Chapter1-Introduction.doc

Chapter 1 Introduction

1. What is Optimization? What it can do?
2. Terminology and basic concepts
3. Mathematical statement of the optimum-structural-design problem
4. Optimization methods

What is Optimization? What it can do?
People optimize. Investors seek to create portfolios that avoid excessive risk while achieving a high rate of return. Manufacturers aim for maximum efficiency in the design and operation of their production processes. Engineers adjust parameters to optimize the performance of their designs.
Nature optimizes. Physical systems tend to a state of minimum energy. The molecules in an isolated chemical system react with each other until the total potential energy of their electrons is minimized. Rays of light follow paths that minimize their travel time.
Optimization is an important tool in decision science and in the analysis of physical systems.
Optimization is the mathematical discipline which is concerned with finding the maxima and minima of functions, possibly subject to constraints.
Application: Architecture/Nutrition/Electrical circuits/Economics/Transportation/etc.

Terminology and basic concepts
1． Deign variables
The design variables of an optimum-structural-design problem may consist of the number sizes, parameters that describe the structural configuration and mechanical or physical properties of the material, as well as other quantifiable aspects of the design. The simplest design variable is the “size” of a member representing the cross-sectional area of a truss member, the moment of inertia of a flexural member, or the thickness of a plate. Many practical structures have fixed geometry and material properties. Configuration variables, often represented by the coordinates of element joints, are next in order of difficulty, followed by material properties.
The ith design variable is designed herein as xi, and the full set of variables for a given structure is listed in the vector x. The design space is described by axes representing the respective design variables. Figure 1.1 shows a three-variable, and consequently three-dimensional design space, for example, the three-bar truss. The number of design variables n is generally very much greater than three, and therefore defines illustration: the n-dimensional space is termed a hyperspace.
Many of the design algorithms to be discussed employ the strategy of a direct search, in which a series of directed design changes are made between successive points in design space. A typical move is between the kth and k+1 th point, given by the equation:

The vector defines the direction of the move and gives its amplitude.
x1
x2
x3
Xk
Xk+1
αkdk
k
k+1
Figure1.1 Three-variables design space
.

2. Objective function
The objective function, also termed the cost function or merit function, is the function whose least( or greatest) value s sought in an optimization procedure, and constitutes a basis for the selection of one to several alternative acceptable designs. The objective function is a scalar function of the design variables. It represents the most important single property of a design, such as cost or weight, but it is also possible to represent the objective function as a weighted sum of a number of desirable properties.
It is useful to illustrate the linear objective function in design space. A linear function in three-dimensional space is a plane, representing here the locus of all design points with a single value. In n-dimensional space, the surface so defined is a hyperplane, when the objective function has non-linear design variables; a hypersurface is described in design space.
There is an important concept is that of the gradient of the objective function, the gradient is a vector composed of the derivatives of the objective function with respect to each of the design variables. For the linear objective function, the gradient is constant, but for non-linear objective function, the gradient is the function of the design variables. The gradient vector derives its utility from the fact that it defines the direction of the design change or travel, in which the objective function is increased most rapidly for given amplitude of change. Our interest is principally in the reduction of the objective function value, representing the negative of the gradient vector. This search algorithm is the method of steepest descent.

3. Constraints
A constraint, in any class of problem, is a restriction to be satisfied in order for the design to be acceptable. It may take the form of a limitation imposed directly on a variable or group of variables (explicit constraint),or may represent a limitation on quantities whose dependence on the design variables can not be stated directly(implicit constraint).
The constraints have equality constraint and in equality constraint, side and behavior constraints etc.
An equality constraint, which maybe either explicit or implicit, is designed as

In theory, each equality constraint is an opportunity to remove a design variable from the optimization process and thereby reduce the number of dimensions of the problem. However, as the elimination procedure may be awkward and algebraically complicated, the approach is not always adopted.
An inequality constraint is of the form:

The idea of an inequality constrain is of major importance in optimum structural design. If equality constraints only were stipulated in a design limited by stresses alone, all procedure would lead to fully stresses designs.
The side constraint is a specified limitation (minimum or maximum) on a design variable, or a relationship which fixes the relative value of a group design variables. The side constraints are therefore explicit in form.
The behavior constraints in structural design are usually limitation on stresses or displacements but they may also take the form of restrictions on such factors as vibrational frequency or buckling strength. Explicit and implicit behavior constraints are both encountered in practice.

Mathematical statement of the optimum-structural-design problem
Mathematically speaking, optimization is the minimization or maximization of a function subject to constraints on its variables. We use the following notation:
- x is the vector of variables, also called unknowns or parameters;
- f is the objective function, a (scalar) function of x that we want to maximize or minimize;
- gi are constraint functions, which are scalar functions of x that define certain equations and inequalities that the unknown vector x must satisfy.
Using this notation, the optimization problem can be written as follows:
subject to and
Here I and E are sets of indices for equality and inequality constraints, respectively.

As a simple example, consider the problem:
subject to ,

Figure shows the contours of the objective function, that is, the set of points for which f (x) has a constant value. It also illustrates the feasible region, which is the set of points satisfying all the constraints (the area between the two constraint boundaries), and the point x∗, which is the solution of the problem. Note that the “infeasible side” of the inequality constraints is shaded.

EXAMPLE: A TRANSPORTATION PROBLEM
We begin with a much simplified example of a problem that might arise in manufacturing and transportation. A chemical company has 2 factories F1 and F2 and a dozen retail outlets R1, R2, . . . , R12. Each factory Fi can produce ai tons of a certain chemical product each week; ai is called the capacity of the plant. Each retail outlet Rj has a known weekly demand of bj tons of the product. The cost of shipping one ton of the product from factory Fi to retail outlet Rj is cij .
The problem is to determine how much of the product to ship from each factory to each outlet so as to satisfy all the requirements and minimize cost. The variables of the problem are xij , i _ 1, 2, j _ 1, . . . , 12, where xij is the number of tons of the product shipped from factory Fi to retail outlet Rj ; see Figure . We can write the problem as:

Subject to