In this regard, we write the non convex functions in a difference of concave functions
(DC) form, then DC approximation is applied to convert the optimization problem to a standard form of convex optimization.
The key features of this method are as follows: it has the possibility of simultaneous optimization of a large number of parameters; it has a high level of efficiency and convergence even for bad-conditioned problems; it can overcome local minimums; it has the capability of minimization of concave functions
, that is, functions with negative definite Hessian matrix; and finally, when the second-order Taylor expansion is used as an approximating function, the problem of the objective function minimization reduces to the problem of minimizing a quadratic functional .
In Section 2 we study a characterization of concave functions
defined on a Thspider and we are dealing with the localization of minimum and maximum of a concave function
defined on a Thspider.
By , [[alpha].sub.i][I.sub.i](x) + [[alpha].sub.i+1][I.sub.i+1] (x) is a strictly increasing concave function
. The infimum of concave functions
is also concave.
Therefore, this technique is not comparable with genetic algorithm, since it could be applied for only one field of problems and is not applicable for concave functions
, although genetic algorithm has no limits.
The three are variants of Jensen's inequality, which is concerned with convex and concave functions
. The purpose ofthis paper is to measure the size and statistical significance of these three inequalities.
The algorithm was developed as a particular case of the simplified algorithm for minimizing separable concave functions
over linear polyhedra (see Falk and Soland).
If [[J[double prime].sub.1] + [J[double prime].sub.2]] are non-positive, and R ([center dot]) and [Alpha] ([center dot]) are concave functions
, then the optimal number of inspections in a production run is one.
have a unique maximal point whereas convex functions have a unique minimal point.
As the product of real number and concave function
and the sum of several concave functions
are both concave, it is easy to deduce that the objective function R([lambda], t) is concave too.
Let f : [a, a + [eta](mb,a)] [right arrow] (0,[infinity]) be an integrable harmonic (p, h, m)-preinvexfunction, and let h be a concave function
The first two conditions ensure that the probability density function will map a concave function
onto another concave function