# Deriving Controllable Local Optimal Solutions through an Environment Parameter Fixed Algorithm
## 1 Introduction  
Optimization can arrive at the most suitable solution from among various solutions by finding the parameters that maximize or minimize the value of the function in an objective function [1]. Constraint optimization is a branch of optimization which deals with problems where the objective function is subject to certain constraints [2]. Another type of optimization problem is based on the specific definition of the objective function, such as quadratic programming (QP), which optimizes a quadratic objective function [3]. These different types of optimization problems find applications in solving engineering problems, including optimizing space vehicle trajectories, designing cost effective civil engineering structures such as frames, foundations, bridges, towers, chimneys, and dams, and addressing various forms of random loading [4]. However, accurately representing the objective function in non-ideal scenarios influenced by diverse environmental factors poses challenges in optimization problems. To address these challenges, we introduce a novel optimization problem called trained Deep Neural Network (DNN) objective optimization. This approach utilizes a trained DNN as the objective function. Since the DNN reflects environmental constraints as inputs to the model in [5], environment parameters (E) and control parameters (C) have been defined. The term E refers to the diverse set of measured values associated with the environment, such as temperature and humidity. On the other hand, C refer to the factors that enable manipulation and control over the system. Additionally, parameters C and E exhibit an independent relationship. Figure 1 illustrates the relationship between the defined parameters and input parameters. A trained DNN Controllable Local optimal Solution (CLS) is defined as the solution of thetrained DNN objective optimization and represents an optimal control parameter when a specific environment parameter is used. One of the key advantages of defining the objective function as a trained DNN is its flexibility. Not only can functions be modeled-based on data through deep learning-based regression [6; 7; 8; 9; 10], but they can also incorporate non-ideal cases influenced by various environments [5]. Data-driven approaches can accurately capture the effects of diverse environments on both input and output data [11]. Moreover, objective function modeling is adaptable and can be applied to control problems, within the definition of regression. The function value of the objective function can be defined by the user. In Figure 2, the current input is represented by a red dot. In such a situation, the user can identify a position (blue dot) where the function value can be optimized and adjust the input parameter accordingly. This allows the user to acquire data aligned with their desired purpose and then model the objective function for control purposes. To obtain the solution for optimizing the Trained Deep Neural Network (DNN) objective, we propose the Environment Parameter Fixed Algorithm with Controllable Local Optimal Solution (EPFA-CLS). The EPFA-CLS algorithm reduces the original optimization problem to an optimization problem for control parameters by fixing the environment parameters in the DNN defined objective function [12]. Subsequently, the CLS is obtained through the application of Gradient Descent (or Gradient Ascent) optimization techniques. By incorporating environmental constraints and leveraging deep learning-based regression analysis, our proposed EPFA-CLS algorithm offers a promising approach to obtaining controllable local optimal solutions for complex engineering problems. Figure 1: The relationship between input parameters, environment parameters, and control parameters. Environment parameters and control parameters are defined as input parameters. Figure 2: Defining the objective function as a trained DNN can provide an example of how it can be used for control problems.