a) leads to the positive definiteness of the first summand, assumption (b) leads to the positive definiteness of the last summand, while assumption (c) leads to the positive definiteness of .

From this, the strict convexity under the given assumptions readily follows.

Clearly, the variables ( , , ) are unique in an optimum of the problem under consideration. Using (9), (10), and (14) and induction over , one easily concludes that optimal variables ( ) are unique, too.

The continuity is a classic result from parametric optimization, see, for example, [18], and follows from the strict convexity of the objective in and from the fact that the set of feasible points is a polyhedron.

Remark 1.

The following remarks hold with respect to Lemma 1.

(1)The second result of Lemma 1 can be weakened a bit. In case is not strictly convex in, say, , it suffices to assume that the diagonal entries of the matrix are sufficiently large. That amounts to saying that either is sufficiently large or that is sufficiently large, that is, there is certain amount of monotonicity build into the objective function ( , , ).

(2)The convexity assumptions of Lemma 1 do *not* amount in assuming that
is convex, as the example
shows.

### C. Proof of Lemma 2

Proof.

( ) See Theorem 1 and [9].

( ) This follows by noting that the solution to the problem at hand is uniquely characterized by the fixed-point equation , the latter being a linear system in . The corresponding solution depends continuously on .

Remark 2.

In Lemma 2, strict monotonicity cannot be replaced by monotonicity. However, uniqueness of the results of Lemma 2 hold again if only the (unique) -solution of the optimization problem under consideration is considered.

### D. Proof of Theorem 2

Proof.

*closed*if = and = imply = . Clearly, continuous mappings are closed, and therefore the two mappings mentioned above are closed. (See also [19, page 123],. Actually, closedness is a property usually associated with point-to-set mappings, but we only consider point-to-point mappings here. For point-to-point mappings, continuity is sufficient for closedness, and these two notions are equivalent if the set of arguments is compact.) The rest of the theorem follows the convergence proof of the coordinate descent method [19, Section ], replacing the set = with the set of feasible points for which there does not exist a feasible direction of descent: since is compact, the coupling constraint implies that is compact, too, and therefore the whole set of feasible points is compact. Accordingly, the composition of maps

that is, the set of feasible points from which the algorithm does not improve objective function values any more. In the convex case this is a set of KKT-points which is the same as the set of the optimal points. It is easy to see that every converging subsequence of converges to a point in , more precisely to a fixed point of (see also [20]), which is thereby a point for which no feasible direction of descent exists.

Remark 3.

- (1)
The convexity assumptions within the theorem have mainly be imposed to guarantee that the map is well-defined (i.e., the solution to the corresponding optimization problem is unique). If we simply assume this well-definedness (or enforce it by, say, computing the least-squares optimal solution), we can drop the corresponding assumptions on instead.

- (2)
(2)In principle, strict monotonicity of in is required to obtain that is continuous. However, what usually happens is that in step 5 the power iteration is used, which results in the computation of the unique -solution to . In case the power iteration is replaced by another algorithm, uniqueness of the corresponding solution has to be guaranteed in a different way.

## Authors’ Affiliations

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