ABSTRACT
Optimization problems occur for a diverse range of topics. Perhaps the simplest type of optimization problem involves a scalar function of several variables. For example, the cost of a product having several ingredients may need to be minimized. This problem can be represented by a function F (x) which depends on the vector x = [x1;x2; . . . ;xn] in n-dimensional space. Function F is called the objective function and cases where the independent variables x ı can vary arbitrarily are considered unconstrained. Most problems have constraints requiring x ı to remain within given bounds or satisfy other functional equations. Different analysis procedures exist for solving problems depending on whether they are linear or nonlinear, constrained or unconstrained. General solutions are available to handle linear objective functions with linear equality and inequality constraints. The discipline devoted to such problems is known as linear programming [41] and applications involving thousands of independent variables can be analyzed. 1 Although this class of linear problems is important, it does not offer the versatility of methods used to address nonlinear problems (which are more compute intensive for problems of similar dimensionality).2 The material in this chapter addresses nonlinear problems with a few independent variables which are either constrained or restricted to lie within bounds of the form
aı ≤ xı ≤ bı. This type of constraint can be satisÞed by taking
xı = aı + (bı − aı) sin2(zı) and letting zı vary arbitrarily. The MATLAB intrinsic functions fminbnd and fminsearch are employed for solving this class of problems. The following Þve examples are presented to illustrate the nature of nonlinear optimization methods:
1. Computing the inclination angle necessary to cause a projectile to strike a stationary distant object;
3. Determining components of end force on a statically loaded cable necessary to make the endpoint assume a desired position;
4. Computing the shape of a curve between two points such that a smooth particle slides from one end to the other in the minimum time;
5. Determining the closest points on two surfaces.
