# Fast, Lightweight, and Efficient Cybersecurity Optimization for Tactical-Operational Management
## 2 Problem Modeling and Formulation
### Formulation of the Multi-Objective Optimization Problem  
Let \(\mathbb{P}\) be the multi-objective optimization problem [23; 24]. Let \(S=(\overrightarrow{x}_{1},\overrightarrow{x}_{2},\ldots,\overrightarrow{x}_ {n})\) be the set of feasible solutions for the optimization problem. Let \(\overrightarrow{x}=(x_{1},\,x_{2},\,\ldots,\,x_{n})\) be the vector representation of an asset's cybersecurity status, where each element \(x_{k}\) denotes the degree to which the required cybersecurity measures have been implemented to achieve the expected outcome \(k\). The length of the vector is determined by \(n\), which varies depending on the cybersecurity criticality of the asset. We define \(f1(\overrightarrow{x})\) as a real-valued function that quantifies the number of strategic cybersecurity constraints satisfied by the vector \(\overrightarrow{x}\). Similarly, we define \(f2(\overrightarrow{x})\) as a real-valued function that captures the similarity between the current asset's cybersecurity state and a previously recorded state. Finally, we define \(f3(\overrightarrow{x})\) as a real-valued function that characterizes the overall level of cybersecurity achieved by the asset, as determined by its current cybersecurity status represented by \(\overrightarrow{x}\).  
Formally, we express \(f1(\overrightarrow{x})\), \(f2(\overrightarrow{x})\), and \(f3(\overrightarrow{x})\) as functions that belong to the set of real numbers (\(\mathbb{R}\)), such that \(0.0\leq f1(\overrightarrow{x})\), \(f2(\overrightarrow{x})\), \(f3(\overrightarrow{x})\leq 1.0\). These functions are designed to satisfy mathematical properties that allow for their effective use in the optimization process, which ensures that their values are meaningful and can be used to compare different solutions in a mathematically rigorous manner.  
FLECO computes the individual fitness by means of a scalarization function, as shown in Equation (1). Specifically, the weighted sum scalarization function used is \(f(\overrightarrow{x})=\sum_{i=1}^{3}f_{i}(\overrightarrow{x})\cdot\omega_{i}\), where \(\omega_{i}\) is the weight associated to each objective (\(\sum_{i=1}^{3}\omega_{i}=1.0\)). The values of \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) were determined after an extensive analysis process. During this period, hundreds of FLECO executions were performed with different initial statuses and various strategic constraints. These executions were supervised by the organizations' decision-makers, who worked together with experts and the rest of the team to tune the weights until the convergence time of FLECO was deemed acceptable, and the generated solutions met the requirements of the organization. Finally, the values of \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) were established as \(\omega_{1}=0.94\), \(\omega_{2}=0.05\), and \(\omega_{3}=0.01\), which were deemed to be the optimal weights for the FLECO algorithm.  
We define the multi-objective problem as follows:  
\[\begin{split}\textit{maximize}&\quad f( \overrightarrow{x})=0.94\cdot f1(\overrightarrow{x})+0.05\cdot f2( \overrightarrow{x})+0.01\cdot f3(\overrightarrow{x})\\ \textit{subject to}&\quad f1(\overrightarrow{x}) =1.00\;\forall\overrightarrow{x}\in S\end{split} \tag{1}\]  
The function denoted by \(f1(\overrightarrow{x})\) serves to amalgamate the set of strategic cybersecurity constraints [25]. This approach was chosen to provide guidance to the algorithm towards generating a high-quality set of feasible solutions. Consequently, while non-feasible solutions persist within the population, they are not regarded as solutions.