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Enhancements on the hyperplane arrangements in mixed integer
techniques
Florin Stoican, Ionela Prodan, Sorin Olaru,
AbstractThe current paper addresses the prob-lem of optimizing a
cost function over a non-convexand possibly non-connected feasible
region. A classicalapproach for solving this type of optimization
problemis based on Mixed integer technique. The
exponentialcomplexity as a function of the number of binary
vari-ables used in the problem formulation highlights theimportance
of reducing them. Previous work whichminimize the number of binary
variables is revisitedand enhanced. Practical limitations of the
procedureare discussed and a typical control application,
thecontrol of Multi-Agent Systems is exemplified.
I. Introduction
Collision avoidance plays an important role in thecontext of
managing multiple agents. In the same time isknown to be a
difficult problem, since certain constraintsare non-convex. For
example, the evolution of a dynami-cal system in an environment
presenting obstacles can bemodeled in terms of a non-convex
feasible region. Moreprecisely, it is possible to set up an
optimization problemto optimize the agent state trajectory in order
to avoida convex region, representing an obstacle (static
con-straints) or another agent (dynamic constraints - leadingin
fact to a parametrization of the set of constraints withrespect to
the current state).
A popular framework for the treatment of a suchoptimization
problem is represented by Mixed-Integer-Programming (MIP),
described in [1]. This method hasproved to be very useful due to
the ability to includenon-convex constraints and discrete decisions
in the op-timization problem. Research on these types of
problems,using MIP techniques has focused on optimization ofagent
trajectories [2], multi-vehicle target assignmentand intercept
problems [3], or on coordinating the ef-ficient interaction of
multiple agents in scenarios withmany sequential tasks and tight
timing constraints [4].In [5], the authors used a combination of
MIP and ModelPredictive Control (MPC) to stabilize general
hybridsystems around equilibrium points.
However, despite its modeling capabilities and theavailability
of good solvers, MIP has serious numericaldrawbacks. As stated in
[6], mixed-integer techniques arein the NP-hard computation class.
Consequently, thesemethods may not be fast enough for real-time
control ofsystems with large problem formulations.
SUPELEC Systems Sciences (E3S) - Auto-matic Control Department,
Gif sur Yvette,
France{florin.stoican,ionela.prodan,sorin.olaru}@supelec.fr DISCO
Team INRIA, France
There has been a number of attempts in the literatureto reduce
the computational requirements of MIP for-mulations in order to
make them attractive for real-timeapplications. In [7] an iterative
method for including theobstacles in the best path generation is
provided. Otherworks, like [8], consider a predefined path
constrainedby a sequence of convex sets. In all of these papersthe
original decision problems are reformulated in asimplified MIP
form.
The negative influence of the increased number ofbinary
variables in the problem formulation highlightsthe importance of
reducing them. In [9], we introducea novel linear constraints
expression for reducing thenumber of binary variables necessary in
describing theexterior of convex sets.
In the present paper, we revise these preliminaryresults and
introduce enhancements in the descriptionof non-connected convex
sets (or their complement). Welist some of the noteworthy aspects
of our approachrepresenting also the main contributions of this
paper: a convex representation in the extended space of
state plus binary variables using a hyperplane ar-rangement;
reduced complexity of the problem upon mergingtechniques;
a notable property of optimal association betweenregions and
their binary representation leading tothe minimization of the
number of constraints.
The method presented here can be used in severalfields of
application. We choose to exemplify here withthe control of an
agent operating in a dynamic environ-ment with obstacles. The agent
is required to maneuversuccessfully in a hostile environment. The
obstacles aredesigned as convex polyhedral regions. In this context
thereduction technique is embedded within an MPC pathplanning for
multiple agents.
The rest of the paper is organized as follows. InSection II the
preliminaries are presented, the main ideabeing detailed in Section
III. Discussions based on thecontrol of multiple agents operating
in a hostile environ-ment are presented in Section IV, while the
conclusionsare drawn in Section V.Notation: The following notation
will be used through-
out the paper. The closure of a set S, cl(S) is the
intersec-tion of all closed sets containing S. The collection of
allpossible N combinations of binary variables will be noted{0, 1}N
= {(b1, . . . , bN ) : bi {0, 1} , i = 1, . . . , N}.The ceiling
value of x R denoted as dxe is the smallestinteger greater than x.
We denote |I| as the cardinal of
2011 50th IEEE Conference on Decision and Control andEuropean
Control Conference (CDC-ECC)Orlando, FL, USA, December 12-15,
2011
978-1-61284-799-3/11/$26.00 2011 IEEE 3986
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set I. lp(n, d) (qp(n, d)) denotes the complexity of solvinga
linear (quadratic) program with n constraints and dvariables.
II. PreliminariesThe main objective of this paper is to provide
a
technique for solving an optimization problem over anon-convex
and possibly non-connected region of thestate space. We recall here
methods from [9] which usehyperplane arrangements and mixed integer
techniquesin order to provide an equivalent (in an extended
spaceaugmented with auxiliary binary variables) optimizationproblem
over a convex region.A. Polyhedral notions
In the following we define a bounded polyhedral set,P Rn through
its implicit half-space description:
P = {x Rn : hix ki, i = 1 . . . N} (1)with (hi, ki) R1n R and
its complement, as:
CX(P ) , cl(X \ P ) (2)with the reduced notation C(P ) whenever
X is presumedknown or is considered to be the entire space Rn.By
definition, every affine subspace which defines P
Hi = {x : hix = ki} (3)will partition the space into two
disjoint regions:
R+(Hi) = {x : hix ki} (4)R(Hi) = {x : hix ki} (5)
with i = 1 . . . N . R+i and Ri denote simplified notationfor
region (4) and (5), respectively, associated to the ithinequality
of (1).B. Non-convex and non-connected region
Without restricting the problem let us consider ournon-convex
and non-compact region as the complementof an union of convex
(bounded polyhedral ) sets P =l
Pl:CX(P) = cl(X \ P) (6)
with Pl =Klkl=1
R+ (Hkl) and N ,l
Kl.This type of regions arises naturally in the context of
obstacle/collision avoidance when there is more than asingle
object to be taken into account.
In order to deal with the complement of a non-convexregion in
the context of mixed-integer techniques severaladditional
theoretical tools need to be introduced.Definition 1 (Hyperplane
arrangements [10]). A col-lection of hyperplanes H = {Hi}i=1:N will
partition thespace in an union of disjoint cells defined as
follows:
A(H) =
l=1,...,(N)
Ni=1Rl(i)(Hi)
Al
(7)
where l {,+}N denotes feasible combinations ofregions (4)(5)
obtained for the hyperplanes in H.
Several computational aspects are of interest. Thenumber of
feasible cells, (N), (in relation with the spacedimension d and the
number of hyperplanes N) isbounded by Bucks formula ([11]):
(N) di=0
(N
i
)(8)
with equality satisfied if the hyperplanes are in
generalposition and X = Rn.
We note that there exists a subset {Bl}l=1,...,b(N)of feasible
cells from (7) (with b(N) (N)) whichdescribes region (6):
CX(P) =l
Bl, (9)
such that, for any l there exists i such that Bl = Ai and
Bl {Ai A(H) : Ai
P = } . (10)Mixed integer programming (MIP) allows to express
theunion (9) as a polyhedra in an extended spaceX{0, 1}Nof state +
auxiliary binary variables as follows:
...l(1)h1x l(1)k1 +Ml
...l(N)hNx l(N)kN +Ml
Bl...
(11)
with M a positive scalar chosen appropriately (that
is,significantly bigger than the rest of the variables in theright
hand side of the inequalities) and (1, . . . , N ) {0, 1}N the
auxiliary binary variables.It is straightforward to note that any
of the regions
Bl can be obtained from (11) with an adequate choice ofbinary
variables:
l = (1, . . . , 1, 0l
, 1, . . . , 1). (12)
The number of binary variables negatively influencesthe
computation time, in the worst case, an exponentialbound is
reached. In [9] it was noted that a set
(1, . . . , N0) {0, 1}N0 (13)with N0 = dlog2(N)e permits a
reduced representationwhere any variable l is written as a linear
combinationin the space of variables (1, . . . , N0) {0, 1}N0 :
i = i (1 . . . N0) . (14)
As a prerequisite for explicitly defining relation (14)
weassociate a tuple
i ,(i1 . . .
iN0
)(15)
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to each region Bi. We mention that this association isnot
unique, and various possibilities can be considered;in the
following, unless otherwise specified, the tuples willbe appointed
in lexicographical order. We can now recallthe next result:
Proposition 1. A mapping i() : {0, 1}N0 R whichverifies that
i(i) = 0 and i(j) 1 for any j 6= i isgiven by:
i() =N0k=1
tik, where tik ={k, if ik = 01 k, if ik = 1
.
(16)where k denotes the kth variable and ik its value for
thetuple associated to region Ai.
Proof: See the proof of Proposition 1 in [9].Note that
Proposition 1 gives a linear mapping i()
which will have value 0 only for the tuple associated toregion
Bi and values 1 for any other tuple. This provesthat (14) can be
used to extract region Bi as in (12).Note that if a tuple is
unallocated (i.e., has no asso-
ciated region), the substitution of its binary values inthe
extended polyhedra (11) will result in a degenerateprojection onto
X which for all intents and purposes en-compasses the entire space.
To counteract this undesiredbehavior we need to add constraints
which will force theunallocated tuples to be infeasible. This
reasoning leadsthe following corollary:
Corollary 1. Let there be a tuple i {0, 1}N0 . Thepoint it
describes is made infeasible with respect to theconstraint:
N0k=1
tik (17)
with tik defined as in Proposition 1 and (0, 1) a scalar(for
simplicity, in the rest of the paper we will use 0.5).
Proof: The left side of the inequality (17) will vanishonly at
tuple i and for the rest of the tuples in thediscrete set {0, 1}N0
will give values greater of equal to1. Thus, the only point made
infeasible by inequality (17)is i.
Using Corollary 1 it follows that by adding inequalitiesof form
(17) for each unallocated tuple in (11) we obtaina complete
representation of (6).
C. ExemplificationConsider the following example depicted in
Fig. 1
where the complement of the union of two triangles(P = P1 P2)
represents the feasible region. We takeH = {Hi}i=1:4 a collection
of N = 4 hyperplanes (givenas in (3)) which define P1, P2 as
follows:
P1 = R+1 R+2 R+3P2 = R1 R2 R+4 .
We observe that the bound given in (8) is reached,that is, we
have 11 cells (obtained as in the arrangement
H1 H2
H3
H4
B1 (0, 0, 0, 0)
B2 (0, 0, 0, 1)B3 (0, 0, 1, 0)B4 (0, 0, 1, 1)
B5 (0, 1, 0, 0)
B6 (0, 1, 0, 1)
B7 (0, 1, 1, 0)
B8 (0, 1, 1, 1)
B9 (1, 0, 0, 0)
P1
P2
Fig. 1: Exemplification of hyperplane arrangement
(7)). From them, a total of 9, which we denote here asB1, . . .
, B9, describe the non-convex region (6). To eachof them we
associate a unique tuple from {0, 1}N0 as seenin Fig. 1 with N0 =
dlog29e = 4.As per Proposition 1 and (11), we are now able to
write
the following set of inequalities:
h3x k3h4x k4 +M ( 1 + 2 + 3 + 4)
}B1
h2x k2h3x k3h4x k4
+M (1 + 1 + 2 + 3 4)
B2h1x k1h2x k2
h3x k3+M (1 + 1 + 2 3 + 4)
B3h1x k1h3x k3 +M (2 + 1 + 2 3 4)
}B4
h1x k1h2x k2h3x k3h4x k4
+M (1 + 1 2 + 3 + 4)
B5h2x k2
h4x k4 +M (2 + 1 2 + 3 4)}
B6
h1x k1h2x k2
h4x k4+M (2 + 1 2 3 + 4)
B7h1x k1h3x k3
h4x k4+M (3 + 1 2 3 4)
B8h1x k1h2x k2h3x k3h4x k4
+M (1 + 1 2 + 3 + 4)
B9.
(18)
Note that in the above set we simplified the descriptionby
cutting the redundant hyperplanes in a cell repre-sentation(e.g.,
for cell A1, 2 hyperplanes suffice for acomplete description).
Since only 9 tuples, from a total number of 16 areassociated to
cells, we need to add constraints to theproblem such that remaining
7 unallocated tuples will
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never be feasible: (2 1 2 + 3 + 4) 0.5 (3 1 2 3 + 4) 0.5 (3 1 2
+ 3 4) 0.5 (4 1 2 3 4) 0.5 (2 1 + 2 3 + 4) 0.5 (3 1 + 2 3 4) 0.5 (2
1 + 2 + 3 4) 0.5. (19)
III. Main ideaAs seen in [9], palliatives for reducing the
computa-
tional load exist but ultimately, the computation time isin the
worst case scenario is exponentially dependent onthe number of
binary variables which in turn dependson the number of cells of the
hyperplane arrangements(see (8)). We conclude then, that the
problem becomesprohibitive for a relatively small number of
polyhedrain P and that any reduction in the number of cells
isworthwhile and should be pursued.
This can be accomplished in two complementary ways.Firstly, we
note that bound (8) is reached for a givennumber of hyperplanes
only if they are in general posi-tion. As such, particular classes
of polyhedra may some-what reduce the actual number of cells in
arrangement(7) and consequently, the number of auxiliary
binaryvariables.
The other direction, which we chose to pursue in therest of the
paper is the merging of adjacent cells intopossibly overlapping
regions which describe the comple-ment of our initial union of
polyhedra. Using, the cellsof (7) and some well known notions of
set theory we willdescribe a reduced representation of (6), both in
numberof cells and of interdicting constraints (similar to the
onesdiscussed in Corollary 1).A. Cell merging
Recall that any of the cells of (9) is described by anunique
sign tuple (Bl l). As such, we obtain thatthe cells are disjunct
and cover the entire feasible space.For our purposes we are
satisfied with any collection ofregions not necessarily disjoint
which covers the feasiblespace. In this context we may ask if it is
not possibleto merge the existing cells of (9) into a reduced
numberof regions which will still cover region (6). Note that
byreducing the number of regions, the number of necessaryauxiliary
variables my also decrease substantially.
We can formally represent the problem by requiringthe existence
of a collection of regions,
CX(P) =
k=1,...,c(N)
Ck (20)
which verifies the next conditions: the new polyhedra are formed
as unions of the old
ones (i.e., for any k there exists a set Ik which selectsindices
from 1, . . . , c(N) such that Ck =
iIk
Bi)
the union is minimal, that is, the number c(N) ofregions is
minimal
Existing merging algorithms are usually computation-ally
expensive but here we can simplify the problem bynoting two
properties of the cells in (9): the sign tuples l describe an
adjacency graph since
any two cells whose sign tuples differ at only oneposition are
neighbors
the union of any two adjacent cells is a polyhedraRemark 1. Note
that a region Ck is described by at mostNd hyperplanes where d
denotes the number of indicesin the sign tuples which flip the
sign. It makes sense thento, not only reduce the number of regions,
but also tomaximize the number of disjoint cells that go into
thedescription of a region from (20).
In order to construct (20) we may use merging al-gorithms (see
for example [12] which adapts a branchand bound algorithm to merge
cells of a hyperplanearrangement) or we can pose the problem in the
booleanalgebra framework [13].
B. Constraint reductionNote that,
Nint , 2dlog2c(N)e log2c(N), (21)
the number of unallocated tuples, may have significantvalues. If
we associate to each tuple an inequality in-tended to discard the
combination from the set of feasiblepoints as in Corollary 1, we
negatively influence the speedof the associated optimization
algorithm. This can bealleviated by noting (as previously
mentioned) that theassociation between feasible cells in (7) and
tuples isarbitrary. One could then chose favorable
associationswhich will permit more than one tuple to be
removedthrough a single inequality. To this end, we present
thefollowing proposition.
Proposition 2. Let there be a collection of tuples{i}i1,...,2d
{0, 1}
N0 which completely spans a d-facetof hypercube {0, 1}N0 . Let I
be the set of the N0 dindices which retain a constant value over
all the tuples{i}i1,...,2d composing the facet. Then there exists
the
constraintkI
tk , (22)
which renders the tuples of the given facet (and only theseones)
infeasible.Variables tk and are taken as in Corollary 1 with tk
associated to k, the common value of variable k overthe set of
tuples
{i}i1,...,2d .
Proof: Geometrically, the tuples are extreme pointson the
hypercube {0, 1}N0 and the inequalities we aredealing with are
half-spaces which separate the pointsof the hypercube. If a set of
tuples completely spans ad-facet it is always possible to isolate a
half-space that
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separates the points of the d-facet from the rest of
thehypercube.By a suitable association between feasible cells
and
tuples we may label as unallocated the extreme pointswhich
compose entire facets on the hypercube {0, 1}N0which permits to
apply Proposition 2 in order to obtainconstraints (22).Remark 2. By
writing Nint as a sum of consecutive pow-
ers of 2 (Nint =dlog2Ninte1
i=0bi2i), an upper bound Nhyp
for the number of inequalities (22) can be computed:
Nhyp =dlog2Ninte1
i=0bi dlog2c(N)e 1 (23)
where bi {0, 1}. C. Exemplification
By applying the merging algorithm of Subsection III-A we obtain
that the feasible region (6) is expressed byan union as in (11) and
we depict the result in Fig. 2.
H1 H2
H3
H4
B1
B2B3B4
B5
B6
B7
B8
B9
C1 (0, 0)
C2 (0, 1)
C3 (1, 0)
C4 (1, 1)
P1
P2
Fig. 2: Exemplification of hyperplane arrangement withmerged
regions
As it can be seen, we obtain 4 overlapping regions:C1 = B1 B2 B3
B4, C2 = B4 B5 B6, C3 = B6 B7B8B1 and C1 = B8B9B1B2.
Consequently,we note that N0 = 2 auxiliary binary variables
sufficein coding the regions. As per Proposition 1 and (11), weare
now able to write the following set of inequalities (weattach to
each of the regions a tuple in lexicographicalorder):
h3x k3 +M ( 1 + 2)}
C1
h1x k1h4x k4
+M (1 + 1 2)}
C2
h4x k4 +M (1 1 + 2)}
C3
h1x k1h2x k2
+M (2 1 2)}
C4
(24)
In the reduced representation (24) there are no unal-located
tuples, since the number of tuples coincides with
the number of regions. For the sake of the presentationwe take
the construction from (18) where there remain7 unallocated tuples
and consider Proposition 2. In theoriginal formulation (19) we
required 7 constraints todiscard these tuples. We apply now the
results of Sub-section III-B and observe the following
improvements.For the 7 unallocated tuples, we observe that 4 of
them,(1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0) and (1, 1, 1, 1) form
a2-facet of the hypercube {0, 1}4. Tuples (1, 0, 1, 0) and(1, 0, 1,
1) form an edge and (1, 0, 0, 1) is on a vertex. Wecan now apply
Proposition 2 and obtain the followingconstraints
(2 1 2 ) 0.5 (2 1 + 2 3 ) 0.5 (2 1 + 2 + 3 4) 0.5. (25)
Note that we were able to diminish the number ofinequalities
from 7 in (19) to only 3 in (25): the first4 constraints of (19)
are replaced by the 1st constraintof (25). The same holds for the
next 2 that correspondto the 2nd and for the last that is identical
with the 3rd.
IV. Collision avoidance example
A number of commonly found situations in the controlrelated to
Multi-Agent Systems imply a cost function hasto be minimized, while
in the same time, the agent avoidscollision with obstacles and
other agents.
In this illustrative example we will describe an agentthat has
to navigate its way around a group of fixed ob-stacles. We consider
the dynamics of the agent describedby a LTI system as follows:
k+1 = Ak +Buk. (26)
The agent model is used in a predictive control contextwhich
permits the use of non-convex state constraints forobstacle
avoidance behavior.
An optimal control action u is obtained from thecontrol sequence
u ,
{uk|k, uk+1|k, , uk+N1|k
}as a
result of the optimization problem:
u = argminu(Tk+Nh|kPk+Nh|k +
Nh1l=1
Tk+l|kQk+l|k+
+Nh1l=0
uTk+l|kRuk+l|k)
subject to:{k+l|k = Ak+l1|k +Buk+l1|kk+l|k C(P), l = 1, . . . ,
Nh
(27)
Here Q 0, R > 0 are the weighting matrices, P 0defines the
terminal cost, Nh the prediction horizon andP is an union of
polytopes describing the obstacles.As a practical application we
consider a linear system
(vehicle, pedestrian or agent in general form) whose
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dynamics are described by:
A =
0 0 1 00 0 0 10 0 m 00 0 0 m
, B =
0 00 01m 00 1m
(28)where = [x y vx vy]T , u = [ux uy]T are the state and
theinput of the system. With the components of the statebeing (x,
y), the position, and (vx, vy) the velocities ofthe agent, m is the
mass of the agent and its dampingfactor.
We consider the position component of the agent stateto be
constrained by 4 obstacles as shown in Fig. 3 (inblue). Considering
the 14 hyperplanes which describethe polyhedra associated with the
obstacles we have(14) = 106 regions obtained as in (7).
Additionally weobserve that 10 of the cells will describe the
interdictedregions and the rest, b(14) = 96 will describe the
feasibleregion, as shown in (9). Further, we apply the notionsfrom
Subsection III-A (in this particular situation, theproblem is small
enough to be solved using a Karnaughmap) to obtain a reduced
representation for the feasibleregion as in (20). We observe that
the number of cells issubstantially reduced, from b(14) = 96 to
c(14) = 11which warrants in turn a reduction of the auxiliary
binaryvariables from 7 to 4 that, for a worst case scenario,equals
to an eightfold speed up. In Fig. 3 (a) we depictthe cells of (9)
and the obstacles while in Fig. 3 (b) weshow the covering (20) of
merged cells.
14 12 10 8 6 4 2 0 2 4 6 8 10 12 1415
10
5
0
5
10
15
x1
x2
(a) Partitioning with disjoint cellsobtained as in (9)
15 10 5 0 5 10 1515
10
5
0
5
10
15
x1
x2
(b) Partitioning with merged cells(20)
Fig. 3: Cell partitionings of the feasible region.
We apply the predictive control strategy for horizonN = 3 and
cost matrices Q = 105 I4, R = I2 andP = 105 I4 and obtain the
trajectory depicted in Fig. 4.
V. ConclusionsIn this paper we revisit a technique which
transforms
a non-convex and possibly non-connected region into apolyhedra
in an augmented space (state and auxiliarybinary variables) through
the use of hyperplane arrange-ments. With respect to previous
results we minimizedthe number of cells describing the feasible
region throughmerging methods and discussed an improvement of
theoptimization problem such that the number of
additionalconstraints is minimized. These numerical
improvements
10 8 6 4 2 0 2 4 6 8 1010
8
6
4
2
0
2
4
6
8
10
x1
x2
Fig. 4: Simulations of agent trajectories.
were presented and tested in an obstacle avoidance con-trol
problem.Acknowledgements: The research of Ionela Prodan is
financially supported by the EADS Corporate Founda-tion
(091-AO09-1006).
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