LINEBURG


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A brief description of the computing method is discussed here. First, given
the approximated control to solve the state equation, obtain the state values
taken at the grid-points of the subintervals; second solve the co-state equation
with respect to thirdly, calculate the augmented Lagrangian; and lastly,
calculate the gradients of the cost function respect to This algorithm had
been programmed in FORTRAN. Note that jumps in co-states are not yet im-
plemented. Unconstrained minimization is done using the CONMIN package
[70], which uses the BFGS quasi-Newton algorithm [28, 1980].
Using the augmented Lagrangian allows gradients to be calculated by a sin-
gle adjoint differential equation, and instead of having to code a separated
formula for each constraint. In this algorithm, a time interval [0, T] is divided
into equal subintervals. A diversity of control problems can be solved by this
method, but the equality constraint like presents some
12 OPTIMAL CONTROL MODELS IN FINANCE

problems with OCIM. Several famous models were successfully solved by this
method. They are a damped oscillator (the approaches developed in this re-
search also solved an example of an oscillator problem), the Dodgem problem
[29, 1975], and the Fish problem [12, 1976]. In Craven™s paper, there was
a useful non-linear transformation of the time scale that can effectively give a
good subdivision in big time ranges, without requiring a large number of subdi-
visions, and also avoid computing problems. The details of the transformation
will be described in Section 2.3.

7.2 RIOTS 95
Relating to another computing optimal control software package SCOM,
which will be introduced in Section 1.7.5, the RIOTS (Recursive Integration
Optimal Trajectory Solver 95) package [77, 1997] also runs on MATLAB . It
solves optimal control problems by solving (in various ways) the differential
equations that lead to function values and gradients, and then applying a mini-
mizer package. The computing scheme is similar to the one that was described
by Craven [14, 1995] in Section 6.4.5. This package obtains good accuracy
of the switching times by using a sufficiently small subdivision of the time
interval.

7.3 A computational approach for the cost of changing
control
A computational method based on the control parameterization technique
[84, 1991] was introduced in reference [82, 1992] for solving a class of optimal
control problems where the cost function is the sum of integral cost, terminal
cost, and full variation of a control. The full variation of a control is used to
measure changes on the control action.
This method involves three stages of approximation. In the first stage, the
problem is approximated by a sequence of approximation problems based on
the control parameterization technique. Every approximation problem involv-
ing full variation of control is an optimal parameter selection problem. They
become non-smooth functions in the form of “ norm. In the second stage,
the non-smooth functions are smoothen by a smoothing technique [83, 1988].
Another smoothing technique [42, 1991] is used to approximate the continuous
state inequality constraints by a sequence of conventional canonical constraints
in stage three. All these approximation problems are standard optimal param-
eter selection problems which can be solved by the general optimal control
software package MISER3 [40, 1991]. The method is supported by the con-
vergence properties. A calculation of a fishery harvesting model (which was
computed in reference [41, 1990] with penalty was involved to illus-
trate the method, and the chosen value penalty term was explained in this
Optimal Control Models 13

paper. A similar situation is also discussed in this book, which is called the
chosen cost, in Section 2.8. An application of an optimal control problem in
which the cost function does not have the full variation of control can also
utilize this method by adjusting the penalty constant appropriately to obtain a
smoother control without changing the optimum of the original cost function.
There are some relevant references [55, 1987], [2, 1976], [66, 1977] for this
problem.

7.4 Optimal switching control computational method
In Li [53, 1995], a computational method was developed for solving a class
of optimal control problems involving the choice of a fixed number of switch-
ing time points which divide the system™s time horizon [0, T] into
K time periods. A subsystem is selected for each time periods
from a finite number of given candidate subsystems, which run in that time
period. All the K subsystems and K “ 1 switching times will be taken as de-
cision variables. These decision variables form candidate selection systems ,
which will lead the optimal control problem to a minimum. Here, the optimal
control problem is only considered over a finite time horizon. In this method,
the original problem is transformed into an equivalent optimal control problem
with system parameters by using some suitable constraints on the co-efficient
of the linear combination, which is formed by the candidate subsystems, and
using a time re-scaling technique.
Many control problems related to the system dynamics that are subject to
sudden changes can be put into this model. In recent years, general optimal
switching control problems have been studied. The optimality principle is ap-
plied and existence of optimal controls is discussed. Basically the problems
are formulated as optimal control problems , in which the feasible controls are
determined by appropriate switching functions. There are relevant references
available in the bibliography [53, 1995], [87, 1989], [88, 1991], [27, 1979].
This situation has problems involving both discrete and continuous deci-
sions represented by the subsystems and switching times. A transformed tech-
nique is introduced for solving this mixed discrete and continuous optimal con-
trol problem. The basic idea behind this technique is transforming the mixed
continuous-discrete optimal control problem into an optimal parameter selec-
tion problem [84, 1991], which only deals with continuous decision variables.
Since the transformed problem still involves the switching times located within
subdivisions, which make the numerical solution of such an optimal control
problem difficult. Another transformation is introduced to further transform
the problem into an optimal control problem with system parameters. The
control is taken as the lengths of the switching intervals as parameters. The
second equivalent optimal control problem can be solved by standard optimal
14 OPTIMAL CONTROL MODELS IN FINANCE

control techniques. A numerical problem was solved in Li™s paper [53, 1995]
by using this computational method.
This kind of optimal control problem has sudden changes in the dynamics at
switching time, and therefore has a mixed continuous-discrete nature. Switch-
ing times, a time scaling and a control function are introduced to deal with
the discontinuities in the system dynamics. The control function is a piece-
wise constant function with grid-points corresponding the discontinuities of
the original problem, hence allowing the general optimal control software to
solve the problem.

7.5 SCOM
In Craven and Islam [18, 2001] (See also Islam and Craven [38, 2002]), a
class of optimal control problems in continuous time were solved by a com-
puter software package called SCOM, also using the MATLAB system. As
in the MISER [33, 1987] and OCIM [15, 1998] packages, the control is ap-
proximated by a step-function. Because of the matrix features in MATLAB,
programming is made easier. Finite difference approximations for gradients
give some advantages for computing gradients. In this paper, end-point con-
ditions and implicit constraints in some economic models are simply handled.

Consider an optimal control problem of the form:




subject to:




The end-point term and constraints can be handled by a penalty term; its de-
tailed description will be introduced in Section 4.4. The concern here is only
with how this computational algorithm works. The differential equation (1.39)
with initial condition, determines from Denote
The interval [0,1] is then divided into N equal subintervals, and is ap-
proximated by a step-function taking values on the succes-
sive subintervals. An extra constraint is added to the given problem,
where V is the subspace of such step-functions. Consequently, becomes a
polynomial function, determined by its values at the grid-points
Since there are discontinuities at the grid-points on
the right side of the differential equation which is a non-smooth function of
Optimal Control Models 15

a suitable differential equation solver must be chosen for such functions.
Many standard solvers do not have this feature. Only one of the six ODE
solvers in MATLAB is designed for stiff differential equations. However, this
ODE solver is not used in this book. Instead, a good computer software pack-
age “nqq” is used to deal with the jumps on the right side of the differential
equation in the dynamic system of the optimal control problems. The fourth
order Runge-Kutta method [30, 1965] (a standard method for solving ordinary
differential equations) is slightly modified for solving a differential equation of
the form where is a step-function. The modification
is simply recording the counting number and time to make sure that
always takes the appropriate value not in subinterval
when In the computation, the differential equation is com-
puted forward (starting at while the adjoint equation is solved backward
(starting at
Two steps are introduced to solve such optimal control problems:

1 Compute objective values from the objective function, differential equation,
and augmented Lagrangian, not compute gradients from the adjoint equa-
tion and Hamiltonian. That assumes gradients can be estimated by finite
differences from what to be computed.

2 Compute objective values and gradients from the adjoint equation and Hamil-
tonian.

Implicit Constraints: in some economic models, such as the model [48,
1971 ] to which Craven and Islam have applied the SCOM package in the paper
[18, 2001], fractional powers of the functions (with in the right side of
the differential equation where appear, then some choices of
will lead to causing the solver to crash. The requirement of
forms an implicit constraint. A finite-difference approximation to gradients
is useful as approximations over a wider domain in this case. As mentioned
in Section 1.7.1, linear interpolation can also be used in solving gradients and
co-state functions. Increasing the number of the subintervals N will get better
results. It will be discussed later in Section 2.7 and 2.8.
Two problems were tested using SCOM in a paper by Craven and Islam
[18, 2001], and accurate results were found for both. In Craven and Islam [18,
2001], “constr” estimated gradients by finite differences to compute the opti-
mum. The economic models in this paper [18, 2001] with implicit constraints
are different from the models that were solved by other computer software
packages. However, this computer software also needs to be further developed
for more general use.
16 OPTIMAL CONTROL MODELS IN FINANCE

7.6 Switching Costs Model
An investment model for the natural resource industry was introduced in
Richard and Mihall™s paper [73, 2001] with switching cost. The problem com-
bines both absolutely continuous and impulse stochastic control. In particu-
lar, the control strategy involves a sequence of interventions at discrete times.
However, this component of the control strategy does not have all the features
of impulse control because the sizes of the jumps associated with each inter-
ventions strategy are not part of the control strategy but are constrained to the
pattern...,1, “1,1, “1,..., jumping between two levels. This kind of control
patterns is also considered in this research.
Perthame [68, 1984] first introduced the combination of impulse and abso-
lutely continuous stochastic control problems which have been further studied
by Brekke and B.˜ksendal [3, 1991]. Mundaca and ˜ksendal [62, 1998] and
Cadenillas and Zapatero [6, 2000] work on the applications to the control of
currency exchange rates.

7.7 CPET
Time optimal control problems (which are not considered in this research)
with bang-bang control associated with or without singular arc solutions can
make the calculation difficult. A novel problem transformation called the Con-
trol Parameterization Enhancing Transform (CPET) was introduced in refer-
ence [51, 1997] to provide a computationally simple and numerically accurate
solution without assuming that the optimal control is pure bang-bang control
for time optimal control problems. A standard parameterization algorithm can
calculate the exact switching times and singular control values of the original
problem with CPET. Also, with the CPET, switching points for the control
match the control parameterization knot points naturally, and hence piece-wise
integration can be conveniently and accurately carried out in the usual control
parameterization manner.
Several models used this technique and gave numerical results with ex-
tremely high accuracy. They are F-8 flight aircraft (first introduced in reference
[31, 1977]) [71, 1994], the well-known “dodgem car” problem [29, 1975], and
a stirred tank mixer [34, 1976]. The generalizations of CPET technique to a
large range of problems were also introduced in reference [72, 1999].

7.8 STC
A new control method, the switching time computation (STC) method,
which finds a suitable concatenation of constant-input arcs (or, equivalently,
the places of switchings) that would take a given single-input non-linear sys-
tem from a given initial point to the target, was first introduced in Kaya and
Noakes™ paper [43, 1994]. It was also described in detail of the mathematical
Optimal Control Models 17

reasoning in paper [44, 1996]. The method is applicable to single-input non-
linear systems. It finds the switching times for a piecewise-constant input with
a given number of switchings. It can also be used for solving the time-optimal
bang-bang control problem. The TOBC algorithm, which is based on the STC
method, is given for this purpose. Since the STC method is basically designed
for a non-linear control system, the problem of the initial guess is equally dif-
ficult when it is applied to a linear system. For the optimization procedure,
an improper guess for the arc times may cause the method to fail in linear
systems. However, the initial guess can be improved by experience from the
failures. In non-linear systems, there does not exist a scheme for ˜guessing™ a
proper starting point in general optimization procedures. In general, the STC
method handles a linear or a non-linear system without much discrimination.
The reason is that the optimization is carried out in arc time space and even a
linear system has a solution that is complicated in arc time space. The STC
method has been applied as part of the TOBC algorithm to two ideal systems
and a physical system (F-8 aircraft). They have been shown to be fast and ac-
curate. The comparisons with results obtained through MISER3 software have
demonstrated the efficiency of the STC method, both in its own right in find-
ing bang-bang controls and in finding time-optimal bang-bang controls when
incorporated in the TOBC algorithm. There is also a possibility to generalize
the system from a single-input system to the multi-input system, which needs
more computer programming involving.

7.9 Leap-frog Algorithm
Pontryagin™s Maximum Principle gives the necessary conditions for opti-
mality of the behavior of a control system, which requires the solution of a
two-point boundary-value problem (TPBVP). Noakes [64, 1997] has devel-
oped a global algorithm for finding a geodesic joining two given points on a
Riemannian manifold. A geodesic problem is a special type of TPBVP. The
algorithm can be viewed as a solution method for a special type of TPBVP.
It is simple to implement and works well in practice. The algorithm is called
the Leap-Frog Algorithm because of the nature of its geometry. Application
of the Leap-Frog Algorithm to optimal control was first announced in Kaya
and Noakes [45, 1997]. This algorithm gave promising results when it was ap-
plied to find an optimal control for a class of systems with unbounded control.
In Kaya and Noakes™ paper [46, 1998], a direct and intuitive implementation
of the algorithm for general non-linear systems with unbounded controls has
been discussed. This work gave a more detailed and extended account of the
announcement. A theoretical analysis of the Leap-Frog Algorithm for a class
of optimal control problems with bounded controls in the plane was given in
Kaya and Noakes™ paper [47, 1998]. The Leap-Frog Algorithm assumes that
the problem is already solved locally. This requirement translates to the case
18 OPTIMAL CONTROL MODELS IN FINANCE

of optimal control as the availability of a local solution of the problem. This is
related to the structure of the small-time reachable sets.

7.10 An obstruction optimal control problem
In Craven [17, 1999], an optimal control problem relating to flow around an
obstacle (original proposed by Giannesi [32, 1996]) can be treated as a mini-
mization problem which leads to a necessary condition or an optimal control
problem. In this paper, Craven gave the discretization augment, and proved
that, if an optimal path exists, the Pontryagin principle can be used to calculate
the optimum. The optimum was verified to be reached by a discretization of
the problem, and was also proved to be a global minimum.

7.11 Computational approaches to stochastic optimal
control models in finance
Computational approaches specific to stochastic financial optimal control
models are relatively well developed in the literature. However, computational
approaches to deterministic financial optimal control models are not well doc-
umented in the literature, the standard general computational approaches to
optimal control discussed above are applied to financial models as well. Some
discussion of the computational approaches with specific applications to fi-
nance may be seen in Islam and Craven [38, 2002].

7.12 Comparisons of the methods
While a discussion of the comparisons of the general computational ap-
proaches is provided below, such comparisons are also relevant when the gen-
eral approaches are applied to financial models. In financial optimal control
models, the control function is approximated by a vector on some vec-
tor space of finite dimension in all algorithms for numerical computation of
such an optimal control model 1.38-1.40. There are some examples with
different chosen approximations. The RIOTS 95 package [77, 1997] which
uses MATLAB, uses various spline approximations, solves the optimization
problems by projected descent methods; MISER3 [33, 1987], uses a step-
function to approximate the control function, solves the optimization problems
by sequential quadratic programming; OCIM [15, 1998], uses conjugate gra-
dient methods. Different implementations behave differently especially on the
functions defined on a restricted domain, since some optimization methods
might want to search the area outside the domain. Although a step-function
is obviously a crude approximation, it produces accurate results shown in
many instants in reference [84, 1991]. Since integrating the dynamic equation
to obtain is a smooth operation, the high-frequency
oscillations are attenuated. In Craven [14, 1995], if this attenuation is suffi-
Optimal Control Models 19

ciently rapid, the result of step-function approximations converges to the exact
optimum while It is necessary to have some assumption of this
qualitative kind in order to ensure that the chosen finite dimensional approxi-
mations will permit a good approximation to the exact optimum. The RIOTS
95 package can run faster than SCOM, maybe because of its implementation
in the C programming language. The efficiency of the STC method has been
demonstrated by comparisons with results through MISER3 optimal control
software (a general-purpose optimal control software package incorporating
sophisticated numerical algorithms). MISER3 did not get results as fast as
the STC method did, perhaps because the general-purpose might be hamper-
ing its agility to certain extent because of some default settings regarding the
tolerances for the accuracy of the ODE solver and optimization routine in the
software.
This research is only concerned with pure bang-bang control problems within
a fixed-time period. All the algorithms and transformations are made for this
purpose. The control function is also approximated by a step-function. How-
ever, because the control does not always jump at the grid-points of the subdi-
visions of the time intervals which are usually equally divided in other works,
it is necessary to calculate the optimal divisions of the time horizon. This
research is mainly computing the optimal ranges of the subdivisions in time
period as well as calculating the minimum of the objective function. Situations
when a cost of changing control is involved in the cost function are discussed
as well as how this cost can effectively work on the whole system. Although
the STC method is also concerned with the calculation of the optimal switching
times, it does not include the cost of each switching control.
The limitations of the above computational approaches are summarized in
Chen and Craven [10, 2002]. From the above survey it will also appear that
each of the above computational methods has characteristics which are compu-
tationally efficient for computing optimal control financial models with switch-
ing times. A new approach which can adapt various convenient components
of the above computational approaches is developed in the next section. Al-
though the present algorithm has similarity with CPET, the details of the two
algorithms are different. A new computer package called CSTVA is also devel-
oped here which can suitably implement the proposed algorithm. The present
computational method consisting the STV algorithm and the CSTVA computer
programs does, therefore, provide a new computational approach for modeling
optimal corporate financing. The computational approach can be suitably ap-
plied to any other disciplines as well.
This approach (STV) consists of several computational methods (described
in Chapter 2):
1 The STV method where the switching time is made a control variable opti-
mal value of which is to be determined by the model.
20 OPTIMAL CONTROL MODELS IN FINANCE

2 A piecewise-linear (or non-linear) transformation of time.
3 The step function approach to approximate the control variable.
4 Finite difference method for estimating gradients if gradients are not pro-
vided.
5 An optimization program based on the sequential quadratic programming
(SQP) (as in MATLAB™s “constr” program similar to the Newton Method
for constrained optimization).
6 A second order differential equation to represent the dynamic model.

8. Conclusion
This book is mainly concerned with using the computational algorithms to
solve certain classes of optimal control problems. The next chapter will in-
troduce the computational approach named Switching Time Variable (STV)
algorithm developed in this book that can solve a class of financial optimal
control problems when the control is approximated by a step-function. The
piecewise-linear transformation that is constructed for the computer software
package is described in Section 2.2. Some non-linear transformations, which
were first introduced by Craven [14, 1995] , are also discussed in Section 2.3.
These transformations are used to solve the large time period optimal control
problem in Chapter 4. A computer software package that was developed by
Craven and Islam [18, 2001] and Islam and Craven [37, 2001] is presented in
Section 2.4. The “nqq” function for solving differential equations is quoted
as a part of the computer software package in this research. The thrust of this
book involves a general computer software for certain optimal control prob-
lems. The principal algorithms behind it are introduced in Section 2.6. All the
computing results of an example problem for optimal investment planning for
the economy are shown in graphs and tables in Section 2.7. A cost of changing
control is also discussed in Section 2.8.
Chapter 2

THE STV APPROACH TO
FINANCIAL OPTIMAL CONTROL MODELS




1. Introduction
In this chapter, a particular case of the optimal financial control problems,
which has one state function and one control function that is approximated by

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