Primal Recovery from Consensus-Based Dual Decomposition for Distributed Convex Optimization

Primal Recovery from Consensus-Based Dual Decomposition for Distributed Convex Optimization

Abstract

Dual decomposition has been successfully employed in a variety of distributed convex optimization problems solved by a network of computing and communicating nodes. Often, when the cost function is separable but the constraints are coupled, the dual decomposition scheme involves local parallel subgradient calculations and a global subgradient update performed by a master node. In this paper, we propose a consensus-based dual decomposition to remove the need for such a master node and still enable the computing nodes to generate an approximate dual solution for the underlying convex optimization problem. In addition, we provide a primal recovery mechanism to allow the nodes to have access to approximate near-optimal primal solutions. Our scheme is based on a constant stepsize choice and the dual and primal objective convergence are achieved up to a bounded error floor dependent on the stepsize and on the number of consensus steps among the nodes.

Mathematics Subject Classifications (2010) 90C25 90C30 90C46 90C59

1Introduction

Lagrangian relaxation and dual decomposition are extremely effective in solving large-scale convex optimization problems [1]. Dual decomposition has also been employed successfully in the field of distributed convex optimization, where the optimization problem requires to be decomposed among cooperative computing entities (called in the following simply by nodes). In this case, the optimization problem is generally divided into two steps, a first step pertaining the calculation of the local subgradients of the Lagrangian dual function, and a second step consisting of the global update of the dual variables by projected subgradient ascent. The first step can typically be performed in parallel on the nodes, whereas the second step has often to be performed centrally, by a so-called master node (or data-gathering node, or fusion center), which combines the local subgradient information.

Even though by solving the dual problem, one obtains a lower bound on the optimal value of the original convex problem, in practical situations one would also like to have access to an approximate primal solution. However, even with the availability of an approximate dual optimal solution, a primal one cannot be easily obtained. The reason is that the Lagrangian dual function is generally nonsmooth at an optimal point, thus an optimal primal solution is not a trivial combination of the extreme subproblem solutions. Methods to recover approximate (near-optimal) primal solutions from the information coming from dual decomposition have been proposed in the past [7] (and references therein). In one way or another, all these methods use a combination of all the approximate primal solutions that are generated while the dual decomposition scheme converges to a near-optimal dual solution. A possible choice for the combination is the ergodic mean [11].

Among the dual decomposition schemes with primal recovery mechanism available in the literature, we are interested here in the ones that employ a constant stepsize in the projected dual subgradient update. The reasons are twofold. First of all, a constant stepsize yields faster convergence to a bounded error floor, which is fundamental in real-time applications (e.g., control of networked systems). In addition, the error floor can be tuned by trading-off the number of iterations required and the value of the stepsize. The second reason is that in many situations the underlying convex optimization problem is not stationary, but changes over time. Having in mind the development of methods to update the dual variables while the optimization problem varies [15], it is of key importance to employ a constant stepsize. In this way, the capability of the subgradient scheme to track the dual optimal solutions does not change over time due to a vanishing stepsize approach.

In this paper, we propose a way to remove the need for a master node to collect the local subgradient information coming from the different nodes and generate a global subgradient. The reason is that in distributed systems, the nodes are connected via an ad-hoc network and the communication is often limited to geographically nearby nodes. It is therefore impractical to collect the local subgradient information in one physical location, whereas it is advisable to enable the nodes themselves to have access to a suitable approximation of the global subgradient. We use consensus-based mechanisms to construct such an approximation. Consensus-based mechanisms have been used in the primal domain both with constant stepsizes [18] and with vanishing ones [20], however, to the best of the authors’ knowledge, they have not been used in the dual domain, and not together with primal recovery. An interesting, but different, approach applying consensus on the cutting-plane algorithm to solve the master problem has been very recently proposed in [22]. Our main contributions can be described as follows.

First, we develop a constant stepsize consensus-based dual decomposition. Our method enables the different nodes to generate a sequence of approximate dual optimal solutions whose dual cost eventually converges to the optimal dual cost within a bounded error floor. Under the assumptions of convexity, compactness of the feasible set, and Slater’s condition, the convergence goes as , where is the number of iterations. The error depends on the stepsize and on the number of consensus steps between subsequent iterations . Furthermore, the nodes are exchanging subgradient information only with their nearby neighboring nodes.

Then, since in our method, each node maintains its own approximate dual sequence, we provide an upper bound on the disagreement among the nodes, and we prove its convergences to a bounded value.

Finally, we propose a primal recovery scheme to generate approximate primal solutions from consensus-based dual decomposition. This scheme is proven to converge to the optimal primal cost up to a bounded error floor. Once again, under the same assumptions, the convergence goes as and the error depends on the stepsize and on the number of consensus steps.

Organization. Section 2 describes the problem setting, our main research question, and some sample applications. In Section 3, we cover the basics of dual decomposition to pinpoint the main limitation, i.e., the need for a master node. We propose, develop, and investigate the convergence results of our algorithm in Sections Section 4 and Section 5. All the proofs are contained in Sections Section 6 and Section 7. In Section 8, we collect numerical simulation results. Future research questions and conclusions are discussed in Sections Section 9 and Section 10, respectively.

2Problem Formulation

Notation. For any two vectors , the standard inner product is indicated as , while its induced (Euclidean) norm is represented as . A vector belongs to iff it is of size and all its components are nonnegative (i.e., is the nonnegative orthant). For any vectors , its components are indicated by , . The vector is the column vector of length containing only ones. We indicate by the identity matrix of size . For any real-valued squared matrix , we say or iff the matrix is positive semi-definite or negative semi-definite, respectively. We also write , iff . For any real-valued squared matrix , the norm represents the Frobenius norm, while the trace is indicated by . The symbol is the transpose operator, represents the Kronecker product, stands for map composition, is the convex hull, is the vectorization operator, while is the projection operator onto the set . The -subgradient of a concave function , for the non-negative scalar , at is a vector such that

Furthermore, the collection of -subgradients of at is called the -subdifferential set, denoted by . If the -subgradient is the regular subgradient and we drop the in the notation of the subdifferential.

Formulation. We consider a convex optimization problem defined on a network of computing and communicating nodes. Let the nodes be labeled with and we equip each of them with the local (private) convex function . Let be the stacked vector of all the local decision variables, i.e., . Let the functions be convex. Let be real-valued square and symmetric matrices. Let be convex and compact sets, and let . We are interested in solving decomposable convex optimization problems of the form,

In order to simplify our notation (and without loss of generality) we have chosen to work with scalar decision variables , with one scalar inequality, and with one linear matrix inequality. The following assumptions are in place.

Assumption ? is required to ensure a convex program with compact feasible set. Assumption ? ensures the existence of a solution for the optimization problem . Let be such a (possibly not unique) solution (i.e., a minimizer) and let be the unique minimum. Assumption ? is often required in dual decomposition approaches in order to guarantee zero duality gap and to be able to derive the optimal value of the optimization problem by solving its dual. In addition, Slater condition helps in bounding the dual variables, which is crucial in our convergence analysis. Assumption ? is required to simplify the convergence analysis. One might be able to loosen it and require only asynchronous communications, but this is left for future research since it is not the core idea of this paper. By Assumption ?, we can define an undirected communication graph consisting of a vertex set as well as an edge set . For each node , we call neighborhood, or , the set of the nodes it can communicate with.

The main research problem we tackle in this paper can be stated as follows.

Research problem: we would like to devise an algorithm that enables each node, by communicating with their neighbors only, to construct a sequence of approximate local optimizers , for which their primal objective sequence eventually converges to (possibly) up to a bounded error floor.

Our approach towards this problem is to devise a consensus-based dual decomposition with approximate primal recovery.

Sample applications. Problems as appear in many contexts: the first example we cite is the network utility maximization (NUM) problem, where a group of communication nodes try to maximize their utility subject to a resource allocation constraint [23]. NUM problems are very relevant in communication systems. Generalizations of NUM problems, where the cost function is separable and the variables are constrained by linear inequalities, can also be handled by , and have been considered, e.g., in model predictive controller design [25] (which is one of the workhorse of nowadays control theory). Another sample application is sensor selection, where a set of nodes try to decide which one of them should be activated to perform a certain task based on a given metric. This is in general a combinatorial problem, yet it can be relaxed to a semidefinite program, which is a generalization of , [26]. In the latter example, the constraint plays an important role.

Multi-agent/Multiuser/Networked problems. If the constraints and involve only local functions, that is the sum is only over the neighbors of a particular , then we have what is known as multi-agent (or multiuser, or networked) problem. These problems can be further complicated by the presence of global decision variables. In all these cases, due to the presence of neighborhood constraint functions only, the dual variables associated to them can be computed locally in the neighborhood (we can refer to them as link dual variables). Therefore, by a proper use of dual decomposition, we can devise distributed algorithms that can be implemented on nodes and connecting links. Relevant recent work on these problems is reported in [28]. In our case, the constraints - involve constraint functions from all the nodes, in all the decision variables together; therefore, the proposed methods for multi-agent problems cannot be directly applied in our case. In general, the link dual variables become a network-wide dual variable in our case, and we retrieve the standard dual decomposition scheme with the need for a master node to compute such a global network-wide dual variable.

3Dual Decomposition

The Lagrangian function is formed, as a first step of dual decomposition,

where is the dual variable associated with the constraint , and is the dual variable associated with . Further, the dual function can be defined as

The set is compact, which means that the function is continuous on . Furthermore, the function is concave. For any pair of dual variables , we can compute the value of the primal minimizers and their set:

Given the compactness of and the form of the dual function , we can define the subdifferential of at and as the following sets

Subgradient choices for are therefore

for any choice of . In addition, since is compact and the constraints - are represented by continuous functions, the subgradients are bounded, and we set, for all

where we have defined , and . Finally, the Lagrangian dual problem can be written as

and by Slater condition (Assumption ?), strong duality holds: .

Since the original convex optimization problem is decomposable, the Lagrangian function is separable as

and so is the dual function

and its subgradients.

Dual decomposition with approximate primal recovery as defined in [4] is summarized in the following algorithm.



This algorithm generates a converging sequence as detailed in the following theorem.

The proof follows from [4]. Since our optimization problem involves also a linear matrix inequality, some extra steps are needed in the proof of part (c). To be more specific, by following the same steps in the proof of [4], we arrive at the following inequality

where and are the optimal dual variables. We now need to find an lower bound for the rightmost term of . By similar arguments of the proof of [4], we obtain for all

Given the two positive semi-definite matrices and of dimension , we know that , [36], which means

This implies that for

where we have used Cauchy-Schwarz inequality [37]. By combining and with , we obtain the lower bound

and the claim is proven.

Although, the dual decomposition method of [4] presents several advantages, in practice, the nodes will need to sum the subgradients coming from the whole network in Step 4 in order to maintain common dual variables. This is often not practical in large networks, because it would call for a significant communication overhead.

In the following sections, (i) we propose a consensus-based dual decomposition with primal recovery mechanism to modify Step 4 in order to make it suitable for limited information exchange (i.e., communication only with neighboring nodes); (ii) we prove dual and primal objective convergence of the proposed method up to a bounded error floor which depends (among other things) on the number of communication exchange with the neighboring nodes for each iteration .

4Basic Relations

The lemma follows from [4] with minor modifications. In particular, we use [36] to bound the inner product

and the fact that , [37]. The remaining steps are omitted since similar to [4].

It follows from the result of the preceding lemma that under Slater, the dual optimal set is nonempty. Since , by using Lemma ?, we obtain

Furthermore, although the dual optimal value is not a priori available, one can compute a looser bound by computing the dual function for some couple . Owning to optimality, , thus

This result is quite useful to render the dual decomposition method easier to study. In fact, as in [4], we can modify the sets over which we project in Step 4 by considering a bounded superset of the dual optimal solution set. This means that we can substitute Step 4 in with

for a given scalar . The nice feature of this modification is that both and are now compact convex sets. This does not increase computational complexity, and it is a useful modification, for it provides a leverage to derive the convergence rate results. In the following, for convergence purposes, we will use .

5Consensus-Based Dual Decomposition

We consider now a consensus-based update to enforce the update rule of dual decomposition in to fit the constraint of a limited communication network. Our approach is inspired by the one of [18] but applied to the dual domain. First of all, we define a consensus matrix , with the following properties:

where returns the spectral radius and is an upper bound on the value of the spectral radius. It is a common practice to generate such consensus matrices; a possible choice is the Metropolis-Hasting weighting matrix [38].

A consensus iteration is a linear mapping with the property that the result of its repeated application converges to the mean of the initial vector, i.e., for

This averaging property is ensured, for example, by conditions as the ones in . In addition, given the structure of in , each consensus iteration involves only local communications (only the neighboring nodes will share their local variables), which will be the key point of our modification. In the following, we will study multiple consensus steps, in the sense that the computing nodes will run multiple consensus iterations (each of which involving only local communications) between subsequent iterations ’s. We let the number of consensus steps be . In this case, the consensus mapping will be of the form . Since we will enable each node to generate its own dual variables on which consensus will be enforced, we start by defining local versions of and as and , respectively. Next, we define our consensus-based dual decomposition as the following algorithm.


(CoBa-DD)



We highlight that the proposed algorithm CoBa-DD (or ) involves only local communication. The only communication involved is in the consensus steps, each of which requiring the nodes to share information with their neighbors. Also, note that computing (for the definition of and ) is not a very difficult task, since a Slater vector is usually easy to find by inspection, and both and can be computed by a consensus algorithm run in the initialization step of CoBa-DD.

In order to analyze dual and primal convergence of , we start by some basic results. First, given that the sets and are compact, and that and are picked to be bounded, the dual variables and are bounded for each . In particular, we have

The claim is proven by using the definition of subgradient of a concave function . Since is a concave function, for all ,

For , the claim follows.

The proof follows given the compactness of and (therefore) the boundedness of the subgradients.

We now present the main convergence results.

Theorem ? and Corollary ? specify how the consensus is reached among the nodes on the value of the dual variables while the algorithm is running. Specifically, the consensus is reached exponentially fast to a steady-state bounded error floor. This bounded error depends on (which can be tuned), and on , which can also be tuned by varying . In particular, for , due to the fact that in conditions , then and we obtain back the usual dual decomposition scheme with perfect agreement among the nodes.

Theorem ? implies dual objective convergence up to a bounded error floor. Convergence is even more evident if we remember that, owning to optimality, , and thus, if we define , we obtain

Note that the rightmost term () represents a measure of sub-optimality of the approximate solution.

Theorem ? formulates convergence of the primal cost up to an error bound . The rate of convergence is . We can also distinguish the error terms that come from the constant stepsize and the terms that come from the finite number of consensus steps . In particular, we can write

and see that the term (1) is due to the constant stepsize, while the term (2) is due to the finite number of consensus steps. Furthermore, if , then , and we can set , yielding

This is similar to the error level we obtain for the dual decomposition method in , and Theorem ?. Theorem ? defines the main trade-offs in designing the algorithm’s parameters and . The larger the stepsize is, the faster the convergence is, even though the steady-state error becomes larger. If we increase then the communication effort increases and the error decreases.

6Proof of Theorem and Theorem

6.1Preliminaries

We start our analysis by rewriting Step 4 of in a more compact way. Let be the vector defined as , and let be the stacked vector of all the , . Similarly, let be the vector , and let the stacked vector of all the , . Let be the convex set

and let . The iterations in Step 4 of can be rewritten as

The iteration represents a consensus-based subgradient method to maximize the dual function , i.e, the maximization problem

In particular assigns to each node a copy of , , and enforces consensus among them. Furthermore, by , by triangle inequality, and by ,

The proof is an adaptation of [18]. In particular, we can show that for all

Therefore, if we choose,

and the claim follows from Lemma ?.(a). In order to prove , we proceed as follows.

where extracts the -th component of a vector. Define

Prior to consensus, the distance between the iterates can be bounded as

which also implies . Given that , after consensus, we have

where . As said which means . Thus, by using Lemma ? we can bound as

which is the rightmost term in and the claim is proven.

6.2Proof of Theorem

The quantity is upper bounded by by Lemma ? (inequality ), thus, . Let us choose , , with determined as in Theorem ?. Then, by Lemma ? and , it follows that,

Let , since , then

and by Lemma ?.(b), we derive . By using the non-expansive property of the projection operator, since , for all , we can write

and by Lemma ?.(a) the claim follows.

6.3Proof of Theorem

We define an average value for as . For convergence purposes, we need to keep track of the difference , and thus we define the vectors and as

The main idea of the proof is to show that is updated via an approximate -subgradient method and, then, by using [41] the theorem follows. The first part is formalized in the following lemma.

a

(a) We start by bounding ,

where we have used the inequality to bound the term .

(b) Since and , by the concavity of and the definition of subgradient of a concave function , we can write for all

In particular, we have used the fact that any subgradient vector of is bounded by , and inequality . If we sum the last relation over , we obtain . In addition for any , by using Lemma ?

We use the fact that by construction in Lemma ?, by , by , and by the preceding proof. By using these inequalities, we can bound

and we obtain

which is .

(c) By using the definition of subdifferential , the inequality implies with . Summation over yields,

for any , such that . Since by construction, then we can choose , from which the claim follows.

(d) It is sufficient to write explicitly the update rule for . Starting from the definition of in and the definition of in Lemma ?, we obtain