# Exploring Evolutionary Algorithms for Multi-Objective Optimization in Seismic Structural Design
## 2 Background
### Multi-Objective Optimization  
Multi-objective optimization focuses on solving problems where multiple objectives must be optimized simultaneously. Such problems are often found in fields such as engineering, economics, logistics and artificial intelligence. Optimization problems generally considered in the literature have a single objective. In real life, we usually need to optimize several goals simultaneously. For instance, we would like to increase a product's performance while also attempting to keep costs as low as possible. We are working with multi-objective optimization issues in this particular case. Several different concepts are needed to solve multi-objective optimization problems. Additionally, there may be conflicts between these several aims, necessitating some trade-offs. Multi-objective optimization in seismic design arises from the necessity of optimizing multiple objectives simultaneously for a structure. These objectives usually include basic elements such as cost, safety, and structural performance. While cost optimization aims to minimize the cost of the construction of the structure and the energy dissipation devices used, safety optimization aims to maximize the resistance of the structure against earthquakes and its capacity to protect human life. Structural performance aims to provide minimum damage by optimizing the behaviors of the structure such as vibration, displacement, and acceleration during an earthquake. In multi-objective optimization problems, Pareto-optimal solutions are usually used because a balance must be achieved between these objectives. In other words, improving one objective is not possible without worsening another objective. For example, stronger materials or additional energy dissipation devices used to increase safety may increase the cost. Therefore, multi-objective optimization in structural design aims to find the best solutions that balance both objectives. In this study, genetic algorithms and evolutionary algorithms, particularly particle swarm optimization, were used as effective tools to solve such multi-objective optimization problems. The mathematical foundations of multi-objective optimization are based on the concept of Pareto-optimality and various optimization algorithms. One of the most fundamental concepts of multi-objective optimization is the Pareto optimum. A solution is considered Pareto-optimal when no other solution improves it for each objective. Mathematically, a solution \(x^{*}\) is Pareto-optimal if no other solution \(x\) is better for all objectives. That is, it satisfies the conditions \(f(x^{*})\leq f(x)\) and \(f(x^{*})\
eq f(x)\). The Pareto front represents the set containing all such optimal solutions [6]. Since the objective function of single-objective optimization issues always reaches a single global optimal value, or a scalar, they can be categorized as scalar optimization problems. Since the several objective functions in multi-objective optimization combine to form a vector, the technique is also known as vector optimization. Basically, any multi-objective optimization problem can be expressed as  
\[\min_{x\in\mathbb{R}^{D}}f(x)=[f_{1}(x),\;f_{2}(x),\ldots,f_{M}(x)] \tag{1}\]  
subject to  
\[g_{j}(x)\leq 0\;(j=1,2,\ldots J) \tag{2}\]  
\[h_{k}(x)=0\;(k=1,2,\ldots K) \tag{3}\]  
where the vector of decision variables is denoted by \(x=(x_{1},x_{2},...,x_{D})^{T}\). \(g_{j}(x)\) and \(h_{k}(x)\) are inequality and equality constraints, respectively. Functions \(f_{i}(x)\) often express conflicting objectives. Mathematically, the goal is to minimize or optimize one or more objective functions, but these objectives are often in conflict with each other, so often, the best solution cannot be found for all objectives at the same time. The response space, also known as the objective space, is the space \(\mathbb{R}^{M}\) generated by all possible values of the objective functions.