# Deriving Controllable Local Optimal Solutions through an Environment Parameter Fixed Algorithm
## 2 Background
### Optimization  
In the this section, the concept of optimization and the optimizer of the training algorithm are explained. \[\hat{x}=\underset{x}{argmax}f(x) \tag{5}\]  
In Equation (5) [1], optimization is the set of variables \(x\) for which objective function \(f\) can reach an optimal value. Optimization problems arise in various areas of mathematics, including applied sciences, engineering, economics, medicine, and statistics [1]. These problems involve finding the set of variables, denoted as \(x\), for which an objective function \(f\) can achieve an optimal value. Depending on the scope of the search area, optimization can be classified into global optimization and local optimization [2; 29]. Global optimization aims to find the best solution over the entire search area, while local optimization focuses on finding the best solution within a specific region. When constraints are imposed on the allowable values of variables, such as independent variables, it becomes a constraint optimization problem [2]. Table 1 illustrates various problems that fall under the category of optimization problems. These problems require formulating objective functions and employing algorithms to obtain the optimal solutions. For instance, in linear programming problems, the production cost optimization can be achieved by formulating objective functions using appropriate formulas [30]. In the financial field, quadratic programming offers solutions by modeling the objective function as a quadratic function [31]. Consequently, problems are defined based on the structure of the objective function and the shape of the constraint function. However, the modeling of complex engineering systems is computationally expensive [32; 33]. Thus, this paper focuses on constraint optimization where a data-driven objective function, reflecting multiple environments, is applied within a specific environment. The data-based modeling process is a form of meta-modeling that involves acquiring data and has the advantage of simplifying complex models [33]. We raise a new problem, the trained DNN objective optimization, from the existing problems. Trained DNN objective optimization is constraint optimization in which a data-driven objective function reflecting multiple environments is applied in a specific environment. As a type of metamodelling, [36], deep learning-based regression techniques offer the advantage of modeling situations, even in non-ideal circumstances where traditional modeling approaches face difficulties. For example, in the domain of fish weighing in turbid water, deep learning regression, such as the one proposed by [5], can accurately estimate weights under non-ideal conditions. Trained deep neural network (DNN) objective optimization is a similar approach to modeling with metamodel-based simulation optimization (MBSO) [33; 36] and metamodeling, but it differs in terms of its focus on considering a specific environment. Modeling these non-ideal situations requires redefining the optimization variables. The outcome of deep learning regression, denoted as \(\hat{y}\), is influenced by various inputs, including the environmental conditions during data acquisition. We can solve the problem by defining an environment parameter (**E**) and a control parameter (**C**). The control parameters are defined as optimization variables, while the environment parameter represents the specific environmental conditions and is not subject to optimization. The consideration of **C** and **E** is necessary due to the data-driven modeling approach, and they are independent of each other within the input parameter **X**. \[\textbf{X} =[x_{1},x_{2},\cdots,x_{\varphi}]=[\textbf{E},\textbf{C}] \tag{6}\] \[\textbf{C} =[c_{1},c_{2},\cdots,c_{m}]\] (7) \[\textbf{E} =[e_{1},e_{2},\cdots,e_{n}]\] (8) \[\varphi =n+m \tag{9}\]  
The term \"controllable\" originates from the concepts of environment parameters and control parameters. It refers to a constraint optimization problem with specific environment parameters.