Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques

J Optim Theory Appl (2012) 154:549572DOI 10.1007/s10957-012-0022-9

Enhancements on the Hyperplanes Arrangementsin Mixed-Integer Programming Techniques

Ionela Prodan Florin Stoican Sorin Olaru Silviu-Iulian Niculescu

Received: 4 May 2011 / Accepted: 6 March 2012 / Published online: 24 March 2012 Springer Science+Business Media, LLC 2012

Abstract This paper is concerned with improvements in constraints handling formixed-integer optimization problems. The novel element is the reduction of the num-ber of binary variables used for expressing the complement of a convex (polytopic)region. As a generalization, the problem of representing the complement of a pos-sibly not connected union of such convex sets is detailed. In order to illustrate thebenefits of the proposed improvements, a typical control application, the control ofmultiagent systems using receding horizon optimization techniques, is considered.

Keywords Mixed integer programming (MIP) Not convex constraints Hyperplane arrangements Cell merging

Communicated by Panos M. Pardalos.

I. Prodan () S. OlaruAutomatic Control Department, SUPELEC Systems Sciences (E3S), Gif sur Yvette, Francee-mail: [email protected]

S. Olarue-mail: [email protected]

F. StoicanDepartment of Engineering Cybernetics, Norwegian University of Science and Technology,Trondheim, Norwaye-mail: [email protected]

I. Prodan S.-I. NiculescuLaboratory of Signal and Systems, CNRS-SUPELEC, Gif sur Yvette, France

I. Prodane-mail: [email protected]

S.-I. Niculescue-mail: [email protected]

550 J Optim Theory Appl (2012) 154:549572

1 Introduction

Collision avoidance plays an important role in the context of managing multipleagents (see [1, 2], where different frameworks for the cooperative control of mul-tiple agents are described). In the same time, it is known to be a difficult problem,since certain constraints are not convex, [3]. For example, the evolution of a dynam-ical system in an environment presenting obstacles can be modeled in terms of a notconvex feasible region. More precisely, it is possible to set up an optimization prob-lem such that the agent state trajectory avoids a convex region, in fact, representingan obstacle (static constraints) or another agent (dynamic constraintsleading to aparameterization of the set of constraints with respect to the current state).

A popular framework for the treatment of such an optimization problem is repre-sented by Mixed-Integer Programming (MIP), described in [4]. This method can bevery useful in several fields of applications, due to its ability to include not convexconstraints and discrete decisions in the optimization problem (see, for instance [5]).There is a growing literature about optimization problems, which can be formulatedthrough the use of MIP techniques. For example, [68] focused their work on opti-mization of agent trajectories. Multivehicle target assignment and intercept problemsare studied by [3, 9]. MIP was also useful to coordinating the efficient interaction ofmultiple agents in scenarios with many sequential tasks and tight timing constraints(see [10, 11]). In [12, 13], the authors used a combination of MIP and Model Pre-dictive Control (MPC) (see, for instance, [14, 15], for basic notions in MPC) to sta-bilize general hybrid systems around equilibrium points. [16] introduced MIP in apredictive control framework to plan short trajectories around nearby obstacles. Themixed-integer formulation has also proven to be useful for the convergence of mul-tiple agents toward a tight formation; see [17]. Finally, an application for differentcontrol design problems was reported in [18], where a feasible reference signal whichpermits set membership testing for fault detection was computed over a not convexregion, leading to a MIP formulation.

However, despite its modeling capabilities and the availability of versatile solvers,MIP has serious numerical drawbacks. As stated in [19], mixed-integer techniquesare classified as NP-hard, i.e., the computational complexity increases exponentiallywith the number of binary variables used in the problem formulation. Consequently,these methods may not be fast enough for real-time control of systems with largeproblem formulations. There has been a number of attempts in the literature to re-duce the computational requirements of MIP formulations in order to make themattractive for real-time applications. In [20], an iterative method for including the ob-stacles in the best path generation is provided. Other references, like [21], considera predefined path constrained by a sequence of convex sets. In all of these papers,the binary variables reduction is not tackled at the MIP level, but instead the originaldecision problems are reformulated in a simplified MIP form. The negative influenceof the large number of binary variables in the problem formulation highlights theimportance of reducing them. In [22, 23], we introduced a novel linear constraintexpression for reducing the number of binary variables necessary in describing theexterior of convex sets.

J Optim Theory Appl (2012) 154:549572 551

In the present paper, we resume these results in a unitary description of not con-nected convex sets (or their complement) using auxiliary binary variables. The op-timization problems where the binary variables are used to express a feasible notconvex region over which a (usually quadratic) cost function has to be minimized,are treated in a first stage. The problem is reformulated using a reduced number ofbinary variables through a compact encoding of the inequalities describing the feasi-ble region. Thus, the problem complexity will require only a polynomial number ofsubproblems (Linear Programming (LP) or Quadratic Programming (QP) problems)that have to be solved with obvious benefits for the computational effort. In the sec-ond part of the paper, the technique is extended for the treatment of not connectednot convex regions. In a similar manner, a reduced number of binary variables suf-fices in describing a not convex and not connected region. Some of the noteworthyaspects of the approach which resumes also the main contributions of this paper arethe following:

a convex representation in the extended space of state plus binary variables usingthe associated hyperplane arrangement;

reduced complexity of the problem upon Boolean algebra techniques; a notable property of optimal association between regions and their binary repre-

sentation leading to the minimization of the number of constraints.

The methods presented here can be used in several fields of application. We chooseto exemplify here with the control of an agent operating in a dynamic environmentwith obstacles. More precisely, the agent, which may have an associated polyhedralsafety region, is required to maneuver successfully in a hostile environment. Theobstacles are designed as convex polyhedral regions. In this context, the proposedreduction technique is embedded within an MPC path planning for multiple agents.

The rest of the paper is organized as follows. In Sect. 2, the preliminaries are pre-sented; the main idea being detailed in Sect. 3. Furthermore, in Sect. 4, the methodis extended to not connected, not convex regions. The improvements in the computa-tional time for the approach are detailed in Sect. 6. Discussions based on an exampleare presented in Sect. 7, while the conclusions are drawn in Sect. 8.

2 Preliminaries

The following notation will be used throughout the paper. The closure of a set S,cl(S), is the intersection of all closed sets containing S. The collection of all possiblecombinations of N binary variables will be noted {0,1}N := {(b1, . . . , bN) : bi {0,1},i = 1, . . . ,N}. For a scalar x R, we denote by x the upper integer part,and by Bnp := {x Rn : xp 1} the unit ball of norm p, where xp is the p-normof vector x Rn. Notations lp(n, d) and qp(n, d) represent the complexity of solvinga linear program, quadratic program, respectively, with n constraints and d variables.

For safety and obstacle avoidance problems (to take just a few examples), thefeasible region in the space of solutions is a not convex set. Usually, this region isconsidered as the complement of a convex region, which describes an obstacle and/ora safety region. Due to their versatility and relative low computational complexity,the polyhedra are the instrument of choice in characterizing these regions.


In the following, we define a polytope, P Rn, through its implicit half-spacedescription:

P := {x Rn : hix ki, i = 1, . . . ,N}, (1)

with (hi, ki) R1n R, and its complement asCX(P ) := cl(X \ P), (2)

with the reduced notation C(P ) whenever X is presumed known or is considered tobe the entire space Rn.

By definition, every affine subspace, which is a support hyperplane for P ,

Hi := {x : hix = ki}, (3)will partition the space into two regions:1

R+(Hi ) := {x : hix ki}, R(Hi ) := {x : hix ki}, (4)with i = 1, . . . ,N .

The not convex region C(P ), denoted by (2), may then be described as a union ofregions that cover all space except P :

C(P ) :=

i

R(Hi ), i = 1, . . . ,N. (5)

Therefore, we note that the complement of a bounded polyhedra (1) is covered by theunion of N overlapping regions denoted as Ri (a simplified notation for region (4)associated to the ith inequality of (1)).

In order to obtain a tractable problem formulation, one has to use mixed integertechniques with the aim of defining a polyhedra in the extended space of state +auxiliary binary variables of the form:

hix ki + Mi, i = 1, . . . ,N, (6)i=N

i=1i N 1, (7)

with M a constant chosen appropriately (that is, significantly larger2 than the restof the variables and playing the role of a relaxation constant), and (1, . . . , N) {0,1}N the auxiliary binary variables (which can activate or not the relaxation).

Remark 2.1 The set of solutions for (6)(7) can be projected in the original space Rn,leading to a coverage of the not convex region, which corresponds to the implicit

1The relative interiors of these regions do not intersect, but their closures have as a common boundary theaffine subspace Hi .2There exists a finite M sufficiently large if and only if the polyhedra of type (1) are bounded. In theremaining portion of the paper, all the polyhedra of type (1) are assumed to be bounded for this reason.


definition in (2). A region Ri can be obtained from (6) with an adequate choice ofbinary variables

i :=(

1, . . . ,1, 0i

,1, . . . ,1). (8)

Note also that the converse is false since no choice of binary variables can lead to thedescription of a region R+i as in (4). Indeed, if a binary variable is 1, the corre-sponding inequality degenerates such that it covers any point x Rn (this representsthe limit case for M ). The condition (7) is thus required to ensure that at leastone binary value be 0, and consequently, at least one inequality be verified.

As it can be seen in the representation (6)(7), a binary variable is associated toeach inequality in the description of the polytope (1). Obviously, for a big number ofinequalities, the number of binary variables becomes exceedingly large. Since theirnumber exponentially affects the resolution of any mixed integer algorithm (usuallybased on branch and cut techniques, and thus very sensitive to the number of binaryterms), the goal to reduce their number is worthwhile. A first step would be to elimi-nate from the half-space representation of the polytope all the redundant constraints;see [24]. We suppose that this pretreatment be performed, and we are dealing with anonredundant description of the polyhedral set in (1).

3 Basic Ideas

By preserving a linear structure of the constraints, we propose in the present sectiona generic solution toward the binary variables reduction.

To each of the regions in (5), we have associated in (6) a unique binary variable.Consequently, the total number of binary variables is N , the number of supportinghyperplanes (see (1)). However, a basic calculus shows that the minimum number ofbinary variables necessary to distinguish between these regions is

N0 = log2 N. (9)The value of N0 is reached by observing that each of the N regions in (6) can belinked to a unique number. Taking the numbers successively (starting from zero), itfollows that we need N0 bits to codify them into a binary representation.

The question that arises is the following:

How to describe the regions in a linear formulation similar to (6) through areduced number of binary variables?We impose firstly that the binary expression appearing in the inequalities has to

remain linear for computational advantages related to the optimization solvers. Thisstructural constraint is equivalent with saying that any variable i should be describedby a linear mapping in the form:

i() = ai0 +N0

k=1aikk, (10)


where

(1, . . . , N0) := {0,1}N0 . (11)In the reduced space of , we will arbitrarily associate a tuple

i := (i1 . . . iN0) (12)

to each region Ri . Note that this association is not unique, and various possibilitiescan be considered: in the following, unless otherwise specified, the tuples will beappointed in lexicographical order.

The problem of finding a mapping in , which describes region Ri , reduces thento finding the coefficients (ai0, a

i1, . . . , a

iN0

) for which i(i) = 0 and i(j ) 1,j = i under mapping (10). This translates into the following conditions for anyi, j {0,1}N0 :

ai0 +N0

k=1 aik

ik = 0,

ai0 +N0

k=1 aik

jk 1, j = i,

(13)

with ik the kth component of the tuple, i, associated to Ri .

Remark 3.1 Note that, in (13), the equality constraints for j = i were relaxed toinequalities since the value of Mi(j ) needs only to be sufficiently large (anyi(

j ) 1 being a feasible choice). Furthermore, the condition 1 can be relaxedto an arbitrary small positive constant by means of counterbalancing through an in-crease in constant M .

Nothing is said a priori about the nonemptiness of the set described by (13). Weneed at least a point in the coefficient space (a0, a1, . . . , aN0), which verifies condi-tions (13) in order to prove the nonemptiness. To this end, we present the followingproposition.

Proposition 3.1 A mapping i() : {0,1}N0 {0} [1,[ which verifies (13), isgiven by

i() =N0

k=1tk, where tk =

{k, if ik = 0,1 k, if ik = 1,

(14)

where k denotes the kth variable and ik its value for the tuple, i , associated toregion Ri .

The coefficients (ai0, . . . , aiN0) of the linear mapping (10) can be then obtained as

ai0 =N0

k=1ik, a

ik =

{1, if ik = 0,1, if ik = 1,

k = 1, . . . ,N0. (15)

Proof The claim is constructive; by introducing mapping (15) in (13), it can be veri-fied by simple inspection that the conditions are fulfilled.


Remark 3.2 Note that the problem of finding parameters i is independent of theactual shape of the polytope P . The coefficients obtained in (14) can be used for anytopologically equivalent polytope (that is, with the same number of half-spaces).

3.1 Interdicted Tuples

By the choice of the cardinal N0 as in (9), the number of tuples allowed by the reducedset of binary variables (11) may be greater than the actual number of regions.

The tuples left unallocated will be labeled as interdicted, and additional inequali-ties will have to be added to the extended set of constraints (6). These restrictions arejustified by the fact that, under construction (15), an unallocated tuple will not enforcethe verification of any of the constraints of (6) (see Remark 2.1). It then becomes ev-ident that the single constraint of (7) has to be substituted by a set of constraints thatimplicitly make all the unallocated tuples infeasible.

The next corollary of Proposition 3.1 provides the means to construct an inequalitywhich renders a tuple infeasible:

Corollary 3.1 Let there be a tuple i {0,1}N0 . The N-dimensional point that itdescribes, and exclusively this one by the combinations in (12), is made infeasiblewith respect to the constraint:

N0

k=1t ik , (16)

with t ik defined as in Proposition 3.1 and ]0,1[ a scalar.

Proof The left side of the inequality (16) will vanish only at tuple i , and for the restof the tuples in the discrete set {0,1}N0 will give values greater than or equal to 1.Thus, the only point made infeasible by inequality (16) is i .

The number of unallocated tuples may be significant, an upper bound is given by

0 Nint 2log2 N 2log2 N1 1 = 2log2 N1 1, (17)with the bound reached for the most unfavorable case of N = 2log2 N1 + 1.

If we associate to each of the unallocated tuples an inequality as in Corollary 3.1,we negatively influence the speed of the associated optimization algorithm. This canbe alleviated by noting (as previously mentioned) that the association between re-gions and tuples is arbitrary. One could then choose favorable associations whichwill permit more than one tuple to be removed through a single inequality. To thisend, we present the next proposition.

Proposition 3.2 Let there be a collection of tuples {i}i1,...,2d {0,1}N0 , whichcompletely spans a d-facet3 of hypercube {0,1}N0 . Let I be the set of the N0 d

3d denotes the degree of the facet, ranging from 0 for extreme points to N0 1 for faces of the hypercube.


indices, which retains a constant value over all the tuples {i}i1,...,2d composing thefacet. Then there exists the constraint

kItk , (18)

which renders the tuples of the given facet (and only these ones) infeasible.Variables tk and are taken as in Corollary 3.1 with tk associated to k , the

common value of variable k over the set of tuples {i}i1,...,2d .

Proof Geometrically, the tuples are extreme points on the hypercube {0,1}N0 andthe inequalities we are dealing with are half-spaces, which separate the points of thehypercube. If a set of tuples completely spans a d-facet, it is always possible to isolatea half-space that separates the points of the d-facet from the rest of the hypercube.

By a suitable association between feasible cells and tuples, we may label as un-allocated the extreme points which compose entire facets on the hypercube {0,1}N0 ,which permits to apply Proposition 3.2 in order to obtain constraints (26).

Remark 3.3 By writing Nint as a sum of consecutive powers of 2, i.e.,

Nint =log2 Nint1

i=0bi2i ,

an upper bound Nhyp for the number of inequalities (26) can be computed:

Nhyp =log2 Nint1

i=0bi log2 Nint, (19)

where bi {0,1}.

Remark 3.4 Note that (19) offers an upper bound for the number of inequalities, butpractically the minimal value can be improved depending on the method used forconstructing the separating hyperplanes and of the partitioning of the tuples betweenthe allocated and unallocated subsets.

3.2 Illustrative Example

As an illustration of the notions described in Sect. 3, we take the following square:

0 10 11 0

1 0

[x1x2

]

1111

. (20)

As stated in this section, the number of binary variables (similar to the formulation(6)) is N = 4, equal with the number of half-spaces described in (20). The reduced


Fig. 1 Outer regions and their associated tuples

number of variables will be N0 = log2 4 = 2, according to (9). Following the prob-lem formulation (14), the variables i can be expressed, as in (10), by

i() = ai0 + ai11 + ai22.We associate to each region a tuple of two values (1, 2) in lexicographical order.

The case of the 2nd half-space, associated to tuple (21, 22) = (0,1), is detailed

in Fig. 1(a). Using (13), we obtain, as depicted in Fig. 1(b), the feasible set of thecoefficients described by

a20 + a22 = 0, a20 1, a21 1.This represents a polytopic region in the coefficients space (a0, a1, a2) R3 and,according to (14), the nonemptiness is assured by the existence of at least a feasiblecombination of coefficients leading to the mapping

2() = 1 + 1 2.This means that the region R2 is projected from

[0 1

][x1x2

] 1 + M(1 + 1 2),

by taking (21, 22) = (0,1) (see Remark 2.1).

Furthermore, the same computations will be performed for the rest of the regions,resulting in an extended system of linear inequalities over mixed decision variables:

0 10 1

1 01 0

[x1x2

]

1 + M(1 + 2)1 + M(1 1 + 2)1 + M(1 + 1 2)1 + M(2 1 2)

.

As an exemplification of the considerations in Sect. 3.1, let there be a polytopewith 5 hyperplanes. This means that the number of binary variables has to be N0 =


Fig. 2 Exemplification of separating hyperplanes techniques

log2 5 = 3 and then, Nint = 23 5 = 3 tuples will remain unallocated; we choosethese to be (0,0,1), (1,0,1), and (1,1,1).

By applying Corollary 3.1, we observe in Fig. 2(a) the 3 inequalities that separatethe unallocated tuples from the rest (for simplicity, in the rest of the subsection, wewill use = 0.5):

(1 + 1 + 2 3) 0.5,(2 1 + 2 3) 0.5,(3 1 2 3) 0.5.

We observe in Fig. 2(b) that the tuples are positioned onto 2 edges, and consequently,using Proposition 3.2, 2 inequalities suffice for separation:

(1 + 2 3) 0.5,(2 1 3) 0.5.

Lastly, recalling Remark 3.4, we note that, in this particular case, a single inequality(as seen in Fig. 2(c)), is enough for separating the unallocated tuples from the rest:

(0.321 + 1.762 + 2.133) 0.5.

4 Description of the Complement of a Union of Convex Sets

In the previous section, the basic reduction method was applied for treatment of thecomplement of a convex set. A generic case will be detailed in the following byconsidering the complement of a union of convex (bounded polyhedral) sets P :=

l Pl :

CX(P) := cl(X \ P), (21)


with4

Pl :=Kl

kl=1R+(Hkl )

and

N :=

l

Kl .

This type of regions arises naturally in the context of obstacle/collision avoidancewhen there is more than a single object to be taken into account. In order to deal withthe complement of a not convex region in the context of mixed-integer techniques,several additional theoretical tools related to the arrangements of the hyperplanes willbe introduced in the following. The noninitiated reader may consult on this conceptof the well-known books [25] and [26], some recent related articles [27, 28], and thereferences therein.

Definition 4.1 A collection of hyperplanes H = {Hi}i=1:N will partition the space ina union of cells5 defined as follows:

A(H) =

l=1,..., (N)

(N

i=1Rl(i)(Hi )

)

Al

, (22)

where sign tuple l {,+}N denotes feasible combinations of regions (4) obtainedfor the hyperplanes in H, and (N) denotes the number of feasible cells.

Several computational aspects are of interest. The number of feasible cells, (N),(in relation with the space dimensiond and the number of hyperplanesN ) isbounded by Bucks formula [29]:

(N) d

i=0

(N

i

), (23)

with equality satisfied if the hyperplanes are in general position6 and X = Rn.An efficient algorithm for describing (22), based on reverse search that runs in

O(N (N) lp(N,d)) time and O(N,d) space, was presented in [30] and implementedin [31].

4The + superscript was chosen for the homogeneity of notation, equivalently one could have chosen anycombination of signs in the half-space representation (4) in order to describe the polyhedral regions Pl .5The relative interior of any two cells is disjoint, but their boundaries may have one of the hyperplanes Hias a common element.6We call a hyperplane arrangement to be in general position whenever any small change in the positionof the composing hyperplanes does not change the number of cells.


Note that there exists a subset {Bl}l=1,..., b(N) of feasible polyhedral cells from(22) (with b(N) (N)) which describes region (21):

CX(P) =

l=1,..., b(N)Bl, (24)

such that, for any l, there exists a unique i for which, Bl = Ai and Ai P = .In (6), a single binary variable was associated to a single inequality, but the mecha-

nism can be applied similarly to more inequalities (e.g., the ones describing one of thecells of (24)). Thus, one can describe (22) in an extended space of state + auxiliarybinary variables as follows:

...

l(1)h1x l(1)k1 + Ml...

l(N)hNx l(N)kN + Ml

Bl

...

(25)

and condition

l= b(N)

l=1l b(N) 1, (26)

which implies that at least a set of constraints will be verified.Construction (25)(26) will permit, through projection along the binary variables

l (see (8)), to obtain any of the cells of the union (24).Analogously to Sect. 3, we propose, in the following, the reduction of the num-

ber of binary variables by associating to each of the cells a unique tuple. The binarypart will be computed following the constructive result in Proposition 3.1 and usedaccordingly in (25). Additional inequalities that render infeasible the unallocated tu-ples, are introduced as in Proposition 3.2.

A few remarks relating to the number of hyperplanes and their corresponding ar-rangement are in order.

Remark 4.1 The number of inequalities in (25) can be reduced by observing that notall the hyperplanes of H are active in a particular cell, and thus they can be discardedfrom the final representation.

Remark 4.2 Note that if we discard the linear structure and allow a nonlinear formu-lation involving products of binary variables, the hyperplane arrangements (22) can


be represented as

...

hix ki + M

l=1,..., b(N)l(i)=

l,

hix ki + M

l=1,..., b(N)l(i)=+

l,

...

(27)

for all sign tuples l associated to cells Bl from covering (24). We have used the factthat the cells of (24) use the same half-spaces (up to a sign), and thus they can beconcatenated. The method presented in [32] transforms an inequality with nonlinearbinary components into a set of inequalities with linear binary components. However,this can be made only at the expense of introducing additional binary variables, whichin the end gives a larger problem than the one presented in (25)(26).

4.1 Exemplification of Hyperplane Arrangements

Consider the following illustrative example depicted in Fig. 3, where the comple-ment of the union of two triangles (P = P1 P2) represents the feasible region. Wetake H = {Hi}i=1:4 the collection of N = 4 hyperplanes (given as in (3)) which de-fine P1, P2.

We observe that the bound given in (23) is reached, that is, we have 11 cells (ob-tained as in the arrangement (22)). From them, a total of 9, which we denote hereas B1, . . . ,B9, describe the not convex region (21). To each of them, we associate aunique tuple from {0,1}N0 as seen in Fig. 3 with N0 = log2 9 = 4.

Fig. 3 Exemplification of hyperplane arrangement


As per Proposition 3.1 and (25), we are now able to write the set of inequali-ties (28).

h3x k3h4x k4

+ M( 1 + 2 + 3 + 4)}

B1,

h2x k2h3x k3

h4x k4+ M(1 + 1 + 2 + 3 4)

B2,

h1x k1h2x k2

h3x k3+ M(1 + 1 + 2 3 + 4)

B3,

h1x k1h3x k3

+ M(2 + 1 + 2 3 4)}

B4,

h1x k1h2x k2

h3x k3h4x k4

+ M(1 + 1 2 + 3 + 4)

B5,

h2x k2h4x k4

+ M(2 + 1 2 + 3 4)}

B6,

h1x k1h2x k2

h4x k4+ M(2 + 1 2 3 + 4)

B7,

h1x k1h3x k3

h4x k4+ M(3 + 1 2 3 4)

B8,

h1x k1h2x k2h3x k3h4x k4

+ M(1 + 1 2 + 3 + 4)

B9.

(28)

Note that, in the above set, we have simplified the description by cutting the re-dundant hyperplanes in a cell representation (e.g., for cell A1, 2 hyperplanes sufficefor a complete description).


Since only 9 tuples, from a total number of 16, are associated to cells, we need toadd constraints to the problem such that the remaining 7 unallocated tuples will neverbe feasible. Using Corollary 3.1, we obtain:

(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.

(29)

We observe that for the 7 unallocated tuples, 4 of them, (1,1,0,0), (1,1,0,1),(1,1,1,0), and (1,1,1,1), form a 2-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. We can now apply Propo-sition 3.2 and obtain the following constraints:

(2 1 2) 0.5,(2 1 + 2 3) 0.5,

(2 1 + 2 + 3 4) 0.5.(30)

Note that we were able to diminish the number of inequalities from 7 in (29) to only3 in (30): the first 4 constraints of (29) are replaced by the 1st constraint of (30). Thesame holds for the next 2 that correspond to the 2nd, and for the last that is identicalwith the 3rd.

5 Refinements for the Complement of a Union of Convex Sets

As seen in [22], palliatives for reducing the computational load exist but ultimately,the computation time is in the worst case scenario exponentially dependent on thenumber of binary variables, which in turn depends on the number of cells of thehyperplane arrangements (see (23)). We conclude then that the problem becomesprohibitive for a relatively small number of polyhedra in P, and that any reduction inthe number of cells is worthwhile and should be pursued.

This can be accomplished in two complementary ways. Firstly, we note that bound(23) is reached for a given number of hyperplanes if and only if they are in generalposition. As such, particular classes of polyhedra may somewhat reduce the actualnumber of cells in arrangement (22), and consequently, the number of auxiliary bi-nary variables. In increasing order of their versatility, we may mention hypercubes,orthotopes, parallelotopes, and zonotopes as classes of interest (for a computation ofthe number of cells (see, [33])).

The other direction, which we chose to pursue in the rest of the paper, is reducingthe number of cells that describe (21).


In [22], we have proposed a hybrid scheme which permits to express (21) as aunion of the cells of (24) which intersect P, and of the regions (in the sense of (4))which describe C(P), where P denotes the convex hull of P.

Alternatively, the merging of adjacent cells into possibly overlapping regions,which describe (21), is discussed in [23]. This results in a reduced representation,both in number of cells and of interdicting constraints. In the next subsection, wedetail the merging techniques used and show how the complexity of the problem isreduced.

5.1 Cell Merging

Recall that any of the cells of (24) is described by a unique sign tuple (Bl l).As such, we obtain that the cells are disjunct and cover the entire feasible space. Forour purposes, we are satisfied with any collection of regions not necessarily disjointwhich covers the feasible space. In this context, we may ask if it is not possible tomerge the existing cells of (24) into a reduced number of regions which will stillcover region (21). Note that by reducing the number of regions, the number of binaryauxiliary variables will also decrease substantially.

We can formally represent the problem by requiring the existence of a collectionof regions,

CX(P) =

k=1,..., c(N)Ck, (31)

which verifies next conditions:

the new polyhedra are formed as unions of the old ones (i.e., for any k there existsa set Ik which selects indices from 1, . . . , b(N) such that Ck = iIk Bi ), the union is minimal, that is, the number c(N) of regions is minimal.Existing merging algorithms are usually computationally expensive, but here we

can simplify the problem by noting two properties of the cells in (24): the sign tuples l describe an adjacency graph since any two cells whose sign tuples

differ at only one position are neighbors, the union of any two adjacent cells is a polyhedra.

In order to construct (31), we may use merging algorithms (see, for example, [34],which adapts a branch and bound algorithm to merge cells of a hyperplane ar-rangement), or we can pose the problem in the Boolean algebra framework. Themerging problem of regions from (24) is functionally identical to the minimizationof a Boolean function given in the sum of products form. A cell describing the(in)feasible region (21) corresponds to a 1 (0) value in the truth table at the po-sition determined by its associated sign tuple, whereas infeasible sign tuples corre-spond to do not care values. It is then straightforward to apply minimization algo-rithms (Karnaugh maps, the QuineMcCluskey algorithm, or the Espresso heuristiclogic minimizer) in order to obtain Boolean minterms who describe the merged cellsof (31). We note that a similar approach was proposed in [35] in order to deal withpolyhedral piecewise affine systems.


Remark 5.1 Note that a region Ck is described by at most N f hyperplanes, wheref denotes the number of indices in the sign tuples which flip the sign. It makes sensethen to, not only reduce the number of regions, but also to maximize the number ofcells that go into the description of a region from (31).

In Algorithm 1, we sketch the notions presented in this section.

Algorithm 1 Scheme for representing C(P)Input P

1 obtain the cell arrangement as in (22) for H2 obtain the feasible cells (24) and merge them in representation (31)3 get the number c(N) of feasible regions and the number N0 of auxiliary binary

variables4 partition the tuples of {0,1}N0 such that Proposition 3.2 can be efficiently applied5 create the extended polyhedron (25) and add the constraints (26)

Note that the steps 1 and 2 are the most computationally expensive. For the firststep, an efficient algorithm for describing (22), based on reverse search that runs inO(N (N) lp(N,d)) time and O(N,d) space, was detailed in [30] and implementedin [31]. For the second step, Boolean algebra techniques have been used: for a smallnumber of hyperplanes, exact methods like Karnaugh maps prove to be effective.For higher numbers, the heuristic Espresso logic minimizer can be used. The latterprovides solutions very close to the optimum while being more efficient and reduc-ing memory usage and computation time by several orders of magnitude relative toclassical methods (see [36]).

5.2 Exemplification of Hyperplane Arrangements with Cell Merging

We revisit here the example provided in Sect. 4.1 and apply the results presented inSect. 5.1 in order to show the improvements.

For this simple case, we apply, as seen in Fig. 4, a Karnaugh diagram and obtainthat the feasible region (21) is expressed by a union as in (25).

As seen from the Karnaugh diagram from Fig. 4, we obtain 4 overlapping regions:C1 = B1 B2 B3 B4, C2 = B4 B5 B6, C3 = B6 B7 B8 B1 and C4 =B8 B9 B1 B2, which we depict in Fig. 5. Consequently, we note that N0 = 2

Fig. 4 Karnaugh diagram forobtaining the reduced cellrepresentation


Fig. 5 Exemplification ofhyperplane arrangement withmerged regions

auxiliary binary variables suffice in coding the regions. As for Proposition 3.1 and(25), we are now able to write the following set of inequalities (we attach to each ofthe regions a tuple in lexicographical order):

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.

(32)

Note that, in addition to reducing the number of regions in (32) comparative with(28), we also have reduced the number of hyperplanes appearing in the regions half-space representation (see Remark 3.3).

6 Numerical Considerations

In this section, we will test the computation time improvements for our approachversus the standard technique encountered in the literature. As previously mentioned,a MI problem is NP-hard in the number of binary variables (due to the fact that thealgorithms implement branch-and-bound techniques and as such, in the worst casescenario, they need to iterate through all the branches defined by the binary searchtree variables). Therefore, even a small reduction will render sensible improvements.

The complexity of the MI algorithm with constraints in the classical form (6)(7)will be of the order of O(2N p(N + 1, d)), where p(n,d) denotes either lp(n, d) orqp(n, d), as it is required. Using the alternative formulation proposed in Sect. 3, we


Fig. 6 Comparative test forcomputation time for classicaland enhanced methodtimeaxis in logarithmic scale

Table 1 Numerical values forthe solving of an MIoptimization problem underclassical and enhanced methods

No. ofhyperplanes

5 10 15 20 25 50 100

Classical 9.91 64.06 91.74 511.47 306.04 Enhanced 1.14 0.81 0.59 4.84 4.18 3.66 2.94

obtain the complexity as

O(2log2 N p(N + log2 N 1), d

) = O(N p(N,d)). (33)We consider the worst case scenario for the mixed-integer problem, that is, the

optimization algorithm will need to pass through each of the possible combinationsprovided by the binary variables. In the case of formulation (6)(7), this leads to aNP complexity in the number of lp/qp problems to be solved. On the other hand,the formulation shown in (33), through the reduction of binary variables, provides apolynomial complexity.

To illustrate these computational gains, we will compare the times of executionfor both schemes as follows: the computational time will be measured and averagedfor 10 samples of 2D polytopes with the same number of support hyperplanes; fur-thermore, the procedure will be iterated by changing the number of hyperplanes from4 to 25. The results are depicted in Fig. 6 on a semilogaritmic scale and, as it can beseen, there are significant improvements. In fact, the differences may be even morepronounced since, under default settings, the MI algorithm over the classical methodstopped computing the optimum value after a maximum number of iterations wasreached (the MI algorithm used was the one described in [12]).

Similar results are shown in Table 1, where we observe the evident improvementof our method relative to the classical technique.

In Sect. 4, a method for describing in the MI formalism of the complement ofa possibly not connected union of polytopes was presented. The main drawback isthat in both classical and reduced formulation, the problem depends on the number


of cells. Supposing that the hyperplanes from the hyperplane arrangement (24) be inrandom position, we obtain, for formulation (25)(26), a complexity of order

O(2 b(N) p(N Nd + 1), d), (34)which can be further reduced, using the techniques from Sect. 3, to

O(2log2 b(N) p(N Nd + 1, d)) = O( b(N) p(Nd+1, d)). (35)Again, we observe that the mixed-integer problem becomes polynomial in the num-ber of branches: in the worst case, we have b subproblems to solve, and b which in turn is given by the polynomial expression (23). However, the problem isstill challenging due to the number of cells (see (23)). By reducing the number of cellsas in Sect. 5, it is possible to significantly reduce the computation time. For exempli-fication, take the example depicted in Figs. 3 and 5. We observe that in this particularcase, we were able to reduce the representation from 9 cells to only 4. Presumably,for a higher number of hyperplanes, the gain will be even more pronounced.

7 Collision Avoidance Example

A number of commonly found situations in the control related to Multi-Agent Sys-tems imply a cost function that has to be minimized, while in the same time, the agentavoids collision with obstacles and other agents. To solve this problem, there existsvarious methods. Arguably, they can be gathered in methods which penalize throughthe cost function as the violation of the constraints (e.g., Potential Field Method [37]and Navigation Functions [38]), and methods which impose hard constraints that maynot be broken. The latter group usually employs receding horizon techniques as theynaturally take into account constraints [7, 15]. In this illustrative example, we willdescribe an agent that has to navigate its way around a group of fixed obstacles. Weconsider the dynamics of the agent described by a LTI system as follows:

k+1 = Ak + Buk. (36)The agent model is used in a predictive control (see [14]) context which permits

the use of not convex state constraints for obstacle avoidance behavior.An optimal control action u is obtained from the control sequence u :={

uk|k, uk+1|k, . . . , uk+N1|k}

as a result of the optimization problem:

u = arg minu

(

Tk+N |kP k+N |k +N1

l=1Tk+l|kQk+l|k +

N1

l=0uTk+l|kRuk+l|k

)

,

s.t.:

{k+l|k = Ak+l1|k + Buk+l1|k,k+l|k C(P), l = 1, . . . ,N.

(37)

Here, Q 0, R > 0 are the weighting matrices, P 0 defines the terminal cost andP is an union of polytopes describing the obstacles.


Fig. 7 Interdicted region in thestate space

As a practical application, we consider a linear system (vehicle, pedestrian or agentin general form), whose dynamics are described by

A =

0 0 1 00 0 0 10 0

m0

0 0 0 m

, B =

0 00 01m

00 1

m

, (38)

where = [x y vx vy]T , u = [ux uy]T are the state and the input of the system. Withthe components of the state being (x, y) the position, and (vx, vy) the velocities ofthe agent, m is the mass of the agent and its damping factor.

We consider the position component of the agent state to be constrained by 4obstacles as shown in Fig. 7.

Considering the 14 hyperplanes which describe the polyhedra associated with theobstacles, we have (14) = 106 regions obtained as in (22). Additionally, we observethat 10 of the cells will describe the interdicted regions, and the rest, b(14) = 96will describe the feasible region, as shown in (24). Furthermore, we apply the notionsfrom Sect. 5.1 to obtain a reduced representation for the feasible region as in (31).We observe that the number of cells is substantially reduced, from b(14) = 96 to c(14) = 11, which warrants in turn a reduction of the auxiliary binary variables from7 to 4 that, for a worst case scenario, equals to an eightfold speed up. In Fig. 8(a),we depict the cells of (24) and the obstacles, while in Fig. 8(b) we show the covering(31) of merged cells.

We apply the predictive control strategy for horizon N = 3 and cost matrices Q =105 I4, R = I2 and P = 105 I4, and obtain the trajectory depicted in Fig. 9.

A last aspect of interest is the scalability of the problem for a larger number ofagents/obstacles. Obviously, the results depend greatly on the number of sets, thenumber of hyperplanes defining them and their relative positions. We depict in Table 2a few cases (showing the number of obstacles, hyperplanes, and regions (6)(8) and(19), respectively) and observe that the improvements are (at least for the examplesconsidered) significant.


Fig. 8 Cell partitionings of the feasible region

Fig. 9 Simulations of agenttrajectories

Table 2 Complexity analysiswith increased numbers ofobstacles

#P #Hi #Ai #Bi #Ci

5 19 210 191 137 24 295 266 169 30 496 449 23

8 Concluding Remarks

In this paper, we revisit a technique, which transforms a not convex and possiblynot connected region into a polyhedra in an augmented space (state and auxiliarybinary variables) through the use of hyperplane arrangements. With respect to previ-ous results, we minimized the number of cells describing the feasible region throughmerging methods and discussed an improvement of the optimization problem suchthat the number of additional constraints is minimized. These numerical improve-


ments were presented, tested, and discussed on applications concerning the obstacleavoidance control problem.

Acknowledgements The research of Ionela Prodan is financially supported by the EADS CorporateFoundation (091-AO09-1006). Florin Stoicans work was carried out in Supelec, during the tenure ofa CARNOT C3S fellowship. The authors would like to thank Professor Panos M. Pardalos, as well asthe anonymous reviewers for their useful comments and remarks that helped in improving the overallpresentation of this paper.

References

1. Grundel, D., Murphey, R., Pardalos, P.: Cooperative Systems, Control and Optimization, vol. 588.Springer, Berlin (2007)

2. Grundel, D., Pardalos, P.: Theory and Algorithms for Cooperative Systems, vol. 4. World Scientific,Singapore (2004)

3. Earl, M., DAndrea, R.: Modeling and control of a multi-agent system using mixed integer linear pro-gramming. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida,USA, vol. 1, pp. 107111 (2001)

4. Osiadacz, A.: Integer and combinatorial optimization. In: Nemhauser, G.L., Wolsey, L.A. (eds.)Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1988). In-ternational Journal of Adaptive Control and Signal Processing, 4(4), 333344 (1990)

5. Jnger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W., Reinelt, G., Rinaldi, G.,Wolsey, L.: 50 Years of Integer Programming 19582008: From the Early Years to the State-of-the-Art. Springer, Berlin (2009)

6. Schouwenaars, T., De Moor, B., Feron, E., How, J.: Mixed integer programming for multi-vehiclepath planning. In: Proceedings of the 2nd IEEE European Control Conference, Porto, Portugal, pp.26032608. (2001)

7. Richards, A., How, J.: Model predictive control of vehicle maneuvers with guaranteed completiontime and robust feasibility. In: Proceedings of the 24th American Control Conference, Portland, Ore-gon, USA, vol. 5, pp. 40344040 (2005)

8. Ousingsawat, J., Campbell, M.: Establishing trajectories for multi-vehicle reconnaissance. In: Pro-ceedings of AIAA Guidance, Navigation and Control Conference, Providence, Rhode Island, USA,pp. 21882199 (2004)

9. Beard, R., McLain, T., Goodrich, M., Anderson, E.: Coordinated target assignment and intercept forunmanned air vehicles. Proc. IEEE Trans. Robot. Autom. 18(6), 911922 (2002)

10. Richards, A., Bellingham, J., Tillerson, M., How, J.: Coordination and control of multiple UAVS. In:Proceedings of AIAA Guidance, Navigation and Control Conference. Monterey, CA (2002)

11. Schumacher, C., Chandler, P., Pachter, M.: UAV task assignment with timing constraints. In: Proceed-ings of AIAA Guidance, Navigation and Control Conference, Austin, Texas (2003)

12. Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Automat-ica 35, 407428 (1999)

13. Bemporad, A., Borrelli, F., Morari, M.: Optimal controllers for hybrid systems: stability and piecewiselinear explicit form. In: Proceedings of the 39th IEEE Conference on Decision and Control, Sydney,NSW, Australia, pp. 18101815 (2000)

14. Mayne, D., Rawlings, J., Rao, C., Scokaert, P.: Constrained model predictive control: stability andoptimality. Automatica 36, 789814 (2000)

15. Camacho, E.F., Bordons, C.: Model Predictive Control. Springer, Berlin (2004)16. Bellingham, J., Richards, A., How, J.: Receding horizon control of autonomous aerial vehicles. In:

Proceedings of the 21th American Control Conference, Anchorage, Alaska, USA, vol. 5, pp. 37413746 (2002)

17. Prodan, I., Olaru, S., Niculescu, S.: Predictive control for tight group formation of multi-agent sys-tems. In: Proceedings of the 18th IFAC World Congress, Milano, Italy, pp. 138143 (2011)

18. Stoican, F., Olaru, S., Seron, M., De Don, J.: Reference governor for tracking with fault detectioncapabilities. In: Proceedings of the 2010 Conference on Control and Fault Tolerant Systems, Nice,France, pp. 546551 (2010)


19. Garey, M., Johnson, D.: Computers and intractability. A Guide to the Theory of NP-Completeness.A Series of Books in the Mathematical Sciences. Freeman, San Francisco (1979)

20. Earl, M., DAndrea, R.: Iterative MILP methods for vehicle-control problems. IEEE Trans. Robot.21(6), 11581167 (2005)

21. Vitus, M., Pradeep, V., Hoffmann, G., Waslander, S.L., Tomlin, C.J.: Tunnel-milp: path planning withsequential convex polytopes. In: Proceedings of AIAA Guidance, Navigation and Control Conference,Honolulu, Hawaii, USA (2008)

22. Stoican, F., Prodan, I., Olaru, S.: On the hyperplanes arrangements in mixed-integer techniques. In:Proceedings of the 30th American Control Conference, San Francisco, CA, USA, pp. 18981903(2011)

23. Stoican, F., Prodan, I., Olaru, S.: Enhancements on the hyperplane arrangements in mixed integertechniques. In: 50th IEEE Conference on Decision and Control and European Control Conference,Orlando, FL, USA, pp. 39863991 (2011)

24. Olaru, S., Dumur, D.: Avoiding constraints redundancy in predictive control optimization routines.IEEE Trans. Autom. Control 50(9), 14591465 (2005)

25. Orlik, P., Terao, H.: Arrangements of Hyperplanes, vol. 300. Springer, Berlin (1992)26. Ziegler, G.: Lectures on Polytopes. Springer, Berlin (1995)27. Orlik, P.: Hyperplane arrangements. In: Floudas, C.A., Orlik, P.M. (eds.) Encyclopedia of Optimiza-

tion, pp. 15451547. Springer, New York (2009)28. Pardalos, P.: Hyperplane arrangements in optimization. In: Floudas, C.A., Pardalos, P.M. (eds.) Ency-

clopedia of Optimization, pp. 15471548. Springer, New York (2009)29. Buck, R.: Partition of Space. Am. Math. Mon. 50(9), 541544 (1943)30. Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65(1), 2146 (1996)31. Ferrez, J.A., Fukuda, K.: Implementations of lp-based reverse search algorithms for the zonotope

construction and the fixed-rank convex quadratic maximization in binary variables using the zram andthe cddlib libraries. Tech. rep., Mcgill (2002)

32. Kobayashi, K., Imura, J.: Modeling of discrete dynamics for computational time reduction of ModelPredictive Control. In: Proceedings of the 17th International Symposium on Mathematical Theory ofNetworks and Systems, Kyoto, Japan, pp. 628633 (2006)

33. Zaslavsky, T.: Counting the faces of cut-up spaces. Am. Math. Soc. 81, 5 (1975)34. Geyer, T., Torrisi, F., Morari, M.: Optimal complexity reduction of piecewise affine models based on

hyperplane arrangements. In: Proceedings of the 23th American Control Conference, Boston, Mas-sachusetts, USA, vol. 2, pp. 11901195 (2004)

35. Geyer, T., Torrisi, F., Morari, M.: Optimal complexity reduction of polyhedral piecewise affine sys-tems. Automatica 44(7), 17281740 (2008)

36. Rudell, R., Sangiovanni-Vincentelli, A.: Espresso-mv: algorithms for multiple-valued logic minimiza-tion. In: Proceedings of the IEEE Custom Integrated Circuits Conference, pp. 230234 (1985)

37. Olfati-Saber, R., Murray, R.: Distributed cooperative control of multiple vehicle formations usingstructural potential functions. In: Proceedings of the 15th IFAC World Congress, Barcelona, Spainpp. 346352 (2002)

38. Rimon, E., Koditschek, D.: Exact robot navigation using artificial potential functions. IEEE Trans.Robot. Autom. 8(5), 501518 (1992)

Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming TechniquesAbstractIntroductionPreliminariesBasic IdeasInterdicted TuplesIllustrative Example

Description of the Complement of a Union of Convex SetsExemplification of Hyperplane Arrangements

Refinements for the Complement of a Union of Convex SetsCell MergingExemplification of Hyperplane Arrangements with Cell Merging

Numerical ConsiderationsCollision Avoidance ExampleConcluding RemarksAcknowledgementsReferences

Documents

Enhancements on the Hyperplanes Arrangements in Mixed-Integer Programming Techniques