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An Approximate Dual Subgradient Algorithm for
Multi-Agent Non-Convex Optimization
Minghui Zhu and Sonia Martínez
Abstract—We consider a multi-agent optimization problem where agents
subject to local, intermittent interactions aim to minimize a sum of local
objective functions subject to a global inequality constraint and a global
state constraint set. In contrast to previous work, we do not require that
the objective, constraint functions, and state constraint sets are convex. In
order to deal with time-varying network topologies satisfying a standard
connectivity assumption, we resort to consensus algorithm techniques and
the Lagrangian duality method. We slightly relax the requirement of exact
consensus, and propose a distributed approximate dual subgradient algorithm to enable agents to asymptotically converge to a pair of primal-dual
solutions to an approximate problem. To guarantee convergence, we assume that the Slater’s condition is satisfied and the optimal solution set of
the dual limit is singleton. We implement our algorithm over a source localization problem and compare the performance with existing algorithms.
Index Terms—Dual subgradient algorithm, Lagrangian duality.
I. INTRODUCTION
Recent advances in computation, communication, sensing and actuation have stimulated an intensive research in networked multi-agent
systems. In the systems and controls community, this has translated
into how to solve global control problems, expressed by global objective functions, by means of local agent actions. Problems considered
include multi-agent consensus or agreement [11], [17], coverage control [4], formation control [7], [21] and sensor fusion [24].
The seminal work [2] provides a framework to tackle optimizing
a global objective function among different processors where each
processor knows the global objective function. In multi-agent environments, a problem of focus is to minimize a sum of local objective
functions by a group of agents, where each function depends on
a common global decision vector and is only known to a specific
agent. This problem is motivated by others in distributed estimation
[16], [23], distributed source localization [20], and network utility
maximization [12]. More recently, consensus techniques have been
proposed to address the issues of switching topologies, asynchronous
computation and coupling in objective functions; see for instance
[14], [15], [27]. More specifically, the paper [14] presents the first
Manuscript received October 14, 2010; revised January 25, 2012; accepted
October 08, 2012. Date of publication November 16, 2012; date of current version May 20, 2013. This work was supported by the NSF CAREER Award
CMMI-0643673. Recommended by Associate Editor E. K. P. Chong.
M. Zhu is with the Laboratory for Information and Decision Systems,
Massachusetts Institute of Technology, Cambridge MA 02139 USA (e-mail:
[email protected]).
S. Martínez is with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093 USA (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TAC.2012.2228038
0018-9286/$31.00 © 2012 IEEE
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013
analysis of an algorithm that combines average consensus schemes
with subgradient methods. Using projection in the algorithm of [14],
the authors in [15] further address a more general scenario that takes
local state constraint sets into account. Further, in [27] we develop two
distributed primal-dual subgradient algorithms, which are based on
saddle-point theorems, to analyze a more general situation that incorporates global inequality and equality constraints. The aforementioned
algorithms are extensions of classic (primal or primal-dual) subgradient methods which generalize gradient-based methods to minimize
non-smooth functions. This requires the optimization problems of
interest to be convex in order to determine a global optimum.
The focus of the current technical note is to relax the convexity
assumption in [27]. In order to deal with all aspects of our multi-agent
setting, our method integrates Lagrangian dualization, subgradient
schemes, and average consensus algorithms. Distributed function
computation by a group of anonymous agents interacting intermittently can be done via agreement algorithms [4]. However, agreement
algorithms are essentially convex, and so we are led to the investigation of nonconvex optimization solutions via dualization. The
techniques of dualization and subgradient schemes have been popular
and efficient approaches to solve both convex programs (e.g., in [3])
and nonconvex programs (e.g., in [5], [6]).
Statement of Contributions: Here, we investigate a multi-agent optimization problem where agents desire to agree upon a global decision
vector minimizing the sum of local objective functions in the presence
of a global inequality constraint and a global state constraint set. Agent
interactions are changing with time. The objective, constraint functions, as well as the state-constraint set, can be nonconvex. To deal
with both nonconvexity and time-varying interactions, we first define
an approximate problem where the exact consensus is slightly relaxed.
We then propose a distributed dual subgradient algorithm to solve it,
where the update rule for local dual estimates combines a dual subgradient scheme with average consensus algorithms, and local primal
estimates are generated from local dual optimal solution sets. This algorithm is shown to asymptotically converge to a pair of primal-dual
solutions to the approximate problem under the following assumptions:
firstly, the Slater’s condition is satisfied; secondly, the optimal solution
set of the dual limit is singleton; thirdly, dynamically changing network
topologies satisfy some standard connectivity condition.
A conference version of this manuscript was published in [26], and
an enlarged archived version of this paper is [25]. Main differences are
the following: i) by assuming that the optimal solution set of the dual
limit is a singleton, and changing the update rule in the dual estimates,
we are able to determine a global solution in contrast to an approximate
solution in [26]; ii) we present a simple criterion to check the new sufficient condition for nonconvex quadratic programs; iii) new simulation
results of our algorithm on a source localization example and a comparison of its performance with existing algorithms are performed. Due
to space limitations, details of the technical proofs and simulations can
be found in [25].
II. PROBLEM FORMULATION AND PRELIMINARIES
Consider a networked multi-agent system where agents are labeled
. The multi-agent system operates in a synby
, and its topology will be
chronous way at time instants
,
represented by a directed weighted graph
. Here,
is the adjacency matrix,
for
is the weight assigned to the edge
where the scalar
pointing from agent to agent , and
is the
set of edges with non-zero weights. The set of in-neighbors of agent
at time is denoted by
.
Similarly, we define the set of out-neighbors of agent at time as
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. We here make the
following assumptions on communication graphs:
Assumption 2.1 (Non-Degeneracy): There exists a constant
such that
, and
, for
, satisfies
, for all
.
Assumption 2.2 (Balanced Communication): It holds that
for all
and
, and
for
and
.
all
Assumption 2.3 (Periodical Strong Connectivity): There is a
such that, for all
, the directed graph
positive integer
is strongly connected.
The above network model is standard to characterize a networked
multi-agent system, and has been widely used in the analysis of average consensus algorithms; e.g. see [17], and distributed optimization
in [15], [27]. Recently, an algorithm is given in [9] which allows agents
to construct a balanced graph out of a non-balanced one under certain
assumptions. The objective of the agents is to cooperatively solve the
:
following primal problem
(1)
where
is the global decision vector. The function
is only known to agent , continuous, and referred to as the objec, the state constraint set, is
tive function of agent . The set
are continuous, and the incompact. The function
is understood component-wise; i.e.,
, for
equality
, and represents a global inequality constraint. We
all
and
. We
will denote
.
will assume that the set of feasible points is non-empty; i.e.,
is
Since is compact and is closed, then we can deduce that
compact. The continuity of follows from that of . In this way, the
is finite and
, the set of primal
optimal value of the problem
optimal solutionss, is non-empty. We will also assume the following
Slater’s condition holds:
Assumption 2.4 (Slater’s Condition): There exists a vector
such that
. Such is referred to as a Slater vector of the
.
problem
Remark 2.1: All the agents can agree upon a common Slater vector
through a maximum-consensus scheme. This can be easily implemented as part of an initialization step, and thus the assumption that
the Slater vector is known to all agents does not limit the applicability
of our algorithm; see [25] for an algorithm solving this problem.
In [27], in order to solvethe convex caseof the problem
(i.e.; and
are convex functions and is a convex set), we propose two distributed
primal-dual subgradient algorithms where primal (resp. dual) estimates
move along subgradients (resp. supergradients) and are projected onto
convex sets. The absence of convexity impedes the use of the algorithms
in [27] since, on the one hand, (primal) gradient-based algorithms are
easily trapped in local minima; on the other hand, projection maps may
not be well-defined when (primal) state constraint sets are nonconvex. In
the sequel, we will employ Lagrangian dualization, subgradient methods
and average consensus schemes to design a distributed algorithm which
.
can find an approximate solution to the problem
Towards this end, we construct a directed cyclic graph
where
. We assume that each agent has a unique
in-neighbor (and out-neighbor). The out-neighbor (resp. in-neighbor)
(resp. ). With the graph
, we will
of agent is denoted by
:
study the following approximate problem of problem
(2)
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013
where
, with a small positive scalar, and is the column
, and
vector of ones. Problem (2) provides an approximation to
. In particular, the approximate
will be referred to as problem
when
. Its optimal value
problem (2) reduces to the problem
and
, respecand the set of optimal solutions will be denoted by
,
is finite and
.
tively. Similarly to the problem
can be replaced by any strongly
Remark 2.2: The cyclic graph
connected graph . Given , each agent is endowed with two inand
, for
equality constraints:
each out-neighbor . This set of inequalities implies that any feasible
of problem
satisfies the approximate consolution
. For simplicity, we will use
sensus; i.e.,
, with a minimum number of constraints, as the
the cyclic graph
initial graph.
A. Dual Problems
Before introducing dual problems, let us denote by
,
,
and
. The dual problem
associated with
is given by
,
,
, where
and
is given as
,
is not separable since
.
depends on
B. Dual Solution Sets
The Slater’s condition ensures the boundedness of dual solution sets
for convex optimization; e.g., [10], [13]. We will shortly see that the
Slater’s condition plays the same role in nonconvex optimization. To
as
achieve this, we define the function
follows:
Let be a Slater vector for problem
. Then
with
is a Slater vector of the problem
. Similarly to (3)
and (4) in [27], which employ Lemma 3.2 of [27], we have that for any
, it holds that
(5)
(3)
where
Here, the dual function
Note that
neighbor’s multipliers
where
in (5). This leads to the upper bound on
. Let
be zero
.
(6)
is the Lagrangian
function
where
note
can be computed locally. We de-
(7)
We denote the dual optimal value of the problem
by
and
. We endow each agent with
the set of dual optimal solutions by
and the local dual
the local Lagrangian function
defined by
function
In the approximate problem
, the introduction of
, renders the and separable. As a result, the global
dual function can be decomposed into a simple sum of the local dual
functions . More precisely, the following holds:
Since and are continuous and is compact, then
is continuous; see Theorem 1.4.16 in [1]. Similarly, is continuous. Since
is bounded, then
.
in the
Remark 2.3: The requirement of exact agreement on
is slightly relaxed in the problem
by introducing a
problem
. In this way, the global dual function is a sum of the
small
, as in (4);
is non-empty and uniformly
local dual functions
bounded. These two properties play important roles in the devise of
our subsequent algorithm.
C. Other Notation
Define the set-valued map
; i.e., given , the set
lutions to the following local optimization problem:
as
are the so(8)
Notice that in the sum of
, each
for
appears in two terms: one is
, and the
any
. With this observation, we regroup the
other is
terms in the summation in terms of , and have the following:
(4)
Here,
is referred to as the marginal map of agent . Since is comare continuous, then
in (8) for any
. In
pact and
the algorithm we will develop in next section, each agent is required to
obtain one (globally) optimal solution and the optimal value the local
optimization problem (8) at each iterate. We assume that this can be
, or and
easily solved, and this is the case for problems of
being smooth (the extremum candidates are the critical points of the
objective function and isolated corners of the boundaries of the constraint regions) or having some specific structure which allows the use
of global optimization methods such as branch and bound algorithms.
, we define the distance between a point
In the space
to a set
as
,
as
and the Hausdorff distance between two sets
.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013
We denote by
and
where
.
III. DADS ALGORITHM
In this section, we devise a distributed approximate dual subgradient
algorithm which aims to find a pair of primal-dual solutions to the ap.
proximate problem
For each agent , let
be the estimate of the primal soluat time
tion to the approximate problem
be the estimate of the multiplier on the inequality constraint
(resp.
)1 be the estimate of the multiplier associated with the collection of the local
(resp.
),
inequality constraints
. We let
,
for all
to be the collection of dual estimates of agent . And
for
where
denote
and
are convex combinations of dual estimates of agent and its neighbors
at time .
At time , we associate each agent a supergradient vector
defined as
, where
,
has components
,
, and
for
, while the components of
are given by:
,
, and
, for
. For each agent , we define the set
for some
where
. Let
is
to be the projection onto the set
. It is easy to check that
is well-defined.
closed and convex, and thus the projection map
The Distributed Approximate Dual Subgradient (DADS) Algorithm
is described in Table I.
Algorithm 1 The DADS Algorithm
Initialization: Initially, all the agents agree upon some
in the
. Each agent chooses a common Slater
approximate problem
and obtains
through a
vector , computes
is given in (7). After that, each
max-consensus algorithm where
and
.
agent chooses initial states
Iteration: Every , each agent executes the following steps:
, given
, solve the local optimization
1) For each
and the dual
problem (8), obtain a solution
.
optimal value
, generate the dual estimate
according
2) For each
to the following rule:
(9)
where the scalar
3) Repeat for
is a step-size.
.
Remark 3.1: The DADS algorithm is an extension of the classical
dual algorithm, e.g., in [19] and [3] to the multi-agent setting and nonconvex case. In the initialization of the DADS algorithm, the value
serves as an upper bound on
. In Step 1, one solution in
is needed, and it is unnecessary to compute the whole set
.
To assure primal convergence, we assume that dual estimates converge to the set where each has a single optimal solution.
1We will use the superscript to indicate that
of some global variables.
and
are estimates
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Definition 3.1 (Singleton Optimal Dual Solution Set): The set of
is the set of such that the set
is a singleton,
for each
.
where
The primal and dual estimates in the DADS algorithm converge to
. We
a pair of primal-dual solutions to the approximate problem
formally state this in the following theorem:
Theorem 3.1: (Convergence Properties of the DADS Algorithm):
and the corresponding approximate problem
Consider the problem
with some
. We let the non-degeneracy assumption 2.1, the
balanced communication assumption 2.2 and the periodic strong connectivity assumption 2.3 hold. In addition, suppose the Slater’s con. Consider the dual sequences of
dition 2.4 holds for the problem
,
,
and the primal sequence of
of the
satdistributed approximate dual subgradient algorithm with
,
,
isfying
.
1) (Dual estimate convergence) There exists a dual solution
where
and
such that the
following holds for all
2) (Primal estimate convergence) If the dual solution satisfies
, i.e.
is a singleton for all
, then there is
such that, for all
IV. DISCUSSION
Before outlining the technical proofs for Theorem 3.1, we would
like to make the following observations. First, our methodology is motivated by the need of solving a nonconvex problem in a distributed
way by a group of agents whose interactions change with time. This
places a number of restrictions on the solutions that one can find. Timevarying interactions of anonymous agents can be currently solved via
agreement algorithms; however these are inherently convex operations,
which does not work well in nonconvex settings. To overcome this, one
can resort to dualization. Admittedly, zero duality gap does not hold in
general for nonconvex problems. A possibility would be to resort to
nonlinear augmented Lagrangians, for which strong duality holds in
a broad class of programs [5], [6], [22]. However, we find here another problem, as a distributed solution using agreement requires separability, as the one ensured by the linear Lagrangians we use here. Thus,
we have looked for alternative assumptions that can be easier to check
and allow the dualization approach to work.
More precisely, Theorem 3.1 shows that dual estimates always converge to a dual optimal solution. The convergence of primal estimates
requires an additional assumption that the dual limit has a single optimal solution. We refer to this assumption as the singleton dual optimal
solution set (SD for short). The assumption may not be easy to check
a priori, however it is of similar nature as existing algorithms for nonconvex optimization. In [5] and [6], subgradient methods are defined
in terms of (nonlinear) augmented Lagrangians, and it is shown that
every accumulation point of the primal sequence is a primal solution
provided that the dual function is required to be differentiable at the
dual limit. An open question is how to resolve the above issues imposed by the multi-agent setting with less stringent conditions on the
nature of the nonconvex optimization problem.
In the following, we study a class of nonconvex quadratic programs
for which a sufficient condition guarantees that the SD assumption
holds. Nonconvex quadratic programs hold great importance from both
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 6, JUNE 2013
theoretic and practical aspects. In general, nonconvex quadratic programs are NP-hard, and please refer to [18] for detailed discussion. The
aforementioned sufficient condition only requires checking the positive
definiteness of a matrix.
Consider the following nonconvex quadratic program:
(10)
and
are real and symmetric matrices.
where
is given by
The approximate problem of
Lemma 5.2 (Lipschitz Continuity of ): There is a constant
such that for any
, it holds that
.
In the DADS algorithm, the error induced by the projection map
is given by
A basic iterate relation of dual estimates in the DADS algorithm is the
following.
Lemma 5.3 (Basic Iterate Relation): Under the assumptions in Thewith
for all
,
orem 3.1, for any
we have for all
(11)
as before. The local LaWe introduce the dual multipliers
can be written as follows:
grangian function
where the term independent of is dropped and is a linear function
. The dual function and dual problem can be defined
of
:
as before. Consider any dual optimal solution . If for all
is positive definite;
(P1)
(P2)
;
then the SD assumption holds. The properties (P1) and (P2) are easy
is obtained. We
to verify in a distributed way once a dual solution
would like to remark that (P1) is used in [8] to determine the unique
global optimal solution via canonical duality when is absent.
V. CONVERGENCE ANALYSIS
This section outlines the analysis steps to prove Theorem 3.1. Please
refer to [25] for more details. Recall that is continuous and is comsuch that
and
pact. Then there are
for all
. We start our analysis from the computation of supergradients of .
, then
Lemma 5.1 (Supergradient Computation): If
is a supergradient of
at . That is, for any
(12)
A direct result of Lemma 5.1 is that the vector
is a supergradient of
; i.e., the following supergradient inequality holds for any
:
at
(14)
Asymptotic convergence of dual estimates is shown next.
Lemma 5.4 (Dual Estimate Convergence): Under the assumptions
in Theorem 3.1, there exists
such that
,
, and
.
The remainder of the section is dedicated to characterizing the convergence properties of primal estimates.
Lemma 5.5 (Properties of Marginal Maps): The set-valued marginal
is closed. In addition, it is upper semicontinuous at
;
map
, there is
such that for any
,
i.e., for any
.
it holds that
Lemma 5.6 (Primal Estimate Convergence): Under the assumptions
, it holds that
in Theorem 3.1, for each
where
.
The main result of this technical note, Theorem 3.1, can be shown
next. In particular, we will show complementary slackness, primal feasibility of , and its primal optimality, respectively.
Proof for Theorem 3.1:
,
Claim 1:
and
.
Proof: Rearranging the terms related to in (14) leads to the folwith
lowing inequality holding for any
for all
:
(13)
It can be seen that (9) of dual estimates in the DADS algorithm is
a combination of a dual subgradient scheme and average consensus
is Lipschitz continuous
algorithms. The following establishes that
with some Lipschitz constant .
(15)
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Summing (15) over [0, ], and dividing by
(16)
We now proceed to show
for each
.
,
for
and
,
in
Let
is not summable but square summable, and
(16). Recall that
is uniformly bounded. Take
, and then it follows
from Lemma 5.1 in [27] that:
(17)
On the other hand, since
, we have
given the
. Let
where
fact that is an upper bound of
. Then we could choose a sufficiently small
and
in (16) such that
where is given in the definition of
and is given by:
,
for
. Following the same lines toward (17), it gives that
. Hence, it holds that
. The
rest of the proof is analogous and thus omitted.
The proofs of the following two claims can be found in [25].
Claim 2: is a primal feasible solution to the approximate problem
.
.
Claim 3: is a primal solution to the problem
VI. CONCLUSION
We have studied a distributed dual algorithm for a class of multiagent nonconvex optimization problems. The convergence of the algorithm has been proven under the assumptions that i) the Slater’s condition holds; ii) the optimal solution set of the dual limit is singleton; iii)
the network topologies are strongly connected over any given bounded
period. An open question is how to address the shortcomings imposed
by nonconvexity and multi-agent interactions settings.
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