# Qom--A New Hydroologic Prediction Model Enhanced with Multi-Objective Optimization
## 2 Background on Multi-Objective Optimization with Metaheuristics  
This section is devoted to providing a basic background of multi-objective optimization with metaheuristics, which is needed to understand the calibration scheme we propose. Many real-world optimization problems can be formulated as the maximization and/or minimization of two or more conflicting functions or objectives at the same time. These problems are known as multi-objective optimization problems, and they can be found in many disciplines such as civil engineering ([PERSON], 1997), economics ([PERSON], 1997), telecommunications ([PERSON], 1997), bioinformatics ([PERSON], 1997),agriculture [20], and so forth. A multi-objective optimization problem can be defined as follows (we assume minimization of all the functions, without loss of generality):  
**Definition 1** (Multi-objective optimization problem). : _A multi-objective optimization problem is a tuple \((S,f,g,h)\), where \(S\
eq\emptyset\) is called the solution space (or search space), \(f=[f_{1},f_{2},\ldots,f_{k}]\), with \(k>=2\), is a vector of functions, where \(f_{i}:S\rightarrow\mathbb{R}\), are the objective functions and \(g=[g_{1},g_{2},\ldots,g_{m}]\) and \(h=[h_{1},h_{2},\ldots,h_{p}]\) are also vectors of functions, where \(g_{i}:S\rightarrow\mathbb{R}\) and \(h_{i}:S\rightarrow\mathbb{R}\) are the constraint functions. Thus, solving an optimization problem consists of finding a set of solutions \(X^{*}\subseteq S\) such that, for all \(x^{*}\in X^{*}\):_  
\[f_{j}(x^{*})\leq f_{j}(x),\ \ \ \ \forall\ \ x\in S, \tag{1}\]  
_for some \(1\leq j\leq k\), subject to:_  
\[g_{i}(x^{*}) \leq 0\ \ \ i=1,2,\ldots,m \tag{2}\] \[h_{i}(x^{*}) = 0\ \ \ i=1,2,\ldots,p, \tag{3}\]  
_where \(g_{i},h_{j}:S\rightarrow\mathbb{R}\), \(i=1,...,m\), \(j=1,...,p\) are the constraint functions of the problem._  
The set of all values satisfying all the equality and inequality constraints defines the _feasible region_\(\Omega\), so any point \(\vec{x}\in\Omega\) is a _feasible solution_. For simplicity, in the following definitions we will consider \(\Omega=S\). A key difference between mono- and multi-objective optimization is that in the latter, vectors of values (instead of a single value) must be compared to determine which is the best of them. In this sense, a basic concept in multi-objective optimization is _Pareto dominance_, which is defined as follows:  
**Definition 2** (Pareto dominance). : _Given two vectors \(\vec{x},\vec{y}\in\mathbb{R}^{k}\), we say that \(\vec{x}\leq\vec{y}\) if \(x_{i}\leq y_{i}\) for \(i=1,...,k\), and that \(\vec{x}\) dominates \(\vec{y}\) (denoted by \(\vec{x}\prec\vec{y}\)) if \(\vec{x}\leq\vec{y}\) and \(\vec{x}\
eq\vec{y}\)._  
Given two solutions in objective space, Pareto dominance indicates whether one _dominates_ the other one or, on the contrary, that the compared solutions are _non-dominated_, that is, none of them is strictly better than the other in all the objective values. This way, solving the multi-objective problem can be viewed as the process of finding the set of solutions that dominates every other point in the solution space and all the solutions of that set is said to be _Pareto optimal_ for that problem. Formally:  
**Definition 3** (Pareto Optimality). : _A solution \(x^{*}\in\mathcal{S}\) is Pareto optimal if it is non-dominated by any other solution \(x^{\prime}\in\mathcal{S}\)._  
The set of solutions composed of all the Pareto optimal solutions is known as _Pareto optimal set_ (or Pareto set):  
**Definition 4** (Pareto Optimal Set). : _The_ **Pareto Optimal Set**\(\mathcal{P}^{*}\) is defined as:_  
\[\mathcal{P}^{*}=\{x\in\mathcal{S}|x\text{ is Pareto optimal.}\}\]  
The solutions in the Pareto set belong to the variable space (\(S\)), and their correspondence in the objective space (\(\mathbb{R}^{k}\)) is a set known as _Pareto front_:  
**Definition 5** (Pareto Front).