EXACT GLOBAL OPTIMIZATION.

Constrained optimization problems are problems for which a function f (x) is to be minimized or maximized subject to constraints ф (x) . Here f : Rn → R is called the objective function and F(x) is a Boolean-valued formula. In Mathematica the constraints ф (x) can be an arbitrary Boolean combination of equations g (x) →0 , weak inequalities g (x) >=0, strict inequalities g(x)> 0, x is integer and x>0 statements. A point u ∈ Rn is said to be a global minimum of f subject to constraints F if u satisfies the constraints and for any point v that satisfies the constraints, f (u) ≤ f(v). A value a∈ (- ∞,∞) is said to be the global minimum value of f subject to constraints F if for any point v that satisfies the constraints, a ≤ f(v). A value a∈ (-∞,∞) is said to be the global minimum value of f subject to constraints F if for any point v that satisfies the constraints, a ≤ f (v). The global minimum value a exists for any f and ф. The global minimum value a is attained if there exists a point u such that ф (u) is true and f(u)= a. Such a point u is necessarily a global minimum. If f is a continuous function and the set of points satisfying the constraints ф is compact (closed and bounded) and nonempty, then a global minimum exists. Otherwise a global minimum may or may not exist. Here the minimum value is not attained. The set of points satisfying the constraints is not closed. Exact global optimization problems can be solved exactly using Minimize and Maximize. [ABSTRACT FROM AUTHOR]