# Hybrid Optimization and Explainable Deep Learning for Breast Cancer Detection
## 2 Proposed Methodology
### Optimization Algorithms  
#### 2.4.1 Algorithm for Dingo Optimization  
DOA [59] is a new bio-inspired metaheuristic based on the more advanced predation and survival techniques of Australian dingoes to solve complex, nonlinear, high-dimensional optimization issues. Several behaviors, including cooperative group attacks, individual over pursuit and opportunistic scavenging as well as behaviors that are flexible to environmental limitations, are integrated in DOA to have a successful balance between exploration and exploitation phases. DOA is applied in the long-established setting of constrained optimization that can be described as  
\[\underset{x\in R^{\eta}}{min}f(x)\text{ subject to }g_{i}(x)\leq 0,\ h_{j}(x)=0 \tag{1}\]  
where \(\overset{\rightarrow}{x}\in\text{Rn}\) represents the decision variables, \(f(\overset{\rightarrow}{x})\) is the objective function, and \(g_{i}\) and \(h_{j}\) denote inequality and equality constraints, respectively. This formulation follows the canonical representation of constrained optimization problems as outlined in foundational texts such as [PERSON] [60]. Four key dingo-inspired behaviors are mathematically modeled to guide the evolution of candidate solutions through the search space,forming the core operational logic of DOA. The update rules defined in Equations (2)-(5) are core components of the DOA, adapted from gravitational interaction concepts introduced by [61] in the Gravitational Search Algorithm (GSA) framework. These equations do not represent novel contributions but are included here to illustrate the internal mechanics of DOA. All symbols and parameters are defined at first mention to ensure clarity. The Group Attack strategy, which simulates coordinated hunting, updates the position of a dingo (candidate solution) as follows:  
\[\overline{x_{i}}(t+1)=\beta_{1}\Bigg{(}\frac{1}{n_{a}}\sum_{k=1}^{n_{a}} \overline{x_{atk}}(t)-\overline{x_{i}}(t)\Bigg{)}-\overline{x^{*}}(t) \tag{2}\]  
where  
* \(\overline{x_{i}}(t)\): current position of the \(i\)-th candidate;
* \(\overline{x_{atk}}(t)\): positions of \(n_{a}\) randomly selected attackers;
* \(\overline{x^{*}}(t)\): best-known solution so far;
* \(\beta_{1}\)\(\sim\)U(\(-2\), 2): stochastic scaling factor for update magnitude. The Persecution strategy mimics an individual dingo pursuing prey, as follows:  
\[\overline{x_{i}}(t+1)=\overline{x^{*}}(t)+\beta_{1}e^{\beta^{2}}(\overline{x_ {r1}}(t)-\overline{x_{i}}(t)) \tag{3}\]  
where  
* \(\overline{x_{r1}}(t)\): randomly selected peer position;
* \(\beta^{2}\)\(\sim\)U(\(-1\), 1): additional stochastic factor. The Scavenging strategy, simulating opportunistic feeding behavior, updates as  
\[\overline{x_{i}}(t+1)=\frac{1}{2}\Big{[}e^{\beta^{2}}\overline{x_{r1}}(t)-(-1 )\sigma\overline{x_{i}}(t)\Big{]} \tag{4}\]  
where \(\sigma\in\) [0, 1] is a binary variable that introduces randomness in direction. Lastly, the Survival Rate Adaptation strategy determines whether a dingo (solution) should be replaced based on its survival rate, as follows:  
\[\text{Survival}(i)=\frac{f_{max}-f_{i}}{f_{max}-f_{min}} \tag{5}\]  
where \(f_{i}\) represents the current solution's fitness and \(f_{max}\) and \(f_{min}\) represent the population's worst and greatest fitness values, respectively. Individuals with low survival probability are replaced using a hybridized movement. \[\overline{x_{i}}(t+1)=\overline{x^{*}}(t)+\frac{1}{2}[\overline{x_{r1}}(t)-(- 1)\sigma\overline{x_{2}}(t)] \tag{6}\]  
The algorithm employs a probabilistic controller with two switching parameters, P and Q, to decide whether the population enters hunting, scavenging, or survival mode in each iteration. This adaptive mechanism allows DOA to navigate rugged, multi-modal search spaces effectively, avoiding premature convergence while accelerating progress toward global optima.