A Piecewise Linearization Algorithm for Solving MINLP in Intersection
Management
Matthias Gerdts
1 a
, Sergejs Rogovs
1
and Giammarco Valenti
2
1
Institute of Engineering Mathematics, Bundeswehr University Munich, Germany
2
Department of Industrial Engineering, University of Trento, Italy
Keywords:
Connected and Automated Vehicles, Cooperative Driving, Intersection Management, Piecewise Linearization,
Mixed Integer Nonlinear Programming.
Abstract:
In this paper, we propose a linearization algorithm for solving a Mixed Integer NonLinear Problem (MINLP)
for Intersection Management (IM) of Connected Autonomous Vehicles (CAVs). The objective of such problem
is to minimize the time it takes to clear a given arbitrary intersection for all vehicles in the consideration. We
treat the IM problem as a bi-level optimization problem. On the lower level we solve an Optimal Control
Problem (OCP) for each individual vehicle, whereas on the higher level we deal with an optimization problem
of finding the optimal sequence and starting times for every car, which essentially yields a MINLP. An intuitive
linearization technique is presented to solve the emerging MINLP in a reasonable time. The actual controls, if
necessary, are computed a posteriori by minimizing the L
2
-norm of control variables. The algorithm is tested
in different intersection scenarios. Numerical results show that it is suitable for real-time applications.
1 INTRODUCTION
Road intersections play a very important role in trans-
portation networks due to safety and efficiency rea-
sons, especially in urban areas. The rise of automa-
tion in the automotive field, combined with commu-
nication paradigms such as vehicle-to-vehicle (V2V)
and vehicle-to-infrastructure (V2I), gave birth to new
ways of addressing the problem. The concept of Con-
nected Autonomous Vehicles allows to exploit the ca-
pabilities of self-driving vehicles. Optimal coordina-
tion of CAVs at intersections can help a lot in the im-
provement of traffic efficiency and safety aspects. Re-
search on this topic gained a lot of interest in the last
decade with a significant boost in the last three years,
see e.g. (Namazi et al., 2019) or (Khayatian et al.,
2020). Therein, one can find a detailed review of the
literature on the topic. Since there exists a vast variety
of algorithms on this topic, there are several ways how
one can group them. For instance, the approaches can
be grouped by optimization goals. The most common
goals are: safety, efficiency, ecology and passenger
comfort. Another way how to cluster the approaches
is by their cooperation paradigm: distributed and cen-
tralized. In the distributed case, CAVs try to find
a
https://orcid.org/0000-0001-8674-5764
an optimal crossing sequence by communicating with
each other, whereas in the centralized case there is
some managing unit located in the vicinity of the in-
tersection. A examples of distributed approaches can
be found in (Britzelmeier and Dreves, 2020), (Ma-
likopoulos et al., 2018) and (Ahmane et al., 2013).
In the first reference, the problem is represented as a
generalized Nash equilibrium problem which results
from a coupled optimal control problem. Moreover,
to the best of our knowledge this is the first work on
intersection scheduling with a proof of convergence.
In the underlying work, we consider a centralized
approach. If no traffic priorities are given, the inter-
section management problem in this case can be ad-
dressed as a bi-level optimization problem. On the
lower level the motion of each individual vehicle has
to be found in accordance to some optimization goal,
whereas on the upper level one tries to find an opti-
mal crossing order such that no collisions occur, and
some global cost is optimized. Often, this global cost
is simply the total time elapsed to clear an intersection
for all vehicles in consideration. Let us first mention
some existing works with similar modeling/solution
techniques, and then discuss the contribution of our
work. The approaches given in (Fayazi and Vahidi,
2018), (Mohamad Nor and Namerikawa, 2019) ex-
ploit the same vehicle model and the same modeling
438
Gerdts, M., Rogovs, S. and Valenti, G.
A Piecewise Linearization Algorithm for Solving MINLP in Intersection Management.
DOI: 10.5220/0010437104380445
In Proceedings of the 7th International Conference on Vehicle Technology and Intelligent Transport Systems (VEHITS 2021), pages 438-445
ISBN: 978-989-758-513-5
Copyright
c
 2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

techniques for scheduling constraint. The geometry
of the intersection is instead restricted to a specific
use case with no turns, and due to some simplifica-
tions, the upper level problem is modeled as a MILP.
In (Hult et al., 2018), (Hult et al., 2019) a more so-
phisticated car model is considered which results in
a MINLP on the upper level. The emerging MINLP
is then solved using an SQP-type algorithm. See also
(Hult et al., 2020) for a real-world experiment.
In this paper we consider a simple car model as
in (Fayazi and Vahidi, 2018), (Mohamad Nor and
Namerikawa, 2019), however, on the upper level we
end up with a MINLP as in (Hult et al., 2018), (Hult
et al., 2019), and in contrast to Hult et al., we solve the
MINLP with some linearization technique inspired
by (Winston and Goldberg, 2004). Such a choice is
explained by existence of high-performance software
packages for solving MILP. We show that our tech-
nique can be used for real-time applications.
The rest of the paper is organized as follows. In
Section 2 we shape the scope of our considerations
by discussing restrictions and assumptions we make
regarding vehicles and intersection geometry. Section
3 is devoted to the problem formulation, where we
introduce the vehicle model, define scheduling con-
straints, and state the final MINLP. Moreover, in this
section we discuss the case when a car cannot enter an
intersection at an optimal (in local sense) velocity. In
Section 4 a piecewise linearization technique is intro-
duced and a final MILP is stated. Finally, in Sections
5 and 6 we present some numerical experiments and
state conclusions.
2 GENERAL CONSIDERATIONS
AND ASSUMPTIONS
In this section we define the scope of our considera-
tions. Here, we discuss all the restrictions we make
related to vehicles and intersections in our consider-
ation. Moreover, we summarize the most important
quantities and variables we are using in Table 1.
2.1 Vehicles
For the sake of shortness we restrict our considera-
tions to only one vehicle in each lane. This assump-
tion seems to be a quite strict one, however the ap-
proach presented in this work can be easily general-
ized to an arbitrary number of vehicles in each lane
by adding some constraints to guarantee that no rear-
end collisions take place, see e.g. (Hult et al., 2019).
Such a general setting is going to be considered in
our future work. Throughout this paper, we denote by
N the total number of vehicles and the corresponding
index set by V = {1, ...,N}. Moreover, we assume
that the states of each vehicle are known without any
uncertainties. The states can be obtained by solving a
simple optimal control problem for each vehicle, see
Section 3.1.
Assumption 1. There is at most one car in each lane.
Assumption 2. The states of each vehicle are known
without any uncertainties.
The last assumption related to vehicles we make,
states that we deal only with feasible initial condi-
tions. Such an assumption is straightforward. The
upcoming MINLP problem cannot be solved for any
set of initial conditions. The initial conditions that
lead to an infeasible problem (impossibility to respect
all constraints) are not considered. This assumption
does not imply any loss of generality, since infeasible
initial conditions eventually lead to an accident and
they must be a priori excluded.
Assumption 3. Only feasible initial conditions are
considered.
2.2 Intersection Geometry
Now, let us define some assumptions on the intersec-
tion geometry. The approach presented in this work is
conceived to be flexible and can be easily adapted to
any arbitrary intersection.
Lanes: The lanes that enter/exit an intersection
are called entering/exit lanes. It is not allowed to over-
take within the same lane. Moreover, no lane change
is allowed (vehicle is supposed to be already in the
correct lane) in the portion of the intersection consid-
ered in the problem. These assumptions are widely
used for such problems (Namazi et al., 2019).
Assumption 4. No overtake or lane change maneu-
vers are allowed.
Paths: Every entering lane can be connected to
each exit lane with some predefined path. The number
of all paths is denoted by M and the corresponding
index set reads as P = {1,. ..,M}. The corresponding
vehicle-to-path mapping reads
P : V → P ,
where for every i ∈ V there exist exactly one k ∈ P
such that P(i) = k. In other words, to cross an inter-
section each car follows exactly one path which de-
pends on the car’s route. Each path’s trajectory is
assumed to be explicitly known. See an example of
intersection geometry in Fig. 1.
Assumption 5. Each vehicle follows exactly one pre-
defined path.
A Piecewise Linearization Algorithm for Solving MINLP in Intersection Management
439

Conflict Zone: A conflict zone (CZ) is the area of
an intersection that contains all the overlapping points
of the all paths. All the possible trajectory collisions
can take place only in this area. For simplicity it is
usually assumed that a CZ is given by some polygon,
see Fig. 1 where a canonical intersection is consid-
ered, whose CZ is given by a square.
Once a conflict zone is defined, the longitudi-
nal motion of each vehicle is broken down into two
phases. Namely, the approaching phase (fully de-
scribed in Section 3.1) which deals with the vehicle’s
motion from the initial position to the point where the
conflict zone begins, and the intersection phase which
starts where the first phase finishes and ends when a
vehicle leaves the conflict zone. For simplicity rea-
sons the velocity of each vehicle is assumed to be con-
stant during the intersection phase. This assumption
is also widely used in the literature, see e.g. (Namazi
et al., 2019).
Assumption 6. The velocity of each vehicle during
the intersection phase is constant.
Conflict Matrix: In order to capture all possible
combinations of pairwise intersecting paths, which is
essential for collision avoidance reasons, a so-called
conflict matrix is introduced. Entries of such conflict
matrix can take only binary values. We define the con-
flict matrix as follows. For all pairs k, l ∈ P , the cor-
responding entry of the conflict matrix is given by
c
kl
=

0, paths do not overlap,
1, paths overlap, there is conflict.
(1)
Such a matrix is essential for the formulation of
scheduling constraints (7). Note that in order to avoid
duplicate constraints, by convention, the values of the
lower triangle of the matrix must be set to zero.
In a general case, where we have more than one
car in a lane, a conflict matrix will possess a more
complex structure. However, as already mentioned
above, such complex cases are not considered in the
underlying work and will be addressed in one of our
future publications.
3 PROBLEM FORMULATION
This section is the modeling part of the underlying
work. Here, we discuss the tools and techniques we
need to solve the intersection scheduling problem. Let
us briefly discuss the steps we take in this section.
First, we introduce a simple vehicle model and N op-
timal control problems, which maximize the veloc-
ity of each car at the end of the approaching phase.
The choice of such objective function is due to As-
sumption 6. These simple OCPs can be solved an-
Conflict Zone
Lane
Path
Possible paths for each lane
Figure 1: Sample of an intersection geometry. In the upper
part of the figure a classical road intersection is shown. It
consists of 4 entering lanes, 4 exit lanes and 12 possible
paths that are shown in the lower part of the figure. The path
number of the sample vehicle 1 is 12, which corresponds to
P(1) = 12.
alytically by treating the final times (of correspond-
ing approaching phases) as parameters. The results
of this problems are final velocities (6) as functions
of final times. Next, we define scheduling constraints
(7) which are essential for collision avoidance in the
CZ, and state mixed integer nonlinear problem (10),
which gives us an optimal intersection entering se-
quence in terms of total time elapsed. Finally, we
post-regularize the motion of those vehicles that are
subject to velocity saturation.
3.1 Vehicle Model and Motion
For simplicity reasons the selected vehicle model is a
double integrator. The states of each vehicle are given
by its curvilinear coordinates along a predefined path
s
i
(t), i ∈ V , and its derivatives (longitudinal veloci-
ties) v
i
(t), i ∈ V . The coordinates s
i
(t) have to van-
ish at the end of the approaching phase (at the begin-
ning of the intersection phase). The velocity is as-
sumed to be non–negative and bounded from above.
The controls are simply longitudinal accelerations de-
noted by a
i
(t), i ∈ V , which are bounded from below
and above by a
m,i
and a
M,i
, i ∈ V , respectively. Ac-
cordingly, for each i ∈ V the motion model reads as
˙v
i
(t) = a
i
(t),
˙s
i
(t) = v
i
(t)
(2)
with the corresponding box constraints
a
m,i
≤ a
i
(t) ≤ a
M,i
,
0 ≤ v
i
(t) ≤ v
M,i
,
(3)
where a
m,i
, a
M,i
are the control constraints, and v
M,i
denotes the legal speed limit, and the initial conditions
s
i
(t
0
) = −s
0,i
,
v
i
(t
0
) = v
0,i
,
(4)
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
440

Table 1: Quantities and variables.
Symbol Description Sec/Fig
i, j ∈ V Vehicle indices with the corresponding index set 2.1
N Number of vehicles 2.1
k, l ∈ P Path indices with the corresponding index set 2.2
M Number of paths 2.2
P(i) A path corresponding to vehicle i 2.2
c
kl
A k, l entry of a conflict matrix 2.2
s
i
(t) Position vehicle i along its path 3.1
v
i
(t) Velocity of vehicle i 3.1
a
i
(t) Acceleration (control variable) of vehicle i 3.1
t
i
Arrival time of vehicle i 3.1
t
i,min
,t
i,max
Lower and upper bounds of t
i
3.1
v
i
(t
i
) Arrival velocity of vehicle i 3.1
T
i
(t
i
) An inverse of the arrival velocity of vehicle i 3.2
w
i j
Binary decision variable (equals to 1 if vehicle i passes first) 3.2
L
b
i j
Length to travel before overlapping zone of P(i) and P( j) 3.2
L
a
i j
Length to travel to clear overlapping zone of P(i) and P( j) 3.2
L
k
Length of path k in CZ 3.2
where s
0,i
are the distances left to the CZ staring at
time t
0
, and v
0,i
are the corresponding velocities at
time t
0
. Note that the terms s
0,i
, i ∈ V , are posi-
tive, and therefore the sign in front of them should
be inverted.Vehicle geometrical characteristics can be
summarized as follows. The footprint of each vehi-
cle is represented by a rectangle. Each rectangle is
conceived to contain actual planar dimensions of the
corresponding vehicle. The locations given by the
states s
i
(t), i ∈ V , are actually the positions of the ge-
ometrical center of each rectangle. Since our global
goal is to minimize the total time it takes for all vehi-
cles to clear an intersection, and taking into account
Assumption 6, we formulate the following OCPs as
maximization of corresponding final velocities, i.e.
the velocities at which vehicles finish the approach-
ing phase, subject to vehicle model from above. For
each i ∈ V we have
max
a
i
(t)
v
i
(t
i
) (5a)
s.t. (2),(3), (4), (5b)
s
i
(t
i
) = 0, v
i
(t
i
) ≤ v
P(i)
, t ∈ [t
0
,t
i
], (5c)
where t
i
denotes the time at which the i-th car finishes
its approaching phase (starts its intersection phase),
and v
P(i)
is the upper bound on the final velocity. This
upper bound depends on the maximum curvature of
the corresponding path P(i) ∈ P . Due to simplicity of
the underlying model, OCPs (5) can be solved analyt-
ically by treating the times t
i
, i ∈ V , as parameters.
By doing so, one obtains final velocities as functions
of final times
v
i
(t
i
), t
i
∈ [t
i,min
,t
i,max
], i ∈ V , (6)
where t
i,min
denotes the minimal time when the i-th
car can start the intersection phase, and t
i,max
denotes
some reasonable time at which we are sure that the i’s
car will enter the intersection. In Section 3.2 we work
with reciprocal of these functions.
On the one hand, there exists no technique how to
determine sharp upper limits t
i,max
, i ∈ V , and they
should be chosen as a good guess. However, note that
starting from some t
?
i
∈ [t
i,min
,t
i,max
], i ∈ V , the corre-
sponding velocities will be constant. The lower limits
t
i,min
, i ∈ V , on the other hand, can be determined as
a minimal time for which problem (5) becomes feasi-
ble.
In this work we restrict ourselves to final velocity
optimization for each vehicle, however this idea can
be extended to any parametric function of t
i
coming
from an analytical solution of some underlying OCP.
Moreover, if no analytical solution can be found, one
can use a value function approach as e.g. in (Hult
et al., 2018).
3.2 Scheduling Constraints
The goal of scheduling constraints is to ensure that
the vehicle footprints at least do not overlap in a CZ.
By saying ”at least” we mean that we also have to
preserve some safety margin between any pair of cars
whose corresponding paths overlap. Overlaps of ve-
hicle footprints together with a safety margin form a
so-called overlapping zone. Possible choices of over-
lapping zones are discussed in the sequel.
As one can easily judge, scheduling constraints
are safety critical. First, we define these constraints,
A Piecewise Linearization Algorithm for Solving MINLP in Intersection Management
441

and then we give a comprehensive explanation on how
they work. The scheduling constraints read as
t
i
+ L
a
i j
T
i
(t
i
) −t
j
− L
b
ji
T
j
(t
j
) ≤ C
big
(1 − w
i j
),
t
j
+ L
a
ji
T
j
(t
j
) −t
i
− L
b
i j
T
i
(t
i
) ≤ C
big
w
i j
,
w
i j
∈
{
0,1
}
,
∀i, j ∈ V with c
P(i)P( j)
= 1,
(7)
where
T
i
(t
i
) = 1/v
i
(t
i
), t
i
∈ [t
i,min
,t
i,max
], i ∈ V .
The constraints given above are the so-called ”either-
or” constraints (Winston and Goldberg, 2004), due to
the fact that the variables w
i j
, i, j ∈ V , can take only
binary values and the constant C
big
is assumed to be
at least as large as
∑
N
i=1
[t
i
+ T
i
(t
i
)] . Since the latter
quantity is never known in advance, one can choose
just some reasonably large constant.
So, let us now discuss what constraints (7) ac-
tually mean. We consider only the cases where
c
P(i),P( j)
= 1 for some i, j ∈ V , i.e. there is a con-
flict between the paths P(i) and P( j), otherwise there
is just no scheduling constraint. Without loss of gen-
erality let us assume that some w
i j
= 1, which actually
means that the i-th car goes before the j-th one, then
the first line in (7) reads as
t
i
+ L
a
i j
T
i
(t
i
) ≤ t
j
+ L
b
ji
T
j
(t
j
), (8)
where L
a
i j
denotes the length of the path P(i) which
i-th vehicle has to travel in a CZ in order to leave the
overlapping zone with the path of j-th vehicle. The
term L
b
ji
, in turn, denotes the length of the path P( j)
which the j-th vehicle has to travel in a CZ before
entering the overlapping zone with the path of i-th
vehicle. For a better understanding of the quantities
discussed above see Fig. 2. Inequality (8) essentially
states that the i-th vehicle has to leave the overlapping
zone before the j-th vehicle enters it. An analogous
argument holds for the second inequality in (7).
There are different ways how to define an overlap-
ping zone. Let us discuss two limiting cases. The first
one is when an overlapping zone is defined in a way
such that vehicle footprints do not overlap and there is
no additional safety margin, in other words, vehicles
can touch each other at one single point. Of course,
such a choice is not appropriate in any real-world sit-
uation just due to obvious safety issues. So, in a real-
world scenario we need to preserve some safety mar-
gin. Another limiting case is when a safety margin is
the whole CZ. In this case constraints (7) read as
t
i
+ L
P(i)
T
i
(t
i
) −t
j
≤ C
big
(1 − w
i j
),
t
j
+ L
P( j)
T
j
(t
j
) −t
i
≤ C
big
w
i j
,
w
i j
∈
{
0,1
}
,
∀i, j ∈ V with c
P(i)P( j)
= 1,
(9)
where L
P(i)
and L
P( j)
denote the lengths of the P(i)’s
and P( j)’s paths, respectively, see Fig. 2.
3
6
L
b
ij
5
6
L
6
L
5
L
k
L
b
ij
L
4
4
3
6
L
a
ij
L
a
ij
Figure 2: Left and middle: CZ representation with the quan-
tities L
a
i j
and L
b
i j
, respectively. Right: paths 4, 5 and 6 with
the corresponding lengths L
k
.
3.3 MINLP
Now, we are at a position to formulate a mixed in-
teger nonlinear minimization problem. As already
mentioned before, our goal is to minimize the time at
which the last vehicle in our consideration will clear
an intersection. Note that we do not know in advance
which vehicle will be the last one. Therefore, the ob-
jective function is given as a total sum of individual
maneuver durations. The final mixed integer nonlin-
ear problem is given by
min
t
i
,w
i j
N
∑
i=1

t
i
+ L
P(i)
T
i
(t
i
)

,
s.t. (7).
(10)
Note that the problem stated above is a nonlinear
problem, and most of existing software packages, can
handle only linear problems. Therefore, we have to
first linearize the problem, namely the nonlinearities
T
i
(t
i
), i ∈ V , in a way such that a mixed integer linear
problem solver can deal with it. One of possible
linearization techniques is discussed in Section 4.
Once the optimal starting times t
i
, i ∈ V , are
found, we can do a post-regularization step for
the vehicles whose final velocities are saturated by
v
i,M
, i ∈ V . The reason is obvious. If there is no
saturation effect, vehicle’s motion is fully described
by (5). Therefore, there are no additional degrees
of freedom. In the other case, however, there are
infinitely many ways how to reach the CZ at the
prescribed (by the MINLP solution) time. Here, we
choose a simple (but the most comfortable) motion,
namely we minimize the L
2
-norm of the control
variable. The corresponding OCP problem reads as
min
a
i
(t)
1
2
Z
t
opt
i
t
0
a
2
i
(t)dt,
s.t. (5b),(5c),
v
i
(t
opt
i
) = v
P(i)
,
(11)
where in (5c) one has to substitute t
i
by t
opt
i
, which
denotes the optimal staring time (in the global sense),
i.e. t
opt
i
comes from the solution of problem (10).
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
442

4 PIECEWISE LINEARIZATION
As mentioned at the end of Section 3.3, we need to
find some linearization technique which would allow
us to reformulate problem (10) into a linear problem.
A straightforward idea would be to use piecewise lin-
earizations of the nonlinearities T
i
(t
i
), i ∈ V , on some
discretizations of the intervals [t
i,min
,t
i,max
],i ∈ V .
Unfortunately, existing high-performance soft-
ware packages, like for instance GUROBI, which we
use for our computations, do not directly accept piece-
wise linear functions, and therefore, we have to intro-
duce some auxiliary variables and constraints. The
method proposed in this paper is inspired by the lin-
earization technique from (Winston and Goldberg,
2004).
Here, we also have to mention that there exist
other, more sophisticated methods of solving MINLP:
SQP-type algorithms, see e.g (Exler et al., 2012),
or adaptive algorithms, see (Geißler et al., 2012),
(Burlacu et al., 2020). The latter technique is very
similar to the one we exploit in this work, however
due to space restrictions, we do not consider those al-
gorithms in the underlying work.
The idea behind the piecewise linearization ac-
cording to (Winston and Goldberg, 2004) is to use
convex combinations of nonlinear function values at
discretization points, and to let the solver choose the
right segment (where the time variable attains its op-
timal values in the global sense) by introducing some
auxiliary binary variables. As we already did before,
let us first give the formulas, and then discuss them.
For each i ∈ V we have
t
i
=
K
i
∑
k=1
λ
i,k
t
i,k
, (12a)
T
i,pw
=
K
i
∑
k=1
λ
i,k
T
i
(t
i,k
), (12b)
K
i
∑
k=1
λ
i,k
= 1,
K
i
−1
∑
k=1
z
i,k
= 1, (12c)
λ
i,1
≤ z
1
,
λ
i,k
≤ z
i,k
+ z
i,k−1
, k ∈ {2,.. .,K
i
− 1},
λ
i,K
i
≤ z
K
i
−1
,
(12d)
λ
i,k
≥ 0, k ∈ {1, ...,K
i
},
z
i,k
∈
{
0,1
}
, k ∈ {1,.. .,K
i
− 1},
(12e)
where t
i,k
, k ∈ {1, ... ,K
i
}, are some grid points (not
necessarily equidistant) in the interval [t
i,min
,t
i,max
]
with t
i,1
= t
i,min
and t
i,K
i
= t
i,max
, and the constant
K
i
denotes the number of discretization points of the
corresponding time interval. The idea behind (12) is
the following. We introduce the real variables λ
i,k
,
k ∈ {1,..., K
i
}, (lambdas), and the binary variables
z
i,k
, k ∈ {1,. .., K
i
− 1}, (z-variables), where i ∈ V .
The role of the z-variables is to indicate the segment
with the optimal values t
i
∈ [t
i,min
,t
i,max
], i ∈ V . In-
deed, since those variables can take only binary val-
ues and their sum should be equal to one, only one
z-variable will be non-zero. In turn, the role of the
lambdas is nothing else but to build a convex com-
bination over the segment, at which the non-zero z-
variable is pointing. This relation is given by (12d),
which also guarantees that only two lambdas are non-
zero. Additionally, the first sum in (12c) guarantees
that the sum of these two lambdas is one. There-
fore, equalities (12a) and (12b) are linear combina-
tions over the ”optimal” segment.
4.1 MILP Problem
Finally, we can formulate a mixed integer linear prob-
lem as an approximation of (10). It reads as follows
min
t
i
,w
i j
N
∑
i=1

t
i
+ L
P(i)
T
pw,i

,
s.t. (7),(12).
(13)
We solve problem (13) with the GUROBI MILP-
solver (Gurobi Optimization, 2020) which is using
a linear-programming based branch-and-bound algo-
rithm. The next section is devoted to examples and
conclusions.
5 SIMULATIONS
In order to validate the proposed algorithm, we per-
formed simulations in different scenarios, namely, we
consider the following intersection geometries: 8-
lanes intersection, 16-lanes intersection and 10-lanes
irregular intersection as depicted in Fig. 1, 3 and 4,
respectively. For each intersection geometry we con-
struct the corresponding scenario such that it repre-
sents some complex traffic situation in which right-
of-way rule or first-arrive-first-served approaches are
outperformed by our approach. For a better compre-
hension of all scenarios considered in this paper we
encourage you to watch a short video (Gerdts et al.,
2020). Moreover, in order to show the robustness
of the algorithm, for the latter intersection geome-
try we perform 24 simulations with different initial
conditions. In all the upcoming simulations iden-
tical vehicles are considered with a
m,i
= −5 m/s
2
,
a
M,i
= 3 m/s
2
and v
M,i
= 17 m/s, i ∈ V . All sim-
ulations are performed on an Intel quad-core i7 2,9
GHz processor. Before we dive into simulations, we
A Piecewise Linearization Algorithm for Solving MINLP in Intersection Management
443

would like to discuss how grid points for the piece-
wise linearization of the corresponding nonlinearities
T
i
(t
i
), i ∈ V , are chosen. Obviously, we want to mini-
mize the approximation error. In order to do so, some
grid points have to correspond to stationary and kink
points of the nonlinear functions.This ensures both
that all values of the nonlinear functions are repre-
sented by corresponding piecewise linear functions
and that the approximation is exact for the portions
of the nonlinear functions which correspond to sat-
urations. Note thah those portions require only two
grid points in order to be exact. In the simulations we
use 10 grid points, leading to an approximation error
less than one millisecond.
8-lanes Intersection: In this example we consider the
whole conflict zone as an overlapping zone for each
pair of conflicting paths, i.e. scheduling constraints
are given by (9). The initial conditions and optimal
times can be found in Table 2, and the correspond-
ing trajectories in Fig. 5. Vehicles 3 and 4 go before
vehicles 1 and 2 because the latter ones are subject
to velocity limitations due to corresponding maximal
curvatures even though the first pair starts closer than
the second one. Also, the right-turning vehicle goes
before the left-turning one, since the right-turning ve-
hicle’s path is shorter than the one of the other vehi-
cle, even though the left-turning vehicle can perform
its maneuver at a higher velocity.
16-lanes Intersection: This example is constructed
in a way such that it is hard to guess an optimal order
in advance. Moreover, with this example we show
scalability of the proposed algorithm, since the solver
runtime is very similar to the previous case. The in-
tersection geometry is depicted in Fig. 3. The ini-
tial conditions and optimal starting times are given in
Table 2. In this case we define proper overlapping
zones by considering the same footprint for each ve-
hicle with the length of 4 m and the width of 2 m. The
trajectories can be found in Fig. 5. The optimal or-
der is similar to the previous one. Four vehicles, that
have straight paths, go first. The other four vehicles
have to perform pairwise symmetric maneuvers, since
there is not enough room in the intersection to cross it
simultaneously.
10-lanes Irregular Intersection: With this example
we want to demonstrate the performance of the algo-
rithm in a very safety-critical situation at an irregular
intersection. Corresponding geometry, initial condi-
tions and optimal times as well as optimal trajectories
can be found in Fig. 4, Table 2 and Fig. 5, respec-
tively. Also, for such kind of intersection we per-
formed 24 simulations with 4 vehicles. We consid-
ered different combinations of paths and initial condi-
tions. The average solver runtime was 0.0060 s with
1234
5
6
7
8
9 10
11
12
13
14
15
30 m
16
1
5
2
6
3
7
4
8
Figure 3: Geometry of the 16-lanes intersection.
40 m
1
2 3
4
3 2 1
5
6
7
8 9
10
11
Figure 4: Geometry of the irregular 10-lanes.
the maximum of 0.0187 s.
6 CONCLUSIONS AND
OUTLOOK
In this paper we proposed a bi-level optimization al-
gorithm for an intersection management problem. On
the lower level, for each vehicle we considered a sim-
ple OCP, which provided us optimal final velocities as
nonlinear functions of final times. On the higher level,
in turn, we exploited those functions in a MINLP. We
solved that problem by approximating the nonlinear-
ities with piecewise linear functions, which resulted
in a more complex but linear problem. We performed
numerical experiments using GUROBI MILP-solver
to validate the algorithm, which showed that it is ro-
bust and fast. The emerging MINLP can be solved
with a slightly different linearization technique in-
spired by (Burlacu et al., 2020), which gives a con-
0 5 10
-40
-20
0
20
1
2
3,4
1,2,3,4
5 and 7
6 and 8
1
2
3
4
Figure 5: From left to right: 4-lanes, 16-lanes and 10-lanes
solutions trajectories.
VEHITS 2021 - 7th International Conference on Vehicle Technology and Intelligent Transport Systems
444

Table 2: Initial conditions and optimal times for the 8-lanes,
16-lanes and 10-lanes intersections, respectively.
i P(i) s
0,i
v
0,i
t
i
L
P(i)
/v
i
1 7 30 m 10 m/s 3.883 s 2.031 s
2 3 30 m 10 m/s 5.914 s 2.436 s
3 5 35 m 10 m/s 2.706 s 1.176 s
4 11 35 m 10 m/s 2.706 s 1.176 s
Solver runtime 0.0040 s
1 2 30 m 10 m/s 2.244 s 1.195 s
2 6 30 m 10 m/s 2.244 s 1.195 s
3 10 30 m 10 m/s 2.244 s 1.195 s
4 14 30 m 10 m/s 2.244 s 1.195 s
5 4 30 m 10 m/s 6.033 s 2.436 s
6 8 30 m 10 m/s 3.071 s 2.436 s
7 12 30 m 10 m/s 6.033 s 2.436 s
8 16 30 m 10 m/s 3.071 s 2.436 s
Solver runtime 0.012 s
1 6 40 m 10 m/s 3.354 s 1.176 s
2 9 40 m 10 m/s 3.478 s 1.192 s
3 10 40 m 10 m/s 3.680 s 1.213 s
4 3 40 m 10 m/s 4.307 s 2.834 s
Solver runtime 0.0072 s
trol over the error between the nonlinearities and their
approximations. Such linearization technique can be
more efficient in the case of nonlinear functions with
high derivatives.
Our algorithm can be applied to any car model,
even with uncertainties. Just in case if no analyti-
cal solution of the corresponding OCP can be found,
one can exploit a value function approach like in (Hult
et al., 2018). Finally, we would like to announce that
in one of our future works we are going to consider
a more general case with multiple cars in one lane,
where a deep analysis of rear-end collision avoidance
conditions has to be performed.
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