For the case of neglecting the valve effects, the cost function is approximately represented as a second order equation called

convex function but for the case of considering the valve effects, the cost function is approximately represented a nonconvex function, which is a sum of a

convex function and a sinusoidal function similar to (1).

Let f : I [right arrow] R be a

convex function defined on the interval I of real numbers and a,b [member of] I with a < b.

The concepts of

convex function and subgradient are expanded from the real field to the complex field.

The main purpose from here on is to demonstrate that the game induced by a convex/concave function is equivalent to the game induced by a concave/

convex function.

It is an explicit rational, strictly

convex function for all x such that [L.

n] [right arrow] R is a continuously differentiable

convex function.

1] [right arrow] R U {+[infinity]} is a proper

convex function.

Though (16) does not satisfy the condition of a convex optimization in which the left side of a less inequality should be a

convex function, it can not prevent us from finding the minimum value of t ([t.

Considering the problem of the validity of the Euler-Lagrange equation from a different point of view, in [1], the authors introduced a method to avoid the assumption of differentiability of the

convex function L to establish the validity of the Euler-Lagrange equation.

Let [phi] be a

convex function and f [member of] [R.

However, since the energy characteristics of the HVAC systems can be adequately approximated as a

convex function (Ann et al.

2001, Inequalities for generalize weighted mean values of

convex function, Math.