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Feasibility condition: If apl < apl, l ∈ 1;…; np

¼ ξð Þ ½ � 0; 1 where ξ xpl

cp; ap D E

22 Optimization Algorithms - Examples

apl; apl h i � �, cpl <sup>&</sup>gt; <sup>0</sup>, <sup>P</sup>np

We propose a method to compute a constrained equilibrium that has a full support (whenever it exists). We do not use the projection operator. Indeed we transform the domain

Figure 3. Economic power dispatch. Evolution of the population states (generated power), fitness functions ^f að Þ¼ <sup>∇</sup>Eh að Þ ; <sup>ω</sup> , and the costs �Er að Þ ; <sup>ω</sup> . Figures (a)-(c) for case 1, (d)-(f) for case 2, (g)-(i) for case 3, and (j)-(l) for case 4.

< b, the constraint set A is non-empty, convex and compact.

� � <sup>¼</sup> aplxpl <sup>þ</sup> apl <sup>1</sup> � xpl

<sup>l</sup>¼<sup>1</sup> cplapl <sup>&</sup>lt; bp, cp <sup>∈</sup> <sup>R</sup>np

� � <sup>¼</sup> apl: <sup>ξ</sup> is a one-to-one mapping and

<sup>&</sup>gt;<sup>0</sup> and <sup>P</sup>

p∈P

Jian Gao, Yida Xu, Julian Barreiro-Gomez, Massa Ndong, Michalis Smyrnakis and Hamidou Tembine\*

\*Address all correspondence to: tembine@ieee.org

Learning and Game Theory Laboratory, New York University, Abu Dhabi, United Arab Emirates
