Engineering Optimization: Theory and Practice, Fourth Edition

7.2 Characteristics of a Constrained Problem 381

Table 7.1 Constrained Optimization Techniques

Direct methods Indirect methods

Random search methods Transformation of variables technique
Heuristic search methods Sequential unconstrained minimization
Complex method techniques
Objective and constraint approximation Interior penalty function method
methods Exterior penalty function method
Sequential linear programming method Augmented Lagrange multiplier method
Sequential quadratic programming method
Methods of feasible directions
Zoutendijk’s method
Rosen’s gradient projection method
Generalized reduced gradient method

Figure 7.1 Constrained and unconstrained minima are the same (linear constraints).

However, to use these conditions, one must be certain that the constraints are not

going to have any effect on the minimum. For simple optimization problems like

the one shown in Fig. 7.1, it may be possible to determine beforehand whether

or not the constraints have an influence on the minimum point. However, in

most practical problems, even if we have a situation as shown in Fig. 7.1, it will

be extremely difficult to identify it. Thus one has to proceed with the general

assumption that the constraints have some influence on the optimum point.

2.The optimum (unique) solution occurs on a constraint boundary as shown in

Fig. 7.2. In this case the Kuhn–Tucker necessary conditions indicate that the

negative of the gradient must be expressible as a positive linear combination of

the gradients of the active constraints.