Optimization in Internet Information Service Todor Stoilov, Kasimira Stoilova and Zlatka Ivanova Institute of Computer and Communication Systems, Bulgarian Academy of Sciences, 1113 Sofia, Acad.G.Bontchev BL.2 tel.: +(359) 2 979 2774, fax: +(359)2 72 39 05, e-mail:
[email protected]
Abstract. The information service in Internet, which performs on-line optimization calculations, is considered. The optimization concerns evaluation of optimal portfolio in financial investments. Thus the information service supports the on-line decision-making in investments. The evolution of the information services is considered as development of hierarchical Web based client-server software suit. The improvements in the functionality of the services results in increasing the hierarchical levels of the server side algorithmic and software structure. A case of optimal resource allocation in investment process is presented. The goal of the system is to offer in real time solutions of portfolio optimization problems, according to the dynamically changes of the input data, related to the assets’ returns. An optimization portfolio problem is defined and new algorithms are derived, satisfying real time requirements for the problem solution. The algorithms are incorporated in WAN based information system.
1 Introduction The meaning of the term WAN (Wide Area Network) is often used to describe distributed information systems that operate in Internet environment. Although WAN are complex technological and informational systems, the importance and the benefit of WAN origins from the information services, offered by WAN to the users [1,4]. The information services operate through the global network and can be accessed by the wide Internet community of users. Nowadays the Internet users assume that the global network is the environment, which offers different information services, accessed by appropriate computer and communication tools [2,6]. The importance of the information services for the management of different technical, economical, and social systems determines the goal of this research to include the optimization functionality in the WAN based information services. The optimization is one of the most complex procedures for treating the information. It consists stages of problem definition, parameter identification and problem evaluation. Due to the complexity of the information treatment, the information services have to cope real time requirements to provide on-line optimization functionality [3,5]. This research targets the implementation of the optimization of resource allocation in on-line information service, related to the portfolio optimization of investments. Such a system supports the real-time decision making in investment process, following in time the fast and dynamical changes of the performance of the financial assets. The paper identifies the system
tendencies, occurring for the design and development of WAN based information services, by means to perform advanced and complex functionality. System model is derived, which includes the optimization functionality in the Web based client -server software model. The optimization addresses the allocation of the financial investment, subject to the portfolio theory. Appropriate fast computational algorithms are derived for the implementation of the real-time decision making. The results of this research have been applied in the development of the server side software suit, which performs on-line portfolio optimization as Internet based information service. The results cover the intersection of three scientific domains: the portfolio theory, which supports financial investments and resources allocation by the choice of appropriate portfolio of financial assets; the optimization theory, which addresses fast computational algorithms for quadratic programming problems; and the computer science domain for the case of the development of Web based real-time information systems. System and algorithmic models, used by the information services in Internet are intensively developed. Till now three kinds of models could be identified, applied for the design and implementation of WAN based information systems [7]. 1.1 Information system with two-tier system model The client-server model is the basic model widely applied for all Internet based information systems [8], Figure 1. It is based on two-tier system modeling. The first tier is the client, who in general operates under a web browser environment. Particularly the client tier does not perform data processing. The server side is the place where the functionality of the information service is supported; the information service proceeds data and responds to user queries. The server side is implemented as Web Server (Internet Information Server – IIS, Apache, Tomcat, Java Web Server – JWS), which operates in different operational environments (Windows, Unix, Sun, HP). CLIENT Browser: IE Netscape Opera
Internet http/htm l
SERVER Windows Web: OS IIS HTML Linux Apache Unix TOMCAT JWS
Figure 1 Two-tier client server model of WAN based information system The two-tier information model has not wide algorithmic potential for complex information treatment. This model is applicable for general information system that supports and offers predefined (static) information. Such information system is applicable for user scenarios related to business and web presence in Internet, for dissemination of data, messages, and events in the global network. These types of information systems do not perform advanced algorithmic treatment of the service data.
1.2 Information system with three-tier system model The three-tired model is a natural extension of the two-tier one, which addresses the complication of the functionality and data processing. The three-tier model introduces next algorithmic tier for the information system, which performs functionality, supported by a database management system [9], Figure 2. CLIENT Browser: IE Netscape Opera
Internet http/htm
SERVER Windows Web: OS Dynamic Linux IIS Pages Apache Unix ASP (IIS) TOMCAT Php JWS (Apache) SUN
Microsoft Access Microsoft SQL My SQL ORACLE INFORMIX SYBASE
ODBC
Database
Server
CGI
(A h of ) WAN based information system Figure 2 Three-tier client server model
The third tier consists a database engine as Microsoft SQL Server, Microsoft Access, Oracle, Informix, Sybase, MySql, e.t.c. Additionally the second tier supports increasing functionality by the inclusion of the server side programming tools. Thus the server side (the second tier) now possesses bigger potential on data management, according to the server side programming functionality. These programs perform online communications with the databases (third tier) so that all additional functionality in data retrieval, search, edit, and data proceeding, which the databases support, are used for the information services. As a result the information systems in Internet become more functional and cover wider area of applications in business, marketing, system engineering, culture, and science. Particularly each on-line e-business system, e-reservation, e-learning service, on-line catalogue and user interoperability system is implemented by the deployment of the appropriate three-tier information system. 1.3 Information system with four- tier system model The current needs in data processing insist the implementation of more complex algorithms for the web services in data treatment, integration, and knowledge estimation. Additionally the on-line requirements for system management insist fast processing of the information. The system design and the algorithmic solutions, which cope the requirements of complex data processing and on-line system management, introduce new algorithmic level in the system structure of the information systems [10]. This forth level performs a specific and complex data processing, evaluates complex mathematical problems, support on-line control functionality. The forth tier
in the paper is titled “Algorithmic Server”. It performs fast algorithmic information processing without server-side programming tools. The four-tier information systems are implemented in real time process control systems, on-line market research, investment systems, on-line decision-making, and resource allocation systems. The bigger potential of the information systems with four-tier system structure in data processing is the reason to address such Web based system in the on-line portfolio optimization and resource allocation in financial investments. Client Web server
Database
Web server
Database
Web server
Database
Server
Algorithmic Server
Client Server
Algorithmic Server
Client Server
Algorithmic Server
Figure 3 Four-tier client server model of WAN based information system
2
Portfolio theory and optimal allocation of investment resources
The classical portfolio optimization problem deals with one time investment horizon, which narrows the application area of the investment [11]. Here the portfolio problem is complicated by introducing several investment horizons. Two optimization problems are derived, describing different aspects of the investment policies: investment with “short sales” and investment without “short sales”. The first optimization problem deals with the so-called “short sales”, that means that the investment xi can have positive and negative values. For xi>0, the investor has to buy security i with the amount given by the relative value xi according to the total amount of the investment. For the case xi < 0, the investor has to sell security xi. This “short sale” means that the investor has to borrow security i, to sell it and later he should restore it [11,12]. The second optimization problem assumes non-negativity of the investments, xi ≥ 0, which means that the “short sales” are non-feasible: (1) min { φ xT V x - ET x} x
xT 1 = 1 , L ≤ x ≤ U, Tr T S x ≤ Tr h, φ ≥0, where xT|Nx1 = (x1, …, xN) is the vector of relative allocations of the investment per asset; ET|Nx1 = (E1, …, EN) is the vector of the average returns of the assets; V(.)|NxN is the covariation matrix representing the portfolio risk; 1|Nx1 is an identity vector; L|Nx1 is the vector of lower bound constraints of the investment x; U|Nx1 is the vector of upper bound constraints of the investment x; h|3x1 is the vector of the relative alloca-
tion of the investment, for the different time horizons; Tr|3x3 is a triangle matrix,
1 1 1 Tr= 0 1 1 ; S|kxN is a matrix of feasible investment strategies; T|3xk is a matrix of 0 0 1 feasible investments strategies; φ represents the investor risk preference (scalar). Problem (1) can be expressed shortly in canonical optimization problem
min { x
1 T x Q x+RT x} , A x ≤ C, 2
(2)
where the notations held:
1 −1 Q = 2 φ V, R = - E, A= Tr.T .S , −1 1
1 −1 C= Tr .h , −L U
C|mx1 = (Ci), i=1,m is a vector of constraints; A|mxN (N>m) is a matrix of constraints, N
including the general investment constraint
∑
xi=1.
i =1
The portfolio optimization problem related to the “short sales” has the form: , Tr T S x = Tr h min { φ xT V x - ET x } , xT 1 = 1 x
(3)
min { x
1 T x Q x+RT x}, A x = C 2
(4) where the correspondence of the notations between problems (10) and (11) are: Q = 2 φ V,
R = - E,
A=
1 1 , C= . Tr.T .S Tr.h
Both problems (2) and (4) are linear-quadratic mathematical programming ones. They can be solved applying the general methods of the non-linear and/or quadratic programming [13]. But according to the specific requirements to solve (2) and (4) fast computational algorithms are derived, which apply fewer calculations. 2.1 Optimization problem with short sales The portfolio problem with “short sales” has the form (4). It analytical solution is found, which speed up the calculations. The analytical solution results from the multilevel system theory, applying the concept of non-iterative coordination [14]. For the initial problem (4) the analytical solution is :
xopt = -Q-1 [R - AT (A Q-1 AT)-1 (C + A Q-1 R)]. (5) The algorithm of the information service performs sequential implementation of (5). The investor inputs his choice about the risk preferences φ and preferable investment horizons h. The information system, applying the parameters for risks Cov(.) and asset performances E, calculates the optimal investment allocation xi, for the problem (4). Particularly relation (5) is used, which does not insist recursive and iterative computations. The information service is able to evaluate very fast the solution of portfolio problem (4) and to estimate the curve of the “efficient frontier” EpT=EpT(τpT). Hence the analytical solution (5) contributes to the faster solution of the portfolio problem (4). Thus the portfolio optimization can be implemented on-line as information service in Internet. 2.2 Optimization problem without short sales A problem (2) lacks “short sales”, which results in the existence of inequality constraints in the feasible area of (2). Thus direct analytical solution can not be found and appropriate logical and computational sequence has to be derived. Here a new algorithm is worked out, which does not perform recursive calculations as a prerequisite for fast problem solution. This algorithm founds on a sequential computational procedure, applying the right and dual Lagrange problems, related to the initial problem (2). For the initial problem (2) the right Lagrange problem is defined as (6) min {L(x, λ )}, L(x, λ ) = 1 xT Q x+RT x+ λ T (A x-C), x 2 where λ is the dual vector. Problem (6) is an unconstrained optimization one, which is reduced towards the solution of the linear set of equation
∂L = Q x+R+AT λ = 0, or x( λ ) = -Q-1 (R+AT λ ) ∂x
(7)
λ is the dual variable, which is evaluated from the dual Lagrange problem as: (8) max H( λ ), where H( λ ) = L(x( λ ), λ ), λ >0 λ
The dual function H( λ ) can be expressed analytically by a substitution of x( λ ) in H( λ ), but the dual problem is a constrained optimization one and its solution is: λ opt =arg { min [-H( λ )= 1 λ TAQ-1 AT λ +(A TQ-1 R+C) = 1 λ TG λ + λ T h]}, (9) λ ≥0
2
2
where G = A Q-1 AT , h = A Q-1 R + C or ⇒ min β = hT λ (10) λ T (G λ + h)=0 G λ + h = 0, λ ≥ 0. λ ≥0 The initial constrained optimization problem (2) or respectively the simpler dual problem (10) can be solved with the methods of the quadratic programming. But it’s worth to solve the dual (10) analytically. Thus the faster solution of (10) will assist the implementation of the portfolio optimization as Internet on-line service.
2.3 Case 1: the matrix G-1 exists This case determines that the quadratic curve H( λ ) has a central point. Because G-1 exists, hence the dimension of G corresponds to this one of λ and the central point of H( λ ) has a value λ *=G-1 h. Respectively λ * is the unique feasible point, given by the relation G λ +h=0, but it could not be feasible. Hence an analysis of the components of λ * will determine whether λ * is the solution of the dual problem (10), λ * = λ *opt , or λ * has to be modified. If λ * is a nonnegative one, λ *≥0, this is the solution of (9) and λ * = λ opt . If λ * has negative components, λ *i
