Models
 Terminology
 Terminology
 Model Components The objective and constraint functions are scalar quantities that vary with the decision vector. The functions may be nonlinear. General Model Every mathematical programming model can be put in this form. General discussions assume this standard format. Global Minimum A solution that has an objective value less than or equal to all other solutions. Unique Global Minimum A solution that has an objective value less than all other solutions. Not all problems have a strict global minimum. Weak Local Minimum A solution that has an objective value less than or equal to all other solutions in a small neighborhood of the solution. Strong Local Minimum A solution that has an objective value less than all other solutions in a small neighborhood of the solution. Global and Local Maximum Definitions are the same as the definitions of global and local minimum with greater than replacing less than. Convex Function When a straight line is drawn between any two points on a convex function, the line lies on or above the function. The figure shows a one dimensional convex function. In multiple dimensions a convex function has the following property. Concave Function When a straight line is drawn between any two points on a concave function, the line lies on or below the function. The figure shows a one dimensional concave function. In multiple dimensions a concave function has the following property. Feasible Region The feasible region is the set of all solutions that satisfy all of the constraints. Convex Set A set S is convex if any point on the line segment connecting any two points in the set is also in S. The figure shows examples of convex sets in two dimensions. An important issue in nonlinear programming is whether the feasible region is convex. When the all the constraints of a problem are linear or convex, the feasible region is a convex set. If the objective function is a convex and the feasible region defines a convex set, every local minimum is a global minimum. If the objective function is not a convex function, a local minima may or may not be global minimum. A nonlinear programming algorithm may terminate at a solution that is not a global minimum. Nonconvex Set If a set does not satisfy the requirements of convex set, it is a nonconvex set. The figure shows examples of nonconvex sets in two dimensions. If the feasible region of a problem is a nonconvex set, a local minimum may or may not be global minimum. A nonlinear programming algorithm may terminate at a solution that is not a global minimum.

Operations Research Models and Methods
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by Paul A. Jensen