## 3 Hierarchical Coevolution Algorithms for Multi-Criteria Optimization under Uncertainty
### Algorithm of Hierarchical Coevolution Search for Set of -Minimax Solutions to the MOU Problem
_Step 2._ Form a hierarchical population game (10). The first move is made by the coordinating center C\({}_{0}\)--it tells the lower-level player its population strategy \(\widetilde{\mathbf{U}}\subset\mathbf{U}\). _Step 3._ With a fixed population strategy \(\widetilde{\mathbf{U}}\subset\mathbf{U}\), the lower-level player solves problem (10), where  
\[\Gamma_{i}=\left\langle\widetilde{\mathbf{u}}^{-i},\mathbf{Z},\mathbf{R}( \mathbf{u},\mathbf{z}),\mathbf{F}\left(\widetilde{\mathbf{u}}^{i},\widetilde{ \mathbf{Z}}^{-i}\right),\mathbf{\Lambda}\right\rangle \tag{18}\]  
In (18), the player's C vector criterion is given as  
\[\mathbf{F}\left(\widetilde{\mathbf{u}}^{-i},\widetilde{\mathbf{Z}}^{i}\right) =\mathbf{P}_{\mathbf{\Lambda}}\left(\mathbf{R}\left(\widetilde{\mathbf{u}}^{- i},\widetilde{\mathbf{Z}}^{-i}\right)\right) \tag{19}\]  
The optimal solution to problem (18) is a population strategy \(\widetilde{\mathbf{Z}}_{\mathbf{\Lambda}}^{i}\subset\mathbf{Z}\), that maximizes the components of the vector criterion (19). _Step 4._ The coordinating center evaluates the effectiveness of the population strategy \(\widetilde{\mathbf{U}}\subset\mathbf{U}\) by calculating the value of the function \(F_{0}\left(\widetilde{\mathbf{U}}\right)\) in accordance with rules (14)-(16). Step 5 The coordinating center solves the problem (17). An optimal solution to the problem (17) \(\widetilde{\mathbf{U}}_{\max}\) is called the \(\mathbf{R}_{\Omega}\)-guaranteeing population strategy of the coordinating center. **Definition 8**. : _Let the population strategy\(\widetilde{\mathbf{U}}_{\max}\) be the optimal solution to the problem (17), where the function \(F_{0}\Big{(}\widetilde{\mathbf{U}}\Big{)}\) is calculated according to the rules (14)-(16), and the population strategies \(\widetilde{\mathbf{Z}}_{\mathbf{\Lambda}}^{i}\) be the optimal solutions to the problem (18) at \(i=\overline{1,\,\left|\widetilde{\mathbf{U}}\right|}\). Then the set of population strategies \(\Big{\{}\widetilde{\mathbf{U}}_{\max},\,\Big{(}\widetilde{\mathbf{Z}}_{ \mathbf{\Lambda}}^{i},\,i=\overline{1,\,\left|\widetilde{\mathbf{U}}\right|} \Big{)}\Big{\}}\) is called the \(\mathbf{R}_{\Omega}\)-equilibrium solution of the hierarchical population game (9)._  
**Theorem 2**. : _Let:_  
1. \(\left\{\widetilde{\mathbf{U}}_{\max},\,\Big{(}\widetilde{\mathbf{Z}}_{ \mathbf{\Lambda}}^{i},\,i=\overline{1,\,\left|\widetilde{\mathbf{U}}\right|} \Big{)}\right\}\)_--a set of population strategies, which is the_ \(\mathbf{R}_{\Omega}\) equilibrium solution of a hierarchical population game with the right of the first move of the form (9);_
2. \(\widetilde{\mathbf{Z}}_{\mathbf{\Lambda}}^{i}\)_--optimal solutions to the problem (18) when \(i=\overline{1,\,\left|\widetilde{\mathbf{U}}\right|}\);
3. \(\widetilde{\mathbf{U}}_{\max}\) is the optimal solution to the problem (17), where the objective function \(F_{0}\Big{(}\widetilde{\mathbf{U}}\Big{)}\) is calculated in accordance with the rules (14)-(16)._  
_Then the population strategy \(\widetilde{\mathbf{U}}_{\max}\subset\mathbf{U}_{\mathbf{R}_{\Omega}}\), where\(\mathbf{U}_{\mathbf{R}_{\Omega}}\) is the set of \(\mathbf{R}_{\Omega}\)-minimax solutions to the MOU problem of the form (1)._  
The peculiarity of algorithm 2 is that the vector risk function \(\mathbf{R}(\mathbf{u},\mathbf{z})\) is used to solve problem (18). In this case, the calculation of the function \(\mathbf{R}(\mathbf{u},\mathbf{z})\) is a separate problem, for the solution of which the following coevolution algorithm is proposed.