
Submitted by
Assigned_Reviewer_4
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
The paper considers global and local path planning for
multiple agents in 2D with a centralized messagepassing algorithm
derived from the "threeweight" version of ADMM, an established algorithm.
The contributions are clearly stated in the introduction: The authors
decompose global planning optimization into several subproblems they dub
"minimizers," which describe various planning objectives that comprise the
larger overall problem to be solved. Minimizers are derived for avoiding
interagent collisions, avoiding collisions with static obstacles, and for
maximizing/minimizing kinetic energy or velocity. They also apply their
approach to local planning by reformulating joint optimization.
The overall concept for using a distributed optimization algorithm
for multiagent path planning is nice; however the results are a
little lacking. They do not provide comparisons to other wellknown high
dimensional path planning techniques, such as A* variants with
inflated heuristics or samplingbased methods. For many of these problems,
there is also the aspect of replanning given new state or obstacle
information. Would this algorithm be amenable to such dynamic or anytime
implementations? In that regard, it's hard to see how this formulation
would guarantee that certain constraints would always be enforced for
early stopping, for example a hard collision or communication constraints.
These and other issues regarding the planning problem should be discussed.
It's also not clear if this manuscript would appeal to the broader NIPS
audience. It would definitely fit into a dedicated planning/robotics
meeting, but it's hard to see if this would appeal from an algorithmic
viewpoint to other NIPS areas.
There are also a few minor items
that could be improved in the presentation. First of all, please use a
letter other than "n" for the correction messages in your formulations.
"n" is already used to describe the number of interior breakpoints, a
major parameter for the overall algorithm. Figure 2middle and Figure
3left include multiple scales on either the x or yaxis. In the case of
Figure 3left, I initially thought that the convergence time between
optimizing total energy and finding a feasible solution were comparable,
when one is several orders of magnitude larger. These figures seem to be
an attempt to fit the paper within the page limit but could cause
confusion and dilute the impact of the results. Also, with respect to the
convergence times reported for local trajectory planning using the
proposed approach versus MIQP [12], the authors compare implementations in
different programming languages and say the results are not "exactly
comparable." Yet, they directly compare the convergence times in Figure
3right. They also state, "our method does not require approximating the
search domain by hyperplanes, nor does it require an additional
branchandbound algorithm," which causes one to further question the
direct comparison of convergence times. These results could be clarified.
Q2: Please summarize your review in 12
sentences
Interesting application of distributed messagepassing
approach to optimization for multiagent path planning.
Submitted by
Assigned_Reviewer_5
Q1: Comments to author(s).
First provide a summary of the paper, and then address the following
criteria: Quality, clarity, originality and significance. (For detailed
reviewing guidelines, see
http://nips.cc/PaperInformation/ReviewerInstructions)
This paper proposes a messagepassing algorithm for
multiagent trajectory planning. It formulates the planning problem as a
numeric optimization in which the speed and direction of each agent at
each "break point" in time are the variables, and minimizing the energy
required and avoiding collisions are the objective. This optimization is
solved via the threeweights variant of the alternating direction method
of multipliers (ADMM), where ADMM is used to break up the objective with a
consensusoptimization pattern. This paper derives solutions to the
specific subproblems induced by ADMM. The algorithm is evaluated on
standard mutltiagent trajectory planning tasks, and compares favorably
with the original version of ADMM.
This paper thoroughly examines
the problem of multiagent trajectory planning and introduces a novel
solution. It derives minimizers for several useful objective terms: energy
minimization, velocity maximization, velocity minimization, agentagent
collision avoidance, and agentwall collision avoidance. The empirical
results are very convincing. The threeweights variant is significantly
better than the original version of ADMM. The proposed algorithm scales
well to high numbers of agents. It also often finds good solutions to the
problems. Is it correct to assume that the global optimum is a little less
than 300 in the middle of Figure 2? If so, then the solutions are usually
within 33%. Further, the solutions shown in the supplementary material are
indeed graceful.
Minor points:  At the end of Section 2,
the contrast between ADMM and the threeweights variant could be improved.
The intuitions given in point 3 of the algorithm description in the
supplementary material are much better, and the authors should consider
adding some of that material to the main paper.  Boyd et al.,
citation 21, should be cited as a book in a series, not a journal
article. Q2: Please summarize your review in 12
sentences
This paper introduces a threeweight alternating
direction method of multipliers (ADMM) approach to multiagent trajectory
planning. The proposed solution is compelling and performs well.
Q1:Author
rebuttal: Please respond to any concerns raised in the reviews. There are
no constraints on how you want to argue your case, except for the fact
that your text should be limited to a maximum of 6000 characters. Note
however that reviewers and area chairs are very busy and may not read long
vague rebuttals. It is in your own interest to be concise and to the
point.
We thank the reviewers for their valuable feedback.
 To all reviewers We will fix all the identified typos and
try to incorporate all suggestions regarding presentation and
organization.
Regarding the originality of our work, we stress
that the primary contribution of this paper is not an incremental
improvement upon previous trajectoryplanning algorithms, but instead the
introduction of a novel, unified framework that subsumes and expands upon
the desirable properties of many previous approaches. Namely, it (i)
jointly optimizes full continuous trajectories for all agents, (ii) is
easily parallelizable, (iii) can flexibly account for static/dynamic
obstacles and restrictions on energy and velocity, (iv) and has a
messagepassing interpretation that lends to decentralized
implementations.
In addition, given the rising interest in the
NIPS community in distributed optimization, and in particular ADMM, we
believe our work will have an impact beyond the areas of Robotics and
MultiAgent systems. Although there exist applications of ADMM to
nonconvex problems, these only consider objective functions that can be
decomposed into a few convex/concave subproblems (e.g. Zhang et. al 09 or
[21] Chapter 9; typically applied to matrix factorization/completion and
nonlinear datafitting with regularization). As far as we know, our work
is the first to show a successful application of ADMM to a realworld
application where the objective is decomposed into a very large number of
nonconvex subproblems. A notable exception is recent work in [13],
although the applications were primarily of academic interest. This
success could have a major impact on the part of NIPS community interested
in largescale optimization.
 To Reviewer 7 We thank the
reviewer for suggesting a simple baseline algorithm. We agree it can
clarify if our performance comes from our specific optimization
formulation, from our ADMMbased algorithm, or both. We will certainly
test it.
Regarding how the number of agents affects the runtime:
based on [13], we believe that in nonadversarial configurations, in
contrast to CONF1 where deadlocks are likely, the complexity is not
exponential. Our belief is based on a connection between trajectory
planning and disk packing. For example, minimumenergy trajectoryplanning
using piecewise linear trajectories is related to, although not the same
as, packing disks in multiple 2D layers, where the two matched disks
between consecutive layers generate a larger cost when far away from each
other. The numerical results in [13] report that, for disk packing, the
runtime of ADMM and TW is no more than polynomial in the number of disks
and we believe the typical runtime for trajectory planning has a similar
complexity. We interpret the seemingly exponential curve of convergence
time vs p for n = 8 in Fig.2 as an atypical, adversarial scenario. By
comparison, in Fig.3, which assumes randomly chosen initial and final
points and also minimization of energy, the dashedblue curve of runtime
vs p for n = 8 does not appear to exhibit exponential growth.

To Reviewer 5 We cannot be completely certain that the optimal
solution has a value of around 300. It is conceivable that the optimal
solution is at the bottom of a deep and narrow well of the objective
function that is hard to reach even by randomly initializing the
algorithm. This being said, looking at the behavior of the trajectories
found, we are inclined to agree that, for the experiment associated to
Fig.2middle, the optimum should be around that value and hence that the
dispersion is about 33%.
 To Reviewer 4 We agree we did not
discuss some aspects of trajectory planning, including a more detailed
comparison to other methods. Unfortunately, space limitations required us
to make difficult choices of what to include. This discussion is left as
future work.
Our algorithm does not possess anytime guarantees
and, if stopped earlier, the trajectories might have collisions. However,
if stopped early, a suboptimal set of noncolliding trajectories can be
found at very low computational cost by using our algorithm to solve the
feasibility problem in Eq. (2) starting from the state of the algorithm at
stop time.
Dynamic/static obstacles can be seamlessly integrated
into our framework, but a solution must be recomputed (as is the case with
A* or RRT*) if their trajectories/positions change unexpectedly. This
being said, in our algorithm, if a new piece of information is received,
the previous solution can be used as the initial guess, potentially
decreasing convergence time. Note that in some scenarios, a lowcost
localplanning approach, such as the one presented in Section 5, can be
beneficial.
A*search based methods and samplingbased methods
require exploring a continuous domain using discrete graph structures. For
large problems, fixedgrid search methods are impractical. Alternatively,
exploration can be done using random/sampling algorithms with proved
asymptotic convergence to optimality, Karaman et al., 2010. However, as
the dimensionality of the configuration space increases, the convergence
rate degrades and the local planners required by the exploration loop
become harder to implement. In addition, as pointed out in [9], sampling
algorithms cannot easily produce solutions where multiple agents move in
tight spaces, like in CONF1 with obstacles. Some of the disadvantages of
using discrete random search structures are even visible in extremely
simple scenarios. For example, for a single holonomic agent that
needs to move as quickly as possible between two points in
freespace, Karaman et al., 2010, require around 10000 samples on
their RRT* method to find something close to the shortestpath solution.
For our algorithm this is a trivial scenario: it outputs the optimal
straightline solution in 200 iterations and 37 msecs. in our Java
implementation.
These points will be clarified in the text.
 