Engineering Optimization, 5th Edition

5 Nonlinear Programming I: One‐Dimensional Minimization Methods

5.1 Introduction

In Chapter 2we saw that if the expressions for the objective function and the constraints are fairly simple in terms of the design variables, the classical methods of optimization can be used to solve the problem. On the other hand, if the optimization problem involves the objective function and/or constraints that are not stated as explicit functions of the design variables or which are too complicated to manipulate, we cannot solve it by using the classical analytical methods. The following example is given to illustrate a case where the constraints cannot be stated as explicit functions of the design variables. Example5.2 illustrates a case where the objective function is a complicated one for which the classical methods of optimization are difficult to apply.

Example 5.1

Formulate the problem of designing the planar truss shown in Figure 5.1 for minimum weight subject to the constraint that the displacement of any node, in either the vertical or the horizontal direction, should not exceed a value δ.

Figure 5.1 Planar truss: (a) nodal and member numbers; (b) nodal degrees of freedom.

SOLUTION

Let the density ρ and Young's modulus E of the material, the length of the members l, and the external loads Q, R, and S be known as design data. Let the member areas A ₁,