Session name:

Applied Optimization

Organisers:

K.L. Teo, The Hong Kong Polytechnic University and L. Caccetta, Curtin University of Technology

B. Abstracts

A1. Professor S. Banks, University of Sheffield, UK

Title: Exact Boundary Controllability and Optimal Control for a Generalised Korteweg de Vries Equation

Email: s.banks@sheffield.ac.uk

A2. Professor L. Caccetta, Curtin University of Technology, Australia

Title: To be supplied soon

Email: caccetta@cs.curtin.edu.au

A3. Professor X.Q. Cai, The Chinese University of Hong Kong, Hong Kong.

Title: to be supplied.

Email: xqcai@se.cuhk.edu.hk

A4. Professor G.Y. Chen, The Chinese Academy of Science, Peoples Republic of China.

Title: Convergence Results for Mathematical Program with Equilibrium Constraints, by G.Y. Chan

Email: machany@polyu.edu.hk

A5. Dr. S.H. Hou, The Hong Kong Polytechnic University, Hong Kong, China

Title: Second Order Symmetric Duality in Nondifferentiable Programming, by S.H. Hou and X.M. Yang.

Email: mahoush@polyu.edu.hk

A6. Professor P.G. Howlett, University of South Australia, Australia

Title: Constructive Operator Approximation in the Modelling of Realistic Non-linear Systems, by P.G. Howlett, C.E.M. Pearce, and A.P. Torokhti

Email: phil.howlett@unisa.edu.au

A7. Professor M. Iri, Chuo University, Japan

Title: Extraction of Invariants from Digital Elevation Data with Applications to Terrain Topology, by M. Iri, Y. Shimakawa, and T. Nagai

Email: iri@ise.chuo-u.ac.jp

A8. Professor Duan LI, Chinese University of Hong Kong, Hong Kong, China

Title: Existence of Saddle Point and Local Convexification of Lagrangian Function in Nonconvex Constrained Optimization, by D. Li and X.L. Sun

Email: dli@se.cuhk.edu.hk

A9. Dr. C.C. Lim, University of Adelaide, Australia

Title: Support Vector Learning with Adaptive Step-size Barrier projection QP Optimization, by K.N. To, C.C. Lim, K.L. Teo, and M.J. Liebelt

Email: cclim@eleceng.adelaide.edu.au

A10. Dr Y. Liu, The Hong Kong Polytechnic University, China

Title: A Dual Parametrization Algorithm for Positive LQ Semi-infinite Programming Problems, by Y. Liu and K.L. Teo

Email: mayliu@polyu.edu.hk

A11. Professor Rein Luus, University of Toronto, Canada

Title: Use of Luus-Jaakola Optimization Procedure for Singular Optimal control Problems

Email: luus@chem-eng.toronto.edu

A12. Professor Katsumi Moriwaki, The University of Shiga Prefecture, Japan

Title: A Method of Automatic Motion Control with Optimization for Nonlinear Dynamic Systems

Email: moriwaki@mech.usp.ac.jp

A13. Dr. V. Rehbock, Curtin University of Technology, Australia

Title: Enhancing the Robustness of Optimal Controls with respect to Inequality Constraints, by A. Siburian, V. Rehbock, and K.L. Teo

Email: rehbock@cs.curtin.edu.au

A14. Professor S. Sethi, The University of Texas at Dallas

Title: Average Cost Optimal Policy for a Stochastic Two-machine Flowshop with Limited Work-in-process, by E.L. Presman, S.P. Sethi, H.Q. Zhang, and A. Bisi

Email: sethi@utdallas.edu

A15. Professor K.L. Teo, The Hong Kong Polytechnic University, China

Title: A Computational Method for A Class of Optimal Switching Control Problems. by K.L. Teo

Email: mateokl@polyu.edu.hk

A16. Professor K.H. Wong, University of the Witswatesran, South Africa

Title: Control Parametrization Enhancing Technique for Time-Delayed Optimal Control Problems

Email: wong@maths.uwa.edu.au, WONG@cam.wits.ac.za

A17. Professor H.M. Yan, The Chinese University of Hong Kong, Hong Kong, China

Title: Game Theoretic Analysis of a Supply Chain with multiple suppliers and a single customer, by H.M. Yan and H.Q. Zhang

Email: yan@se.cuhk.edu.hk

A18. Professor X.M. Yang, The Hong Kong Polytechnic University, Hong Kong, China

Title: Generalized Invexity and Generalized Monotonicity, by X.M. Yang

Email: 99900212r@polyu.edu.hk

A19. Dr. X.Q. Yang, The Hong Kong Polytechnic University, Hong Kong, China

Title: A New Characterization of Proper Efficiency in Terms of the Stability of a Scalar Optimization Problem, by X.X. Huang and X.Q. Yang

Email: mayangxq@polyu.edu.hk

A20. Professor X.Y. Zhou, The Chinese University of Hong Kong, China

Title: Frequency Conditions for stochastic Linear Quadratic Control, by H. Wu and X.Y. Zhou.

Email: xyzhou@se.cuhk.edu.hk

A21. Dr. Sharlene M. Andrijich, Martime Operations Division, Defence Science and Technology Organization of Australia

Title: Solving the Multisensor Data Association Problem, by Sharlene M. Andrijich and L. Caccetta

Email: sharlene.andrijich@dsto.defence.gov.au

A22. Dr Kusumah, Curtin University of Technology,

Title: Computational Aspects of the Facility Layout Design Problem, by L. Caccetta and Y.S. Kusumah
 

B. Abstracts:
 

Authors:

S.P. Banks, Department of Automatic Control and Systems Engineering, University of Sheffield, UK.

Title:

Exact Boundary Controllability and Optimal Control for a Generalised Koeteweg de Vries Equation,

Abstract:

In this paper, we shall consider the genralised Korteweg de Vries equation

\begin{equation}

\phi_{t}+\phi_{x}+h(\phi )\phi_{x}+\phi_{xxx}=0 \tag{(1)}

\end{equation}

on the domain $(\alpha ,\beta )$, $t\geq 0$ with boundary conditions

\begin{equation*}

\phi (\alpha ,t)=u_{1}(t)\text{, }\phi (\beta ,t)=u_{2}(t)\text{, }\phi_{x}(\beta ,t)=u_{3}(t)

\end{equation*}

where $u_{1}(t)$, $u_{2}(t)$ and $u_{3}(t)$ are control inputs and $h$ is a Lipschitz nonlinearity. We shall also consider a cost functional of the form

\begin{equation}

J=\left\langle \phi (T,.),F(\phi )\phi (T,.)\right\rangle_{H}+ \int_{0}^{T}\left( \left\langle \phi (t,.),Q(\phi )\phi

(t,.)\right\rangle _{H}+\underset{i=1}{\overset{3}{\sum }}%

r_{i}u_{i}^{2}(t)\right) dt \tag{(2)}

\end{equation}

on a suitable Hilbert space H.

We shall first show that (1) is exactly controllable with boundary control of the above form and then by using a sequence of approximations of the form

\begin{equation*}

\phi_{t}^{[i]}+\phi_{x}^{[i]}+h(\phi^{\lbrack i]}) \phi_{x}^{[i]}+ \phi_{xxx}^{[i]}=0

\end{equation*}

\begin{equation*}

J=\left\langle \phi ^{\lbrack i]}(T,.),F(\phi ^{\lbrack i-1]})\phi ^{\lbrack

i]}(T,.)\right\rangle_{H}+\int_{0}^{T}\left( \left\langle \phi ^{\lbrack

i]}(t,.),Q(\phi^{\lbrack i-1]})\phi^{\lbrack i]}(t,.)\right \rangle_{H} +%

\underset{i=1}{\overset{3}{\sum }}r_{i}(u_{i}^{[i]})^{2}(t)\right) dt

\end{equation*}

we shall determine a (local) optimal control for the problem (1) and (2), giving at least a locally optimal feedback control for the nonlinear wave supression problem.
 
 

Authors:

D. LI, Department of Systems Engineering \& Engineering Management, Chinese University of Hong Kong

and

Xiao Ling SUN, Department of Mathematics, Shanghai University

Title:

Existence of Saddle Point and Local Convexification of Lagrangian Function in Nonconvex Constrained Optimization

Abstract.

We examine the p-th power formulation from both the global and local optimization viewpoints. Saddle point is a sufficient condition for optimality. We show that, under some mild conditions, the existence of a saddle point can be ensured in an equivalent p-th power formulation for a general class of nonconvex constrained optimization problems. This result expands considerably the class of optimization problems where a saddle point exists and thus enlarges the family of nonconvex problems that can be solved by dual-search methods. It is well-known that a basic requirement for the development of local duality theory in nonconvex optimization is the local convexity of Lagrangian function. We show further in this talk that the p-th power formulation locally convexifies the Lagrangian function and thus expands the class of optimization problems to which dual methods can be applied.
 

Author:

K.L. Teo

Title:

A Computational Method for A Class of Optimal Switching Control Problems

Abstract:

In this paper, we consider a class of optimal control problems in which the system is to be determined in an optimal way. This class of problems involves the choice of a fixed number of switching time points which divide the system's time horizon into a number of time periods. At the same time, for each of these time periods, a subsystem is selected, from a finite number of given candidate subsystems, to run during that time periods. The choice of the switching points and the selection of the subsystems are carried out in such a way that a given cost functional is minimized. We consider only problems involving ordinary differential equations over a finite time horizon.  A computational method is developed for solving these problems. In our method, the candidate subsystems are combined into a single
system by considering their `linear combination'. By introducing suitable constraints on the coefficients of the linear combination and using a time rescaling technique, the original problem is transformed into an equivalent optimal control problem with system parameters. An algorithm is proposed for solving this transformed problem and the required gradient formulae are derived. To show the effectiveness of the method, a numerical example is solved.
 

Author:

X.M. Yang

Title:

Invexity and Generalized Monotonicity

Abstract:

The notions of several kinds of vector-valued monotone maps are generalized to invariant montone and generalized invariant monotone maps. For gradient maps, these invariant monotonicity and generalized invariant montonicity properties are related to invexity and generalized invexity properties of the underlying function. In this way, first-order characterization of various types of invex and generalized invariant functions are obtained.
 
 

Authors:

G.Y. Chen and Y.N. Wu

Title:

Convergence Results for Mathematical Program with Equilibrium Constraints

Abstract:

In this paper, we study the general mathematical program with equilibrium constraints under the perturbations and obtain some convergence results, for instance, the convergence of the marginal function, the value and the solution set.
 
 

Authors:

K.N. To, C.C. Lim, K.L. Teo and M.J. Liebelt

Title:

Support vector learning with adaptive step-size

Abstract:

We consider a support vector machine training problem involving a quadratic objective function with a single linear equality constraint and a box inequality constraint. Using quadratic surjective space transformation to create a barrier for the gradient method, an iterative support vector learning algorithm is derived. We further derive a stable steepest descent method to find the step-size in order to reduce the number of iterations to reach the optimal solution. We show that the step-size problem can be reduced to a scalar cubic polynomial equation and is solvable analytically. This method offers speed improvement over the fixed step-size gradient method, particular for QP problems with ill-conditioned Hessian
 
 

Authors:

Y. Liu and K.L. Teo

Title:

A Dual Parametrization Algorithm for Positive LQ Semi-infinite Programming Problems

Abstract:

We consider a class of LQ semi-infinite programming (SIP) problems where the objective is psoitive quadratic and the linear infinite constraint functions continuously depend on its index variable on a compact set. It is known that, by using the dual parametrization technique, the SIP problem we are consedering can be transformed into a nonlinear programming problem with result in a solution of the SIP problem. However, so far there is no efficient method for finding the global solution of such nonlinear programming problems.

In this paper, we present an algorithm for numerical solution of this class of LQ semi-infinite programming problems. Effort will be focussed on the global solution of the nonlienar programming problems. Effort will be focussed on global solution of the nonlinear programming problem obtained by the dual parametrization method. Convergence result will be provided and an application example in signal processing will be presented.
 
 

Authors:

M. Iri, Y. Shimakawa and T. Nagai

Title:

Extration of Invariants from Digital Elevation Data with Applicaiton to terrain Topography

Abstract:

In connection with the rapidly developing new technology area of Geographic Information System (GIS), abundant geography-related data, inclduing digital elevation data over a fine grid,are now available. The most basic, or primitive, characteristics that we can extract from these data are local maxima/minima of the elevation. Each of these local maxima/minima corresponds to a point at which the gradient vector field (a covariant vector field) vanishes. These points are sometimes called critical points in mathematics. There are other algebraic invariants which can be defined in terms of the gradient vector field (of the elevation function) and the Hessian field. In the two-dimensional case, these invariants can be expressed in a compact form, relating to the topographical concepts of ridges and basins. In addition, the mathematical concept of critical lines can also be defined by means of these invariants.
 
 
 

Authors:

K. Moriwaki, Department of Mechanical Systems Engineering, School of Engineering, The University of Shiga Prefecture, 2500 Hassaka-Cho, Hikone, Shiga 522-8533, Japan.

Title:

A Method of Automatic Motion Control with Optimization for Nonlinear Dynamic Systems

Abstract:

Steering a car by hand means that the driver plans a path by preview and controls the lateral deviation of the vehicle from the planned path by the steering wheel. In an automatic car steering system, this path following is automated. The deviation is kept small by feedback control via the steering motors. The reference trajectory may be calculated from the data of a CCD camera. In order to study automation of car steering, the extended model of vehicle is introduced. The extended model must include not only velocities, but also the vehicle heading and the lateral position of the sensor with respect to the reference path. This extended model is derived using a nonlinear model that is valid for deviations from a stationary path. The controller design method for the nonlinear model of car steering by optimizing a performance index is considered and demonstrated in the case on active steering of a small electric vehicle, which has one front wheel and two rear wheels. For the motion controller design of the small electric vehicle, the system to be controlled is composed of the dynamic equations of vehicle's motion and the dynamic equations of of steering wheel's motion. In the car steering dynamics, the front wheel steering angle is the input variable and the sideslip angle and yaw rate are the state variables. On the other hand, in the steering wheel's dynamics the steering wheel's angle is the input variable and the front wheel steering angle and the sideslip angle are the state variables. The desired system behavior in the car steering motion is primarily to obtain good damping and an almost disturbance decoupling property. A certain stability margin should be satisfied and the actuator activity should not be too high. This desired system behavior is to be made precise by formulating performance criteria. In the design process

of controller, a design parameter is choosed for the optimization process, which results in good damping, our primary objective is to approximately keep this damping. At the same time the lateral acceleration should be better attenuated in the representative response due to a yaw rate initial value disturbance. Then, the design parameter is modified in part for improving the instationary decoupling property. A better disturbance attenuation should not be gained at the cost of too much actuator activity and therefore the maximal steering angle value should not exceed the prescribed value. Furthermore, for the stability margin constraint, the design parameter is again improved for the optimization evaluation. This process of design iteration scheme will be repeated until no significant improvement of the design result with respect to performance criterion can be obtained.
 

Authors:

P.G. Howlett, Industrial and Applied Mathematics, The Centre for Industrial and Applicable Mathematics, University of South Australia, Adelaide, Australia

C.E.M. Pearce, Department of Applied Mathematics, The University of Adelaide, Adelaide, Australia.

A. Torokhti, the University of South Australia, and the University of Adelaide, Adelaide, Australia

Title:

Constructinve operator approximation in the modelling of realistic
non-linear systems,

Abstract:

A non-linear dynamical system is defined by a mapping that transforms a set of input signals into a corresponding set of output signals. Each signal is normally defined by a set of real number parameters. In practice this set could be uncountably infinite. For a computer simulation every every signal must be represnted by a finite parameter set and the actual mapping must be replaced by a simulated mapping defined by a finite arithmetical process. Nevertheless the output from the simulation system must be a good approxiamtion to the output from the actual system. This paper is particularly concerned with the approximation of mappings that belong to a special class of $\mathcal{R}$-operators which can be said to have realistic properties.
 
 

Author:

Rein Luus, Department of Chemical Engineering, University of Toronto, Toronto, ON M5S 3E5, Canada

Title:

Use of Luus-Jaakola Optimization Procedure for Singular Optimal Control Problems

Abstract:

By using flexible stage lengths and incorporating the recent advances in the determination of region sizes, the Luus-Jaakola optimization procedure is used to sovle several singular optimal control problems. The method is easy tp program, and it is found that the optimal ocntrol policy can be accurately determined in spite of the very sensitivity of the control policy on the performance index. Although a large number of variables must be determined simultaneously, the convergence tends to be systematic and the computation time on the personal computer is reasonable.
 
 

Authors:

A. Siburian and V. REhbock, School of Mathematics and Statisitcs, Curtin University of Technology, Perth, Western Australia, Australia.

K.L. Teo, Department of Applied Mathematics, the Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.

Title:

Enhancing the Robustness of Optimal Controls with Respect to Inequality Constraints

Abstract:

We proposed a new approach to overcome the lack of robustness whcih optimal open loop controls display with respect to the satisfaction of various types of inequality cosntraints. In addition to teh orignal optimal control problem, an auxiliary problem is defined and solved such that its solution displys the desired robustness properties. To deal with the case of multiple constraitns, teh definition of the auxiliary needs to include a measure of the sensitivity of each constraint with respect to the expected uncertainty. The effectiveness of the proposal scheme is illustrated with a practical applicaiton example.
 
 

Authors:

E.L. Presman, Central Economics and Matheamtics Institute, The Russian Academy of Sciences, Moscow, Russia

S.P. Sethi, School of Management, the University of Texas at Dallas, Mail Station J047, Box 830688, Richardson, TX 75083-0688, USA

H.Q. Zhang, Institute of Applied Mathematics, Academia Sinica, Beijing, 100080, P.R. China.

A. Bisi, School Management, the University of Texas at Dallas, Mail Station J047, Box 830688, Richardson, TX 75083-0688, USA

Title:

Average Cost Optimal Policy for a Stochastic Two-machine Flowshop with Limited Work-in-process

Abstract:

We consider a procedure planning problem in a two-machine flowshop subject to breakdown and repair of machines and subject to nonnegativity and upper bound constraints on work-in-process. The objective is to choose machione produciton rates over time to minimize the long-run average inventory/backlog and production costs. For sufficiently large upper bound on the work-in-process, the problem is formulated as a stochastic dynamic problem. We then establsih a verification theorem and a partial characterization of the optimal control policy if it exists.
 
 

Author:

K.H. Wong, Department of Applied and Computational Mathematics, University of the Witswatersrand, South Africa.

Title:

Control Parametrization Enhancing Technique for Time-Delayed Optimal Control Problems

Abstract:

In this paper, we extend the control parametrization enhancing technique (CEPT) devised by H.W.J. Lee, K.L. Teo, L.S. Jennings and V. Rehbock in 1999 to a general class of time-delayed optimal control problems. For the fixed-time problem with time-delayed systems, we only need to use the model transformation approach similar to that used by K.H.Wong in 1998 to convert the time-delayed problem to an optimal control problems involving mixed boundary conditions, but without time-delayed arguments. Then we can use the CEPT to solve this non-delayed problem.

The minimum time problem with time-delayed systems has not been tackled by any existing methods in the literature. In order to solve this problem by the normal control parametrization method or the CEPT, we first need to convert the problem to a fixed time problem and then we derive the gradient of the terminal state equality constraints with respect to the controls, the states and the parameters for the transformed problem.

Several examples have been solved to illustrate the efficiency of the CEPT for the two types of time-delayed problems
 
 

Authors:

H. Wu and X.Y. Zhou, Department of Systems Engineering and Engineering Management, the Chinese University of Hong Kong, Hong Kong.

Title:

Frequency Conditions for Stochastic Linear Quadratic Control

Abstract:

In this paper we consider an infinite horizon stochastic LQ problem with noises in both the state and the control, and with indefinite cost matrices. We establish the equivalence among the unique solvability of the LQ problem, the coercivity of some bilinear operator, and the uniformly positive boundedness of the frequency characteristics.
 
 

Authors:

Houmin Yan, Departent of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong

and

Hanqin Zhang, Institute of Applied Mathematics, Academia Sinica, Beijing, 100080, China

Title:

Game Theoretic Analysis of a Supply Chain with multiple suppliers and a single customer

Abstract:

We consider a supply chain model consisting of two suppliers and one customer, where suppliers compete for customer demand with the individual preference, and the customer uses newsboy-type solution as its inventory control policy. Since each supplier's bidding strategy affects the other's profit, game theory is used to find the optimal bidding strategies. We prove the existence and uniqueness of the Nash solution. It is also shown that the competition leads a lower market clearing price, as a result, the customer is better off.
 

Authors: X.X. Huang and X.Q. Yang

Title:

A New Characterization of Proper Efficiency in Terms of the Stability of a Scalar Optimization Problem

Abstract:

In this paper, nonconvex multiobjective optimization problems are studied. New characterizations of a properly efficient solution in the sense of Geoffrion's are established in terms of the stability of one scalar optimization problem and the existence of an exact penalty function of a scalar constrained program, respectively. One of the characterizations is applied to derive necessary conditions for a properly efficient control-parameter pair of a nonconvex multiobjective discrete optimal control problem with linear constraints.