Sentence examples for prove the convexity of from inspiring English sources

To begin with, we need to prove that φ u) is a convex function which will be applied to prove the convexity of U2.

Next, we prove the convexity of ℱ.

In what follows, we will first prove the convexity of (f v)) on (mathbb{R}).

We further prove the convexity of ASR in the power definition domain of a secondary network.

In order to prove the convexity of (10), let denote the objective function in (10), that is, (24).

The subproblem is also simplified such that it is a function of only one parameter, and we prove the convexity of the resultant single-variable piecewise optimization problem.

Importantly, we proved the convexity of the objective function of the MLE so that Sigma can guarantee to attain a global optimum solution using interior point non-linear programming (NLP) provided in the Ipopt library (Wachter and Biegler, 2006).

Furthermore, by using the same techniques as in the proof of Theorem 3.1, we can prove the convexity (univalence) of and so the details may be omitted.

By noting that 0 < ρ ≦ 1 / 2 from (2.2), we obtain from the condition (2.20) that ψ is a close-to-convex function in U (see [24]), and hence ν f is univalent in U. Furthermore, by using the same techniques as in the proof of Theorem 3, we can prove the convexity (univalence) of F and so the details may be omitted.

For special cases subject to the constraint of 100% service level, we prove the convexity and continuous differentiability of the objective function.

The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector instead of the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case.