## 2 Multiobjective Optimization of Compliant Structures  
Methods for solving MOOPs can be distinguished into classical methods and evolutionary algorithms (EA). In general, classical methods need some information about the preferences of each objective or on the algorithmic parameters to find a single Pareto-optimal solution. Due to the complexity in solving a MOOP, it is usually converted into a single-objective optimization problem. The weighted sum method scalarizes the problem by defining a single objective function that results from the dot product of a user-supplied weight vector and the vector that contains all of the objective functions. This approach is the simplest and most widely used classical method, but it poses an issue that is related to the values to be assigned to the weights. They depend on the importance of each objective function, and, at the same time, on the normalization that must be conducted to make the objectives of the same order of magnitude to be considered equally in the solution. The values of these weights vary from problem to problem and, in general, predicting the appropriate weighting factors is impossible. Two interesting theorems from [PERSON] [27] state that a solution obtained by the weighted sum method is Pareto-optimal if all weights are positive, which has also been confirmed for convex problems. Multiple Pareto-optimal solutions can be found by repeating the same single-objective optimization and each time changing the positive weight vector. This approach does not mean that different weight vectors lead to different Pareto-optimal solutions because some solutions could be dependent on one another. Finally, to find \(m\) Pareto-optimal solutions, at least, if not more than, \(m\) different single-objective optimizations must be performed. Weighted sum methods do not definitively guarantee being able to find the Pareto-optimal solutions in the whole Pareto-optimal domain. In addition to these disadvantages, classical methods are affected by typical problems of direct and gradient-based optimization algorithms: they can fall into a locally optimal solution, cannot be parallelized, and the solution's convergence depends on the initial solution. On the other hand, EAs offer significant advantages when they are used to solve MOOPs, due to their capability to process an entire population of solutions in each generation. Therefore, a population of Pareto-optimal solutions is potentially captured after each generation, and increasing the initial population size allows us to capture as many Pareto-optimal solutions as possible. This eliminates the dependency on external parameters, such as weight vectors, and does not require repetitive use of single-objective optimizations to obtain Pareto-optimal solutions that could be weakly dominated by each other, which wastes computational effort. EAs can preserve the diversity of multiple non-dominated solutions inside the population during the entire process that gradually converges to the Pareto-front, ensuring a good spread and providing many different optimal solutions [24]. The synthesis of compliant mechanisms is the design of a particular type of structure that can convert a displacement or a force from an input point (actuator) to an output point to a different direction, using the deformation work efficiently. It is a mechanical component design problem that is usually solved as a single-objective optimization problem. The objective represents a kinematic requirement, such as the maximization of the motion or the force amplification. However, this kind of problem should be better posed as a minimum two-objective optimization problem. On the one hand, a compliant mechanism designed only to be flexible can meet kinematic requirements through deformation, but it could be too flexible to bear external loads. On the other hand, a compliant mechanism designed only to bear external loads could meet structural requirements but could require too much force to achieve a desired motion, producing high stresses as a result.