Introduction:

Chapter 1

Introduction:

1.1 Overview

The nature of mankind is such that, man has always pursed for the best quality of life with the use of available resources. To achieve this effectively and efficiently, design can be used as a tool. Design tools also helps to quench the thirst of mankind to improve the quality of every aspect of our life. The ever-increasing demand on engineers has prompted them to look for rigorous methods of decision-making, such as optimization methods to design and produce products more economically and efficiently. Optimization techniques, have reached degree of maturity over the past several years, are being used in the wide spectrum of industries, including aerospace, automotive, chemical, electrical and manufacturing industries.

With rapidly advancing computer technology over the past few years, computers are becoming more powerful and this increase in computational capabilities has enabled tremendous advances in all disciplines of engineering. Computers have become an inherent part of engineering design, manufacturing and analysis. Optimization methods, coupled with modern tools of computer-aided design, are also being used to enhance the creative process of conceptual and detailed design of engineering systems.

The current technological advances have caused many researchers to rely on hardware efficiency to solve complex mathematical problems. In the field of numerical optimization, work is constantly being performed to find methods that find solutions to optimal design problems more efficiently. However, many efficient numerical optimization algorithms make use of first and second order derivative information to reach the optima, which requires the objective function and constraints to be expressed in closed form mathematical equations. While finding the solutions quickly, this often results in higher computational costs. However, in many real world problems, the objective function representation is nothing but a black box where only the output corresponding to the given inputs is known. Thus, in those cases obtaining derivative information must be done through approximate methods again increasing the computational cost.

A second issue is that many of these methods are suited for unimodal design problems. However many, real life problems are multimodal in nature. While much work has been done to develop algorithms to handle these types of problems, many rely on the power of the computational hardware to find a solution efficiently. There is a definite need for more elegant mathematical solutions to these problems that can simplify the type of calculations being performed and the manner in which a design space is searched to find a solution.

Thus the objective of this work was to develop an efficient math-based optimization algorithm that locates the global optimum in multimodal problems without requiring derivative information. In order to better understand the research issue and the proposed approach within this thesis, an overview of optimization terminologies, optimization methods are provided in the remaining chapter.

1.2 Optimization

1.2.1 Introduction:

Optimization is the act of obtaining the best result under given circumstances. In the most general terms, optimization theory is a body of mathematical results and numerical methods for finding and identifying the best candidate solutions from a collection of alternatives without having explicitly enumerate and evaluate all possible alternatives [1]. In design, construction, maintenance of any engineering systems etc., engineers have to take many technological and managerial decisions at several stages. The ultimate goal of all such decisions is either to minimize the effort required or to maximize the desired benefit. Since effort required or the benefit desired in any practical situation can be expressed as a function of certain decision variables, optimization can be defined as the process of finding he condition that give the maximum or minimum value of a function. It can be seen from figure 1.1 [2] that if a point X* corresponds to a minimum value of function f (X), the same point also corresponds to the maximum value of the negative of the function, --f (X). Thus without the loss of generality, optimization can be taken to mean minimization since the maximum of the function can be found by seeking the minimum of the negative of the same function. The desired solution is generally the maximization or minimization of the objective function. In optimization, the primary objective of the designer is termed as the objective function. The variables of the problem that the designer has control are termed design variables. The optimization problems can be unconstrained or the design space can be constrained. The unconstrained problems are also termed as box-constrained problems. The problem is called box-constrained if the problem has only side constraints to identify the feasible range or acceptable range. In general there are two types of design constraints, equality constraint, where the constraints have to confirm to a certain fixed value or inequality constraints, where some function of the design variables may be greater or lesser then the prescribed value.

1.2.2 Problem Formulation

An optimization problem can be stated as follows.

Minimize: f (X1, X2, …., Xn)

1.1

Subject to constraints:

gj (X1, X2, …., Xn) ≤ 0j = 1,2,….m

1.2

lj (X1, X2, …., Xn) = 0j = 1,2,….p

1.3

Side constraints:

Xil ≤ Xi ≤ Xiu

I = 1,2,….n

1.4

where, X is an n-dimensional vector called the design vector, f (X) is termed the objective function, and gj (X) and lj (X) are known as inequality and equality constraints, respectively. The problem stated in above is called a constrained optimization problem. In this work, the problems will be expressed in standard form. Even in cases where maximization is desired, the negative of function will be minimized. If we consider F be the feasible design space, the portion of design space where all the constraints are active or satisfied, then the point X* is a global minimum if and only if f (X*) ≤ f (X) for all X which belongs to feasible design space F.

1.2.3 Optimization Algorithms

There are many traditional methods available for unconstrained as well as constrained problems. Powell’s method, the conjugate direction method of Fletcher and Reeves, and the Variable Metric method are few examples for unconstrained problems while, for constrained problems we have Sequential Linear Programming (SLP) method, the Method of Feasible Direction, Sequential Quadratic Programming (SQP) method etc. All these optimization algorithms work on the idea of iterative search in a direction where the objective function is decreasing. The design variables are updated using the Up-date formula [3].

Xq-1 = Xq +

*

q

a

Sq

1.5

where, X is the design variable vector, Sq is the search direction,

*

q

a

is a scalar multiplier determining the amount of change in design variable, and q is the iteration number.

Traditionally, non-linear programming methods were developed to obtain local minima. The methods stated above works well if the problem is unimodal. However if the problem is multi-modal the probability of getting stuck with the local minimum increases drastically. Although there was no mathematical criterion that can prove that a particular local minima is also the global minima. Research work on global optimization has been going on for decades [4,5]. Global optimization methods also take the consideration of local minima. Some analytical and numerical methods tending to converge directly to global minima, while others intend to find local minima and select global minima out of them. This gives designer more insight to the feasible design space and also to consider alternative solution in case the global minimum is unstable as suggested by sensitivity analysis [6].

The multistart approach [4] is a very popular stochastic method that tries to find all local minima by starting a local optimization procedure from a set of random points uniformly distributed over a feasible design space. In general method have two phases local and a global phase. Multistart method if used in its original form, can be quite inefficient as it causes extra execution of local search procedure and same local minima could be hit several times. There are various stochastic methods viz. Pure Random Search [7,8], Controlled Random Search [9,10], Simulated Annealing [11,12,13], Genetic Algorithm [14,15,16] which rely on the power of computational hardware to find a solution efficiently. However the basic multistart method has been quite inefficient because it causes extra execution as described above. Several variants have been proposed to improve the efficiency such as Random Tunneling [17], Domain Elimination [18,19] etc.

1.3 Organization of the Thesis

This part will be added as we will proceed towards completion.

References:

1. Reklaitis, G.V., Ravindran, A., Ragsdell, K.M., “Engineering Optimization- Methods and Applications”, John Wiley and Sons, New York, NY, 1983.

2. Rao, S. Singiresu, “ Engineering optimization -Theory and Practice”, Third Edition, New Age International (P) Ltd., New Delhi, 1998.

3. Vanderplaats, G.N., “Numerical Optimization Techniques for Engineering Design with Applications”, McGraw-Hill Inc., New York, NY, 1984.

4.Dixon, L.C.W. and Szego, G.P.(eds) (1975), Towards Global Optimization II, North Holland, Amsterdam.

5.Dixon, L.C.W. and Szego, G.P.(eds) (1978), Towards Global Optimization II, North Holland, Amsterdam

6. Saigal, Sunil and Mukharjee, Subrata (1990), Sensitivity Analysis and Optimization with Numerical Methods, presented at the winter annual meeting of the American Society of Mechanical Engineers, Dallas, Texas Nov. 25-30, 1990.

7.Brooks, S.H. (1958), “A Discussion of Random Search for Seeking Maxima,” Operations Research, Vol.6 244-251.

8.Anderssen, R.S. (1972), “Global Optimization,” in Anderssen, R.S., Jennings, L.S. and Ryan, D.M. eds., Optimization, University of Queensland Press, 1-15.

9. Price, W.L. (1978), “ A Controlled Random Search Procedure for Global Optimization,” in Dixon, L.C.W., Szego, G.P. (eds.) Towards Global Optimization II, North Holland, Amsterdam.

10.Price, W.L. (1983), “ Global Optimization by Controlled Search, “ JOTA, Vol. 40 No. 3, 333-348.

11.Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P. (1983), “ Optimization by Simulated Annealing,” Science Vol. 220, 671-680.

12.Corana, A., Marchesi, M., Martini, C. and Ridella, S. (1987), “ Minimizing Multimodal Function of Continuous Variables with the Simulated Annealing Algorithm,” ACM Transactions on Mathematical Software, 13, 262-280.

13.Dekkers, A. and Aarts, E. (1991), “ Global Optimization and Simulated Annealing,” Mathematical Programming, Vol. 50, 367-393.

14.Holland, J.H. (1975), Adaption in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Michigan.

15.Goldberg, D.E. (1989), Genetic Algorithm in search, Optimization and Machine Learning, Addison, Weskey.

16.Hussain, M.F. and Al-Sultan, K.S. (1997) “ Hybrid Genetic Algorithm for Non-Convex Function Optimization,” Journal of Global Optimization 11, 313-324.

17.Lucii, S. and Piccioni, M. (1989), “Random Tunneling by Means of Acceptance-Rejection Sampling for Global Optimization,” JOTA, Vol. 2, No. 2, 255-277.

18.Elwakiel, O.A. and Arora, J.S. (1996), “Global Optimization of Structural Systems Using Two New Methods,” Structural Optimization, Vol. 12, 1-10.

19.Elwakiel, O.A. and Arora, J.S. (1996), “Two Algorithms for Global Optimization of General NLP problems,” International Journal for Numerical Methods in Engineering, Vol. 39, 3305-3325.

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