## 1 Introduction
### Overview of \(\mathrm{DQL}\) and Smart \(\mathrm{DQL}\) Method  
Some efforts have been made to hybridize direct search and model-based methods. For example, the SID-PSM method involves combining a search step of minimizing the approximated quadratic model over a trust region with the direct search [13; 14]. The RQLIF method, which we discuss next, provides a more versatile approach [15]. To understand the RQLIF method, we note that there are two common ways to find the next evaluation candidate during a model-based method [10]. First, methods can find the candidate at the minima of the surrogate model within some trust region or constraints. These are referred to as model-based trust-region (MBTR) methods. Second, methods can use the model to predict the descent direction and perform a line-search on the direction. These are referred to as model-based descent (MBD) methods. At each iteration, the RQLIF method searches for an improvement using three distinct strategies without relying on gradient or higher-order derivative information. These steps are referred to as the direct step, quadratic step, and linear step. These three steps correspond to three distinct search strategies from the direct search method, MBTR methods, and MBD methods. Inspired by the structure of the RQLIF method, we propose the DQL method framework. The purpose of this framework is to allow a flexible hybrid method that permits a direct search, quadratic-model search, and line-search all in the same method. Our objective is to design a framework that allows the development of a variety of search strategies and to determine the strategies that perform best. The DQL method is a local method for solving unconstrained BBO problems. We ensure its local convergence by implementing a two-stage procedure. The first stage focuses on finding an improvement in an efficient manner. It accepts an improvement whenever the candidate yields a better solution. We call this stage the exploration stage. The second stage focuses on the convergence to a local optimum; we call this stage the convergence stage. In Section 2, we introduce the DQL method's framework and the search strategies. In Section 3, we conduct the convergence analysis. Provided that the objective function has a compact level set \(L(x_{0})\) and the gradient of the objective function is Lipschitz continuous in an open set containing \(L(x_{0})\), the convergence analysis indicates that there exists a convergent subsequence of iterations with a gradient of zero at its limit. This demonstrates that when the evaluation budget is large enough, the method will converge to a stationary point. Using the framework of the DQL method, we obtain a series of combinations of quadratic and linear step strategies. In order to select the best combination among them, in Section 3, we perform a numerical benchmark across all the possible combinations. The quadratic step strategies are capable of improve the overall performance of the method. However, the linear steps show a mixed performance and there is no clear winner on efficiency or robustness. This inspires the idea that by employing an appropriate strategy in certain circumstances, we may be able to achieve an overall improvement in performance. This idea of allowing the method to make decisions on search strategies in various circumstances leads to the Smart \(\mathrm{DQL}\) method, which we discuss in Section 4. By analyzing the search results from various strategies, we develop decision processes that select the appropriate strategies for the search steps during the optimization. This allows the method to dynamically decide the appropriate strategies for the given information. In Section 4, we perform numerical tests on the Smart DQL method and discover that the Smart DQL method outperforms the DQL methods in terms of both robustness and performance. In Section 5, we apply the Smart DQL method to the problem of design of solid tanks for optical computed tomography scanning of 3D radiation dosimeters described in [16].