Cone Fields And The Cone Projection Method Of Designing Signal Settings And Prices For Transportation Networks

Abstract

This paper builds on ideas in Smale [13] and Smith et. al. [11, 12]. The paper utilises Smale’s cone fields rather than vector fields to impel disequilibrium steady state traffic-price-green-time distributions; and applies these ideas to the design of steady state signal controls and prices on transportation networks. The work is applied within a multi-modal equilibrium transportation model which contains elastic demands and deterministic choices. The model may readily be extended to include some stochastic route-choice or mode choice. Capacity constraints and queueing delays are permitted; and signal green-times and prices are explicitly included. The paper shows that, under natural linearity and monotonicity conditions, for fixed control parameters the set of equilibria is the intersection of convex sets. Using this result the paper outlines a cone field method of calculating equilibria and also an associated method of designing appropriate values for the control parameters; taking account of travellers’ choices by supposing that the network is in equilibrium. The method is shown to apply to certain non-linear monotone problems by linearising about a current point.
A rigorous proof of convergence to the set of equilibria is provided, for linear and some non-linear monotone problems. But only an outline of a potential proof of convergence to a (flow, control) pair which satisfies a Karush-Kuhn-Tucker necessary condition for local optimality is provided.

Full text

1
CONE FIELDS AND THE CONE
PROJECTION METHOD OF DESIGNING
SIGNAL SETTINGS AND PRICES FOR
TRANSPORTATION NETWORKS
Andrew Battye, Arthur Clune, Mike Smith and Yanling Xiang,
York Network Control Group,
Department of Mathematics,
University of York,
Heslington,
York, YO1 5DD.
mjs7@york.ac.uk
Abstract:
This paper builds on ideas in Smale [24] and Smith et. al. [22, 23].
The paper utilises Smale's cone elds rather than vector elds to impel dise-
quilibrium steady state trac-price-green-time distributions; and applies these
ideas to the design of steady state signal controls and prices on transportation
networks. The work is applied within a multi-modal equilibrium transp ortation
model which contains elastic demands and deterministic choices. The model
may readily b e extended to include some stochastic route-choice or mode choice.
Capacity constraints and queueing delays are permitted; and signal green-times
and prices are explicitly included. The paper shows that, under natural linearity
and monotonicity conditions, for xed control parameters the set of equilibria
is the intersection of convex sets. Using this result the paper outlines a cone
eld method of calculating equilibria and also an associated metho d of design-
ing appropriate values for the control parameters; taking account of travellers'
choices by supposing that the network is in equilibrium. The method is shown
to apply to certain non-linear monotone problems by linearising about a current
point.
A rigorous proof of convergence to the set of equilibria is provided, for linear
and some non-linear monotone problems. But only an outline of a potential
proof of convergence to a (ow, control) pair which satises a Karush-Kuhn-
Tucker necessary condition for local optimality is provided.
1

2
INTRODUCTION
Inspiration and motivation
This paper follows up ideas introduced in Smith [18, 19]. These were partially
inspired by cone-elds introduced in Smale [24]. Smale introduced dynamical
systems whose solution trajectories are not uniquely dened; they merely move
in \roughly" the right direction, rather than exactly the right direction; and
argued that such dynamical systems may be more appropriate for the study of
the evolution of economic systems. The \solution tra jectories" in these systems
have their direction of motion at each point conned to be within a cone, instead
of being conned to be in a precise direction.
The ideas here were also motivated by Von Neumann and Morgenstern [26],
Nash [11], Brown and Von Neumann [4], Nikaido [12] and Arrow and Hurwicz
[1]. Arrow and Hurwicz has proved to be a constantly inspiring reference.
Smith [18] formulated an elastic trac equilibrium as a point
x
which satises
a vector eld
?
f
is normal
;
at
x;
to a convex set
E:
(1.1)
The vector eld in [18] was specied by a rather general network excess demand
function and the convex set was the set of ows all of whose coordinates are
non-negative.
Smith [19] specied the rigid demand equilibrium problem in the form (1.1).
Three purposes of Smith [19] were: (i) to introduce the above, \vector eld
normal to a convex set", variational inequality, formulation of inelastic equilib-
rium, (ii) to show that this formulation leads naturally to excellent existence
and uniqueness results (allowing for some interaction between costs on one link
and ows on others), and (iii) to connect this particular formulation of equilib-
rium with non-equilibrium dynamics in a natural way; partially motivated by
Smale's cone elds. Prior to this paper transportation theorists sometimes con-
sidered \user-optimised" ow patterns as \equilibria"; see, for example, Potts
and Oliver [17]. However this \user-optimised" denition could never hold in
some simple examples with interactions between ows on one link and costs on
another.
But the main aim of Smith [19] was to introduce a control parameter into
trac assignment so that good controls could be sought; and to suggest a
reasonable way of beginning the search for good signal controls. The pap er
essentially introduced the
P
0
method of choosing signal timings; this metho d,
by controlling junction interactions appropriately, guarantees that any feasible
inelastic equilibrium problem has a solution consistent with the signal setting
method. The
P
0
method of choosing the signal timings uses only link ow infor-
mation; and indeed only local link ow information. This method is naturally
decentralised: information concerning spare capacity is transmitted through
the network by the trac ows themselves; this is how local traveller decisions
and signal setting actions are \informed" of distant network conditions. The
P
0
method is a generalisation to the interacting link case of the \cost-ow func-
tion positive" requirement in Beckman [3]; without a condition similar to that

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
3
ensured by
P
0
feasible inelastic problems with interacting costs might have no
feasible equilibrium solution. See Smith [20] for a very simple example.
The purpose of this paper
This paper takes these things forward by relating the cone pro jection method
introduced in Smith et. al. [22, 23] and Clegg and Smith [7] to Smale's cone
elds; and extending proofs of convergence given in our earlier papers.
Relationship to other projections
The cone projections discussed here are slightly similar to those discussed in
Dupuis and Nagurney [8], and Nagurney and Zhang [13, 14, 15]. However there
are some very substantial dierences. The main dierences are: (i) throughout
we have and adjust a control vector
p
, whereas Nagurney and Zhang[op. cit.]
for example do not mention controls; (ii) we utilise just monotonicity and so are
able to apply our results to multi-level models like that of Charnes and Cooper
[6], whereas Nagurney and Zhang require strict monotonicity for the majority
of their results and so applications to multilevel models are (at a rst glance at
least) ruled out (since strict monotonicity cannot hold for these models); and
(iii) the cone eld idea of Smale which we have used either explicitly or implic-
itly allows us to assume that cost functions are just continuously dierentiable
and also permits polygonal paths as solutions, whereas in Nagurney and Zhang
cost functions have to have Lipshitz continuous gradients and solutions must
follow the relevant vector eld precisely, as in solving dierential equations.
For particular purposes some of these dierences may be unimportant but if
\good" controls are to be sought within multi-modal multi-level networks then
some of these dierences would appear to be critical.
ACHIEVING THE COMPLEMENTARITY FORMULATION
The notation adopted is shown below; there is a base network and a multi-
copy (Charnes and Cooper [6]) version of this. Within each copy the travellers
or vehicles all have a single destination node. This network structure is very
similar to that in Smith [21]; here we have chosen to give a route-formulation
in which each copy has links which comprise routes in the base network. In this
paper then a link will always be an element of the base network; links in the
multi-copy network will here be called routes since they are routes constructed
from base network links. Route costs on the copies may have components which
are sums of costs on base network links (link delays will add in this way along
routes) but may also have elements which are non-additive with respect to base
network links; so that bus fares and so on may be represented.
The structure also represents multi-mode networks; ow on the routes in a
single copy may represent travellers using a single mode; or vehicles of a certain
type. Within the framework given here multi-modal eects are most easily
represented by thinking of all ows on the multi-copy network in congestion-

4
Table 1.1
Summary of Notation
Variable Description
X
r
ow along route
r
in multi-copy network
C
n
least cost of reaching the destination from node
n
in the multi-copy
network where
C
n
= 0 if
n
is a destination
b
i
bottleneck delay at the exit of link
i
in the base network
P
r
price to be paid for traversing route
r
Y
k
proportion of time stage
k
is green
s
i
saturation ow at exit of base link
i
(may be innite)
S
r
free ow travel cost along route
r
(
S
r
>
0)
B
nr
1 if node
n
is the entrance node of route
r
in the multi-copy network
(
n
before
r
) and 0 otherwise
N
ik
s
i
if link
i
is in stage
k
and 0 otherwise (
k
= 0
;
1
;:::
)
M
ir
1 if route
r
in the multi-copy network contains link
i
and 0 otherwise
y
i
nominal exit capacity of link
i
, equal to
P
k
N
ik
Y
k
W
n
(
C
) demand at node
n
if the cost to destination vector is
C
g
i
(
b
i
; y
i
) maximum possible average ow out of link
i
when the nominal link
capacity is
y
i
and the bottleneck delay in link
i
is
b
i
causing units and dening the demand function
W
(
:
) in table 1.1 appropriately.
All the links in the multi-copy part of the network will have sux
r
(as here
they are all routes) and base network links will have sux
i
.
For each signal-controlled node or junction there is also a copy comprising
just the base network links terminating at that node: to represent signal stages.
Stage
k
(say) will be a subset of base network links with the same exit node
and which may be shown green at the same time.
Notation
The main notation used is described in Table 1.1. Here
X
r
,
C
n
and
b
i
are
variables to be found in the equilibrium problem,
P
r
and
Y
k
are control vari-
ables,
s
i
and
S
r
are xed, given variables and
B
nr
,
N
ik
and
M
ir
describe the
structure of the multi-copy network.
Following Payne and Thompson [16], the b ottleneck delays
b
i
and the node
costs
C
n
are regarded as indep endent variables. Also if
n
is a destination node
then
C
n
= 0 and
W
n
(
C
) = 0 for all
C
. (The demand for, and the cost of, travel
from a node to itself is zero.)

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
5
All links at non-signalised junctions will comprise just one \stage" (the rst
in the list of stages) shown green for all time and so
Y
1
= 1. Also
N
i
1
is (as in the
general case above) to be
s
i
for all
i
such that link
i
is not signal-controlled and
then the nominal link capacity will be still be
y
i
=
P
k
N
ik
Y
k
=
N
i
1
Y
1
=
s
i
as within the notation list above. The formula above for
g
i
(
b
i
; y
i
) will then
determine the maximum possible ow consistent with this and a bottleneck
delay
b
i
.
Equilibrium, demand and capacity constraints
We use Wardrop's [27] condition: more costly routes carry no ow. But we
choose to write this in the following form: for each route
r
the (least) cost to
the destination from the node
B
(
r
) upstream of route
r
is no more than the
least cost to the destination via route
r
, and if it is less then no ow will enter
route
r
, or
X
n
B
T
rn
C
n
?
S
r
?
P
r
?
X
i
M
T
ri
b
i

0
;
and
X
n
B
T
rn
C
n
?
S
r
?
P
r
?
X
i
M
T
ri
b
i
<
0
)
X
r
= 0
:
Here
P
n
B
T
rn
C
n
is just a way of writing
C
B
(
r
)
; the cost at the no de
B
(
r
)
upstream of route
r
. This sum comprises just the single cost at that node at
the entrance of route
r
.
The (elastic or inelastic) demand constraints may be written as: the total
route-ow
P
r
B
T
rn
X
r
out of node
n
equals the demand
W
n
(
C
). Rewriting this
in a slightly weaker and more articial way we obtain:
W
n
(
C
)
?
X
r
B
nr
X
r

0
;
and
W
n
(
C
)
?
X
r
B
nr
X
r
<
0
)
C
n
= 0
:
This is in fact, under natural conditions (which include the Wardrop condition
above), equivalent to the stronger condition above.
For the capacity constraint condition we suppose here that for any average
bottleneck delay
b
i
and nominal link capacity
y
i
there is a maximum possible
ow
g
i
(
b
i
; y
i
) consistent with the delay
b
i
. Then the capacity constraint may
be written:
X
r
M
ir
X
r
?
g
i
(
b
i
;
X
k
N
ik
Y
k
)

0
;
and
X
r
M
ir
X
r
?
g
i
(
b
i
;
X
k
N
ik
Y
k
)
<
0
)
b
i
= 0
As specied here this condition will ensure that congestion costs normally
represented by a cost-ow function will in fact occur as \bottleneck" delays

6
b
i
which arise from equilibrating via the functions
g
i
. The
g
i
may be thought
of as the inverse of given cost-ow functions. Given a nominal link capacity
y
i
;
g
i
(
b
i
; y
i
) delivers a (largest) ow compatible with a given cost or delay
b
i
,
instead of delivering a cost
b
i
for each ow. This formulation allows for explicit
capacity constraints: for example we may set
g
i
(
b
i
; y
i
) =
y
i
(independent of
b
i
) in which case we obtain the simplest strictly capacitated Payne/Thompson
model. This may be thought of as having a \vertical" cost-ow \curve" but
the inverse cost-ow functions
g
i
(
:; y
i
) are then at and may b e exp ected to
have numerical advantages. The inverse delay-ow or cost-ow curves may
be expected to have numerical advantages even if the delay-ow or cost-ow
function is just steep, rather than vertical; because the
g
i
are then shallow. It
is convenient and natural to think of
b
i
as the delay at the link exit and
S
r
as the \free-ow" cost of traversing route
r
when there are no link-exit delays.
There is no real problem in adding increasing functions of link ows to the
S
r
.
We do not do this for simplicity only.
Greater generality is entirely possible in this mo del. The main thing to
preserve if the later results are to hold is monotonicity of
f
below.
The complementarity formulation
First we dene network response functions
f
1
r
,
f
2
n
,
f
3
i
as follows:
?
f
1
r
(
X; C; b; Y ; P
) =
X
n
B
T
rn
C
n
?
S
r
?
P
r
?
X
i
M
T
ri
b
i
;
?
f
2
n
(
X; C; b; Y ; P
) =
W
n
(
C
)
?
X
r
B
nr
X
r
;
?
f
3
i
(
X; C; b; Y ; P
) =
X
r
M
ir
X
r
?
g
i
(
b
i
;
X
k
N
ik
Y
k
)
;
and for xed
Y; P
; writing
f
1
r
for
f
1
r
(
X; C; b
) etc; we rewrite the conditions
above as:
?
f
1
r

0 and
?
f
1
r
<
0
)
X
r
= 0
;
?
f
2
n

0 and
?
f
2
n
<
0
)
C
n
= 0
;
?
f
3
i

0 and
?
f
3
i
<
0
)
b
i
= 0
:
Then we put
x
= (
X; C; b
),
p
= (
Y; b
) and
f
(
x; p
) = (
f
1
(
x; p
)
; f
2
(
x; p
)
; f
3
(
x; p
))
to obtain:
x
2
IR
n
+
and
?
f
(
x; p
) is normal
;
at
x;
to IR
n
+
; or (1.2)
x
2
IR
n
+
;
?
f
(
x; p
)

0; and
?
f
i
(
x; p
)
<
0
)
x
i
= 0
:
(1.3)
(The
i
and the
n
are \new" suces and are unrelated to the previous
i
and
n
.)
These conditions are similar to a Tobin economic model; are of the form
(1.1), and suggest \following" a cone eld like the \half-space eld" in Smith
[19].

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
7
The equilibrium set (with xed
p
) as the intersection of convex sets
Let
p
be a feasible control vector (in IR
m
) xed for the moment. The feasibility
and equilibrium conditions (1.2) or (1.3) are plainly equivalent to:
x
i

0
; i
= 1
;
2
;:::;n;
?
f
i
(
x; p
)

0
; i
= 1
;
2
;:::;n;
(1.4)
E
=
X
i
x
i
f
i
(
x; p
)

0
:
The rst two of these contraints will be called feasibility constraints. Supp ose
that for each xed
p
the function
f
(
:; p
) is monotone and linear in
x
. Then
f
(
x; p
) =
A
p
x
+
a
p
for some square positive semi-denite matrix
A
p
and some
n
-vector
a
p
which both depend on
p
. Since
A
p
is positive semidenite (that is
x
T
A
p
x

0 for each
n
-vector
x
),
x
T
A
p
x
is a convex function of
x
(the positive
semi-denite 1
=
2(
A
p
+
A
T
p
) is the Hessian of this function); and so
x
T
f
(
x; p
)
is a convex function of
x
for each xed
p
. By linearity each
?
f
i
is convex too.
Thus each inequality in (1.4) species a convex set and the set of equilibria is
the intersection of the 2
n
+ 1 convex sets in (1.4).
Including control constraints
Since we wish to vary
p
in IR
m
we need constraints on the set of possible
p
values. Let these constraints be
e
j
(
p
)

0. (Capacity constraints, which will
sometime involve
x
and
p
together, are included in the conditions
?
f
i

0 in
(1.4).) Now we let
e
j
(
x; p
)

0 (for
j
= 1
;
2
;:::;J
) stand for
x

0; and the
\old"
e
j
(
p
)

0 and
h
k

0 stand for
?
f
i
(
x; p
)

0 and
E
=
P
x
i
f
i
(
x; p
)

0 .
Then the control-augmented set of constraints is:
e
j
(
x; p
)

0 for
j
= 1
;:::;J
and
h
k
(
x; p
)

0 for
k
= 1
;
2
;:::;K:
(1.5)
Assumptions required
1. All
f
i
are dierentiable, their gradients in (
x; p
) space are continuous and
non-zero and the
e
j
are all linear,
2.
f
is monotone in
x
for each xed
p
,
3.
f
is linear (or, more properly, ane).
MIN-MAX CONDITION FOR EQUILIBRIUM
Given a feasible (
x; p
) put
M
(
x; p
) = max
f
0
; h
k
(
x; p
);
k
= 1
;:::;K
g
= max
f
h
k
(
x; p
);
k
= 1
;:::;K
g
Then a feasible (
x; p
) satises all the constraints in (1.5) if and only if
M
(
x; p
) = 0; or
x
minimises
M
(
:; p
) and
M
(
x; p
) = 0
:
(1.6)

8
So to nd an equilibrium we need only: (i) nd a feasible (
x; p
) i.e. one which
satises
e
j
(
x; p
)

0 for
j
= 1
;:::;J
; (ii) maintain this feasibility throughout;
(iii) reduce
M
(
x; p
) to a minimum; and (iv) check that this
M
(
x; p
) = 0.
For each xed
p
let MM
p
= inf
f
M
(
x; p
); (
x; p
) is feasible
g
.
Theorem 1
Let the assumptions (1,2,3) above hold, and let
M
(
x; p
)
>
MM
p
.
Then there are directions at
(
x; p
)
which reduce
M
(
x; p
)
and maintain feasibil-
ity.
Proof.
Since
f
is linear and monotone all the constraints in (1.5) are convex
and so
M
(
x; p
) is convex in
x
for each xed
p
. Since
M
(
x; p
)
>
MM
p
there
is a feasible (
y; p
) such that
M
(
y; p
)
< M
(
x; p
). Since
M
(
:; p
) is convex and
the feasibility constraints are convex the line joining (
x; p
) and (
y; p
) is an
M
-
reducing feasible direction. Thus there are
M
-descent directions at (
x; p
) which
preserve feasibility. Q.E.D.
Now suppose
f
is monotone but non-linear. Linearise
f
at
x
and let ML(
x
0
;
x; p
)
denote
M
(
x; p
) for the new
f
, linearised at (
x
0
; p
0
); and
MML(
x
0
; p
) = inf
f
ML(
x
0
;
x; p
); (
x; p
) is feasible
g
:
If assumption 1 holds, a direction is a feasible direction which reduces
M
(
x
0
; p
)
>
0 if and only if it is a direction which reduces ML(
x
0
;
x; p
) at
x
=
x
0
; because
all functions are dierentiable and their gradients are continuous. This yields
the following theorem.
Theorem 2
Let assumptions 1 and 2 hold and let
ML(
x
0
;
x
0
; p
)
>
MML(
x
0
; p
)
.
Then there are directions at
(
x
0
; p
)
which reduce
M
(
x
0
; p
)
and maintain feasi-
bility.
Proof.
By theorem 1 there is a direction which reduces ML(
x
0
;
x; p
) at
x
=
x
0
. This direction will then be a feasible descent direction for the original
M
(
:; p
) at
x
0
because all gradients are continuous, using the remark preceding
the statement of the theorem. Q.E.D.
Corollary
Let assumptions 1 and 2 hold and suppose that each linearised
problem has a solution. Then away from equilibrium there are feasible direc-
tions which reduce
M
(
:; p
).
Proof.
The existence of solutions to the linearised problems implies that
ML(
x
0
;
x
0
; p
)
>
MML(
x
0
; p
) away from a solution of (1.5) and theorem 2
then says that there are directions which reduce
M
(
:; p
) and maintain feasi-
bility. Q.E.D.
A CONE-FIELD METHOD OF CALCULATING EQUILIBRIA
In the light of theorem 1 (and theorem 2) it becomes natural to dene the
cone eld
F
to be the function which assigns to any feasible (
x; p
) the cone of
directions in which
M
(
x; p
) does not increase at (
x; p
).

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
9
Suppose (
x
(
t
)
; p
(
t
)) is dened for all
t

0 and has a right derivative ( _
x
(
t
)
;
_
p
(
t
))
at all
t

0. Suppose further that
_
M
((
x
(
t
)
; p
(
t
))
<
0
8
t

0 such that( _
x
(
t
)
;
_
p
(
t
))
6
= 0
:
(1.7)
In this case we shall call the tra jectory (
x
(
:
)
; p
(
:
)) an assignment pro cess fol-
lowing Smith [18] who followed Smale [24].
Thus an assignment process is a path along which
_
M <
0. This notion
of an assignment process is a generalisation of the solution of a dierential
equation. It is very useful here b ecause (i) it p erhaps more precisely reects
travellers likely choices on the basis of current information by being less precise
and (ii) it permits a wide class of \solution trajectories", including polygonal
trajectories to be dened b elow, even if the underlying cone-eld is not very
smooth. We are able here to get away with just continuity of the constraint
gradients rather than standard Lipshitz continuity ordinarily imposed on vector
elds to ensure the existence of solutions of dierential equations. Our cone
eld is just piecewise continuous.
Given
D
(
x; p
)
2
F
(
x; p
), for all (
x; p
), satisfying:
_
M
((
x; p
) +
tD
(
x; p
))
<
0
for 0

t <
1 if the relative interior of
F
(
x; p
) is nonempty; and
D
(
x; p
) = 0 if
the interior of
F
(
x; p
) is empty. Given also a starting point (
x
0
; p
0
) we may then
dene a polygonal assignment process which b egins at (
x
0
; p
0
) and then goes to
(
x
1
; p
1
)
;
(
x
2
; p
2
)
;:::
where (
x
1
; p
1
) = (
x
0
; p
0
) +
D
(
x
0
; p
0
) etc. By construction:
M
(
x
0
; p
0
)
> M
(
x
1
; p
1
)
> M
(
x
2
; p
2
)
> M
(
x
3
; p
3
)
>

unless the process stops. Usually we obtain an innite sequence
f
(
x
n
; p
n
)
g
.
Gain
The gain
w
(
x; p
) at (
x; p
) is dened by
w
(
x; p
) =
M
(
x; p
)
?
M
((
x; p
) +
D
(
x; p
)).
COMPLETE ASSIGNMENT PROCESSES
Smale dened a complete process as one such that if (
x; p
) was approached
by it then there would automatically be no non-trivial process starting from
(
x; p
). In our context an assignment process will be complete if it is such that
any point (
x; p
) approached by it is automatically an equilibrium.
A class of complete assignment processes
Theorem 3
Let
D
(
:; :
)
satisfy the above condition and also be such that the
gain
w
(
x; p
)
is a continuous function of
(
x; p
)
. Then any bounded assignment
process
f
(
x
n
; p
n
)
g
generated by
D
(
:; :
)
will be complete. (These conditions may
be made weaker. In particular it is sucient to suppose that
w
is bounded below
by a continuous function which is positive at non-equilibria.)
Proof.
Let
D
be such that
w
is continuous. It is automatically true that
w
will be positive at non-equilibria. Let an assignment process arising from
D
be

10
bounded. Then if it has only nitely many positive steps it must terminate at
(
x
m
; p
m
) (say) where
D
(
x
m
; p
m
) = 0 and so
w
(
x
m
; p
m
) = 0 and we are at an
equilibrium.
On the other hand let the b ounded polygonal path contain an innite number
of corners
f
(
x
n
; p
n
)
g
. These corners are all distinct as
M
decreases. Hence the
set
f
(
x
n
; p
n
)
g
is innite, and being bounded has a limit point (
x
0
; p
0
). We shall
show that such a limit point is an equilibrium.
Suppose that (
x
0
; p
0
) is a limit point and that this (
x
0
; p
0
) is not an equi-
librium. Then the interior of
F
(
x
0
; p
0
) is non-empty,
D
is non-zero and so
M
(
x
0
; p
0
)
?
M
((
x
0
; p
0
) +
D
(
x
0
; p
0
)) =
w
(
x
0
; p
0
)
>
0
:
Since (
x
0
; p
0
) is a limit point there is a subsequence of
f
(
x
n
; p
n
)
g
converging
to (
x
0
; p
0
). Call this subsequence
f
(
x
n
; p
n
)
g
. Now
w
(
x
n
; p
n
) tends to
w
(
x
0
; p
0
) =
w
0
>
0 as
n
tends to innity since
w
is continuous. Hence there is an
n
0
such
that
w
(
x
n
; p
n
)

w
0
=
2
>
0 for all
n

n
0
.
Thus
M
is reduced by at least
w
0
=
2
>
0 between (
x
n
; p
n
) and (
x
n
+1
; p
n
+1
)
for
n
=
n
0
,
n
0
+ 1,
:::
,
n
0
+
k
?
1 (for any choice of
k
). Over
k
steps
M
is thus
reduced by at least
kw
0
=
2 which may be made more than
M
(
x
0
; p
0
) (the initial
value of
M
) by choosing
k
suciently large. However
M
decreases at each step
and so if
k
is so chosen then
M
(
x
n
0
+
k
; p
n
0
+
k
)
< M
(
x
n
0
; p
n
0
)
?
kw
0
=
2
< M
(
x
0
; p
0
)
?
kw
0
=
2
<
0
:
However M is intrinsically positive and so we have a contradiction.
This contradiction arises from the assumption that
f
(
x
n
; p
n
)
g
has a limit
point (
x
0
; p
0
) which is a feasible non-equilibrium. Thus all limit points of the
sequence
f
(
x
n
; p
n
)
g
must be equilibria and the assignment pro cess dened by
D
is complete. Q.E.D.
Notes.
1. For practical implementation
D
(
x; p
) must be specied.
2. Only continuity at the limit point (
x
0
; p
0
) is required.
3. Even then continuity is only required with reference to a subsequence of
points of the original
f
(
x
n
; p
n
)
g
; this will be important later.
THE CONE-PROJECTION METHOD
Now we specify a
D
(
x; p
) which not only ensures convergence to an equilibrium
(that is gives rise to a complete assignment process) but also seeks to do the
best for any given smooth objective function.
Suppose given a smooth (continuously dierentiable) ob jective function
Z
=
Z
(
x; p
); where
x
is the vector of ows, delays and costs, and
p
is the vector of
signal green-time proportions and prices (including any feasible road prices).

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
11
The general form of \the cone-pro jection method" is to begin at any feasible
(
x; p
) and continually follow a polygonal path which at each step follows a di-
rection
D
which reduces
M
(
x; p
), while \approximately doing the best for" the
given
Z
. As motivated here such a tra jectory may (under natural conditions)
be expected to converge to equilibrium and a weak variety of local-optimality
simultaneously.
Let (
x; p
) be feasible but not an equilibrium. If
h
k
(
x; p
)

M
(
x; p
)
=
2
>
0 we
will say that constraint
k
is very violated; otherwise it is not very violated.
Let
C
0
(
x; p
) be the cone of directions in (
x; p
)-space which do not cause any
non-
h
constraint (in (1.5)) to become violated and let
C
1
(
x; p
) be the cone of
those directions at (
x; p
) along which no very violated
h
-constraint (in (1.5))
becomes more violated. Then
C
1
(
x; p
) is the cone of directions (in (
x; p
)-space)
along which
_
h
k
(
x; p
)

0 if
h
k
(
x; p
)

M
(
x; p
)
=
2. Directions in the relative
interior of
C
1
(
x; p
) (int
C
1
) reduce the violation of all very violated constraints
in (1.5), at (
x; p
), simultaneously and so reduce
M
.
Let the vector
d
(
x; p
) be the \centre-line" of the cone
C
0
(
x; p
)
\
C
1
(
x; p
) (the
zero vector if and only if the cone is empty), desc
Z
(
x; p
) =
?
Z
0
(
x; p
)
=
jj
Z
0
(
x; p
)
jj
and desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
(
C
1
(
x; p
)) be desc
Z
(
x; p
) projected onto the cone
C
0
(
x; p
)
\
C
1
(
x; p
).
The cone projection method follows the assignment process generated by
D
where
D
has the following form:
D
(
x; p
) =
D

(
x; p
)
=
d
(
x; p
) +

desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
C
1
(
x; p
))
:
(1.8)
At each step

and

(both p ositive) are to b e chosen so that
M
decreases
each step. Now
D
(
x; p
) is non-zero unless it is forced to be zero by having
d
(
x; p
) = 0 (and hence
M
(
x; p
) = 0) and also desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
C
1
(
x; p
)) =
0 and hence desc
Z
(
x; p
) is normal at (
x; p
) to
C
0
(
x; p
)
\
C
1
(
x; p
); and so a
trajectory following
D
may be continued (it has a direction to go in) unless
both the above conditions hold.
The solution method is thus in outline to follow a direction
D
(
x; p
) at each
(
x; p
). This is intended to be a renement of the bi-level method proposed in
Smith et. al. [21, 22, 23]; replacing half-spaces with cones to narrow the search
region and reduce numerical/computational problems.
So far, however we have not sp ecied the centre-line
d
(
x; p
) and we have
not dened desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
C
1
(
x; p
)). The \centre-line" of the cone
C
0
(
x; p
)
\
C
1
(
x; p
) =
C
(
x; p
) is obtained by solving the problem
P
(
x; p
) shown
below (for simplicity of notation (
x; p
) is omitted from the statement of the
problem) :
Problem
P
(
x; p
) Minimise
jj
P

i
(
?
e
0
i
) +
P

k
(
?
h
0
k
)
jj
subject to
1.

i

0
8
i
,

i
= 0 if
e
i
<
0 and if there is a
j
such that
e
j
= 0 then
P

i
= 1 and,
2.

k

0
8
k
,

k
= 0 if
h
k
< M=
2 and if there is a
j
such that
h
j
> M=
2
then
P

k
= 1.

12
Now at each feasible non-equilibrium (
x; p
) let
d
(
x; p
) =
P

i
(
?
e
0
i
(
x; p
)) +
P

k
(
?
h
0
k
(
x; p
)) where

and

solve
P
(
x; p
). By virtue of theorem 1 ab ove
d
(
x; p
) is non-zero at non-equilibria.
Consider also problem
Q
(
x; p
) below.
Problem
Q
(
x; p
) Minimise
jj
desc
Z
+
P

i
(
?
e
0
i
) +
P

k
(
?
h
0
k
)
jj
subject to
1.

i

0 and

i
= 0 if
e
i
<
0, and
2.

k

0 and

k
= 0 if
h
k
< M=
2.
Now desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
C
1
(
x; p
)) is dened by
desc
Z
(
x; p
)
j
(
C
0
(
x; p
)
\
C
1
(
x; p
)) = desc
Z
(
x; p
) +
X

i
(
?
e
0
i
(
x; p
))
+
X

k
(
?
h
0
k
(
x; p
))
where

and

solve
Q
(
x; p
).
Direction
D
(
x; p
) may now be dened as a weighted sum of these two vectors
as in (1.8) This generalises the corresponding direction in Smith et. al. [21,
22, 23] which involved projections onto half-spaces. Smith et. al. [22] give an
initial result of the method using half-spaces.
A SIMPLE METHOD
Suppose a feasible starting point (
x
0
; p
0
) is given,
D
(
x; p
) is given as in (1.8)
above, and
w
is continuous. Let (
x
1
; p
1
) = (
x
0
; p
0
) +
D
(
x
0
; p
0
) etc. as before.
Thus we follow a standard \
M
-reducing" p olygonal tra jectory. We obtain a
(usually innite) sequence
f
(
x
n
; p
n
)
g
.
An outline proof that the simple metho d converges to the set of equilibra.
The proof relies on and is similar to theorems 1 and 2. Let the p olygonal
path be bounded. Then if it has only nitely many steps it must terminate at
(
x
m
; p
m
) (say) where
D
(
x
m
; p
m
) is zero. At such a point
w
= 0 and (
x
m
; p
m
)
is an equilibrium.
On the other hand let the bounded p olygonal path have an innite number
of corners
f
(
x
n
; p
n
)
g
, where
n
= 0
;
1
;:::
. Since the path is bounded the set
f
(
x
n
; p
n
)
g
, b eing innite (no duplicates as
M
declines), has a limit point (
x
0
; p
0
).
We shall show that every such limit point is an equilibrium.
Suppose that (
x
0
; p
0
) is any such limit point and that this (
x
0
; p
0
) is not an
equilibrium. For each
n
let
A
(
n
) be the set of those suces
i
and
k
such that
h
k
and
e
i
occur in
C
0
(
x
n
; p
n
)
\
C
1
(
x
n
; p
n
). (These suces correspond to active
constraints.) Then there is at most a nite number of possible
A
(
n
) and some
A
(
n
) must be rep eated innitely often. Consider a subsequence of
f
(
x
n
; p
n
)
g
which converges to (
x
0
; p
0
) and for which
A
(
n
) =
A
(say) is always the same
set of active suces. Call this subsequence
f
(
x
n
; p
n
)
g
.
Then in this subsequence
C
0
(
x
n
; p
n
)
\
C
1
(
x
n
; p
n
) all involve the same con-
straints and the constraint functions are continuous; and so
f
C
0
(
x
n
; p
n
)
\

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
13
C
1
(
x
n
; p
n
)
g
converges (in an obvious sense) to
C
0
0
\
C
0
1
(say) with again the
same active constraints. Now, since the constraint functions are continuous,
C
0
(
x
0
; p
0
)
\
int
C
1
(
x
0
; p
0
)

C
0
0
\
int
C
0
1
so (
x
0
; p
0
) being a non-equilibrium implies
C
0
(
x
0
; p
0
)
\
int
C
1
(
x
0
; p
0
) is non-empty
which implies
C
0
0
\
int
C
0
1
is non-empty.
Hence
M
must decline in direction
D
A
(
x
0
; p
0
) determined as
D
(
x
0
; p
0
) is
determined but using only those
e
and
h
constraints with suces in
A
.
Let
w
be the greatest reduction in
M
possible in direction
D
A
(
x
0
; p
0
) begin-
ning at (
x
0
; p
0
). Then
w
(
x
0
; p
0
)
>
0 and it follows that, since
w
(
x
n
; p
n
) tends to
w
(
x
0
; p
0
) as
n
tends to innity, there is an
n
0
such that
w
(
x
n
; p
n
)

w
(
x
0
; p
0
)
=
2
for all
n

n
0
.
Thus
M
is reduced by at least
w
0
=
2 between (
x
n
; p
n
) and (
x
n
+1
; p
n
+1
) for
n
=
n
0
; n
0
+ 1
;:::;n
0
+
k
?
1 (for any choice of
k
). Over
k
steps
M
is thus
reduced by at least
kw
0
=
2 which may be made more than
M
(
x
0
; p
0
) (the initial
value of
M
) by choosing
k
suciently large. However
M
decreases at each step
and so if
k
is so chosen then
M
(
x
n
0
+
k
; p
n
0
+
k
)
< M
(
x
n
0
; p
n
0
)
?
kw
0
=
2
< M
(
x
0
; p
0
)
?
kw
0
=
2
<
0
:
M
is intrinsically positive and so we have a contradiction.
This contradiction arises from the assumption that
f
(
x
n
; p
n
)
g
has a limit
point (
x
0
; p
0
) which is a non-equilibrium. Thus all limit points of the sequence
f
(
x
n
; p
n
)
g
must be equilibria.
PROBLEMS WITH THE SIMPLE METHOD
The step lengths are chosen solely to reduce
M
to zero and so there is no reason
to think that the polygonal path will converge to a minimum of
Z
within the
set of equilibria.
This problem may be resolved in several ways but here we propose to inter-
rupt the previous
M
reducing scheme periodically and to follow a modied di-
rection with a constant small step length aiming to reduce
Z
. This interruption
may instead use a standard constrained minimisation procedure: minimising
Z
subject to a relaxed equilibrium condition, allowing
M
to increase somewhat
as
Z
is minimised. Thus we come by the implementation proposed below.
Implementation and outline justication
We have dened
M
(
x; p
) but now we put
N
(
x; p
) =
jj
desc
Z
(
x; p
)
j
(
C
o
(
x; p
)
\
C
1
(
x; p
)
jj
=
jj
desc
Z
(
x; p
) +
X

i
(
?
e
0
i
(
x; p
)) +
X

k
(
?
h
0
k
(
x; p
))
jj
N
(
x; p
) is a measure of the degree to which (
x; p
) departs from satisfying a
Karush-Kuhn-Tucker (KKT) condition. (We say that (
x; p
) satises a KKT
condition i
N
(
x; p
) = 0)

14
We follow the two-stage method in Smith et. al. [21] and Clegg and Smith
et. al. [7]. Beginning at (
x
0
; p
0
), let
M
0
=
M
(
x
0
; p
0
)
>
0 and
N
0
= 1. In
the rst stage we follow the polygonal path above each step of which b egins
in direction
D
(
x; p
) given by (1.8); until
M
(
x; p
)

M
0
=
4. Then in the second
stage we follow
D

given by (1.8), with
=
small, so that eventually
N
(
x; p
)

N
0
=
2 and
M
(
x; p
)

M
0
=
2
:
This will be a polygonal path and both

and

must be chosen so that both
conditions hold at the termination of stage 2. We are condent that this may
be done by choosing
=
suciently small but we have no rigorous proof as
yet.
Repeating these two stages yields a sequence
f
(
x
n
; p
n
)
g
. Let (
x

; p

) be any
limit point of this sequence. Then
1.
M
(
x
n
; p
n
)

M
0
=
2
n
and so
M
(
x

; p

) = 0 and (
x

; p

) belongs to the
equilibrium set; and
2.
N
(
x; p
)

N
0
=
2
n
and it follows by continuity that (
x

; y

) is an asymp-
totic KKT point.
Conjecture and alternative second stage
We conjecture that for appropriate

,

the second stage of the algorithm does
halve
N
(at each iteration) without losing control of
M
. An alternative here
would be to use a constrained minimisation algorithm instead of the direction
D

above.
CONCLUSION
The paper has outlined a metho d for calculating signal timings and prices
in an urban transportation network which uses directions conned to cones.
The method applies to linear monotone multi-modal deterministic elastic and
inelastic problems; and may be extended to include stochastic elements. The
linearity may be relaxed for some problems by linearising the given problem to
establish the non-emptyness of certain cones. Monotonicity is however essential.
We have proved convergence to equilibria and we have given an outline
justication of the method; suggesting that the method yields an equilibrium
(
x; p
) which satises an asymptotic Karsh-Kuhn-Tucker condition or a Karsh-
Kuhn-Tucker condition. This is a weak necessary condition: a truly optimal
(
x; p
) must satisfy it.
Further work is needed (i) to assess the practical eciency of the cone projec-
tion method, comparing with other design methods, (ii) to convert the \outline
justication" given here to a rigorous proof of convergence to a local optimum
or at least a critical point, and (iii) to relax certain other conditions sp ecied
in this paper.

THE CONE PROJECTION METHOD OF DESIGNING SIGNAL SETTINGS AND PRICES
15
Acknowledgments
We are grateful for nancial support from: DETR, EPSRC, ESRC, DGVII (E2), and
for the support from our LINK partners at UCL and MVA.
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