# Exploring Evolutionary Algorithms for Multi-Objective Optimization in Seismic Structural Design
## 2 Background
### Heuristics and Metaheuristics  
Heuristic and metaheuristic algorithms play an important role in solving multi-objective optimization problems. These algorithms have been developed to find efficient solutions, especially in large and complex problem spaces. While heuristics are based on strategies specific to a particular problem structure, metaheuristic algorithms incorporate general problem-solving strategies and can be applied to a wide range of problems. Heuristic algorithms are based on rules or heuristics that produce fast and often good results for a given problem. The advantage of these algorithms is that they usually explore a large search space efficiently. However, heuristics do not guarantee an optimal solution and can get stuck at local optima. For example, the Greedy Algorithm proceeds by making the best available choice at each step. However, this method usually does not provide separate solutions for each objective in multi-objective optimization problems and is therefore limited in finding Pareto-optimal solutions [54]. Metaheuristic algorithms provide a general optimization strategy and can be used for a wide range of problems. These algorithms often overcome the shortcomings of heuristics and provide a more comprehensive search strategy. Some of these algorithms can be found in the literature, as mentioned in our paper. We can mention some of the basic ideas of these algorithms. Genetic algorithms (GAs) are inspired by biological evolution processes and are a population-based search method. GAs are often used to explore the Pareto front in multi-objective optimization problems. The mathematical foundation of these algorithms lies in its search within a solution space using genetic operators (selection, crossover, mutation). Genetic algorithms are effective in finding Pareto-optimal solutions due to their ability to explore a wide solution space in parallel [6]. Particle swarm optimization (PSO) is inspired by the swarming behavior of birds or fish in nature. PSO is a metaheuristic in which each member of a population (particle) searches for the global optimum by moving through a solution space. In this method, each particle acts based on its own experience and the experience of its neighbors. The mathematical model of PSO is described using differential equations involving velocity and position updates [55]. Mathematically, in PSO, the position and velocity of each particle are updated with the following equations:  
\[v_{i}(t+1)=wv_{i}(t)+c_{1}r_{1}\Big{(}p_{i}^{best}-x_{i}(t)\Big{)}+c_{2}r_{2} \Big{(}g^{best}-x_{i}(t)\Big{)} \tag{5}\]  
\[x_{i}(t+1)=x_{i}(t)+v_{i}(t+1) \tag{6}\]  
where \(v_{i}(t)\) is the velocity of the particle, \(x_{i}(t)\) is the position of the particle, \(p_{i}^{best}\) t is the best position of the particle, and \(g^{best}\) represents the best position of the swarm. PSO can also be applied to multi-objective optimization problems and can produce various Pareto-optimal solutions [7]. The standard simulated annealing (SA) approach can be extended in a variety of ways to address multi-objective optimization issues. It works similarly to the annealing process of material during heat treatment and tries to find solutions close to the global optimum without getting stuck in local minima. The SA algorithm has an important place in multi-objective optimization problems and attracts attention with its mathematical foundations. The simulated annealing algorithm mimics the process of heating and slowly cooling metals. In this process, initially at high temperatures, the atoms move more freely, and the energy of the system is high. As it gradually cools, the system transitions to lower-energy states and eventually stabilizes at the global minimum energy level. This process forms the basis of the SA algorithm, which aims for global optimization through a random search mechanism. Mathematically, the SA algorithm is based on a solution space \(S\) that obtains a new solution \(S^{\prime}\) from the existing solution \(S\).