## 3 Multiobjective Optimization Problem  
### General Information  
Multiobjective optimization problems are characterized by more than one objective function. Objective functions are usually contradictory, and therefore, cannot be improved simultaneously. The objective functions should be defined in such a way that they correspond to the real demands asked of the system under consideration. In case of electrothermal microactuators it is often desired of them to have low volume, generate high force and generate low stress under working conditions, to name a few demands which can be expressed as objective functions. The determination of objective function values can be done analytically, experimentally or numerically. Analytical methods can only be applied to very simple mechanical systems (which is not the case when considering actuators of an arbitrary shape) and multiple experimental determinations of function values may be too time-consuming or even impossible, as it would require manufacturing multiple microactuators of different shapes and a complex method of measuring values related to objective functions. For those reasons, when optimizing real mechanical systems, the values of objective functions are usually determined numerically using CAE (computer aided engineering) software.  
In multi-criteria problems, as opposed to single-criteria problems, where a single solution is sought, the aim is to find a set of solutions rather than a single solution. The set of solutions found should meet the design assumptions defined, as restrictions imposed on the problem and the values of the objective function should be at a level acceptable for the designer; e.g., maximal equivalent stress in the microactuator under working conditions cannot be higher than a value acceptable for a particular material used. Such an approach to the design problem is called Pareto optimization. The aim of multicriteria optimization is to find the vector of design variables describing the system, such as its geometry, material properties or boundary conditions (e.g., width of a microactuator beam, rounding radius between actuator beams and central shaft and pre-bending angle), so that the values of the functions are approaching extremes (e.g., value of vertical deflection and value of force generated). In the general case, the problem of multicriteria optimization can be presented as search of a vector of design variables.  
\[\mathbf{x}=\left[x_{1},x_{2},\ldots,x_{n}\right]\,, \tag{17}\]  
under \(m\) equality constraints,  
\[\forall i\in\{1,2,\ldots,m\}:\;\;g_{i}(\mathbf{x})=0, \tag{18}\]  
and \(p\) inequality constraints  
\[\forall j\in\{1,2,\ldots,p\}:\;\;h_{j}(\mathbf{x})\leq 0, \tag{19}\]  
to minimize \(k\) objective functions.  
\[f_{i}(\mathbf{x})=[f_{1}(\mathbf{x}),\,f_{2}(\mathbf{x}),\,\ldots,f_{k}( \mathbf{x})]. \tag{20}\]The decision domain is, thus, n-dimensional, and the solution domain is k-dimensional. Therefore, a set of non-dominated (or Pareto-optimal) solutions is sought, also called Pareto front. Considering two vectors \(\mathbf{x}\) and \(\mathbf{y}\) in the decision domain, vector \(\mathbf{x}\) strongly dominates \(\mathbf{y}\) if:  
\[\forall i\in\{1,2,\ldots,k\}:\;\;f_{i}(\mathbf{x})<f_{i}(\mathbf{y}) \tag{21}\]  
solution \(\mathbf{x}\) weakly dominates \(\mathbf{y}\), if:  
\[\forall i\in\{1,2,\ldots,k\}:\;\;f_{i}(\mathbf{x})\leq f_{i}(\mathbf{y})\;\; \wedge\;\;\;\exists j\in\{1,2,\ldots,k\}:\;\;f_{j}(\mathbf{x})<f_{j}(\mathbf{y}) \tag{22}\]  
and solution \(\mathbf{x}\) is neutral (incomparable) to the \(\mathbf{y}\), if:  
\[\exists i,j\in\{1,2,\ldots,k\}:\;\;f_{i}(\mathbf{x})<f_{i}(\mathbf{y})\;\; \wedge\;\;f_{j}(\mathbf{y})>f_{j}(\mathbf{x}) \tag{23}\]