Efstratios N. Pistikopoulos - Multi-parametric Optimization and Control

The closed line segment joining the points , is defined as:

(1.2)

A set is a convex set, if the closed line segment joining any two points in the set belongs to the set for each such that .

Let and be convex sets defined in . Then

1 The intersection of is a convex set.

2 The summation of two convex sets is a convex set.

3 Let be a real number. The product is a convex set.

Examples of convex sets include lines, polytopes and polyhedra, and open and closed halfspaces.

Let be a convex subset, and the real function defined in . The function is a convex function if for any , ,

(1.3)

If strict inequality holds in expression ( 1.3) for , then is a strictly convex function.

Let be a convex subset, and the real function defined in . The function is a concave function if for any , ,

(1.4)

If strict inequality holds in expression ( 1.4) for , then is a strictly concave function.

1 Let be convex functions defined on a convex subset . Their summation(1.5) is convex, and if at least of one is a strictly convex function, then their summation is strictly convex.

2 Let a be a positive number and be a (strictly) convex function defined in a convex subset . Then the product is (strictly) convex.

3 Let be a (strictly) convex function defined in , and be an increasing convex function defined on the range of in . Then, the composite function defined in is a (strictly) convex function.

4 Let be convex functions defined on a convex subset . If these functions are bounded from above, their pointwise supremum(1.6) is a convex function on .

5 Let be concave functions defined on a convex subset . If these functions are bounded from below, their pointwise infimum(1.7) is a concave function on .

We introduce the following definitions for the solution of general nonlinear optimization problems:

is called a local minimum if there exists ball of radius around , , such that

(1.8)