## 3 Relevant Research in Wind Farm Design and Optimization
### Solution Methods
#### 3.3.1 Calculus-Based Optimization  
Calculus-based optimization (or exact optimization) encompass all deterministic and non-deterministic algorithms and iterative methods that rely on the first and second order derivative of the objective function to explore the search space with the aim of finding the optimal solution(s). Calculus-based algorithms require that the objective function is Lipschitz continuous (_i.e._, it satisfies the Lipschitz continuity condition) and is differentiable. In addition, it is required that the search space is convex/concave and bounded to guarantee convergence to the optimal solution. The most common calculus-based optimization algorithms are the gradient-based, Newton-based, branch-and-bound, branch-and-cut, Extended Cutting Plane (ECP), Generalized Benders Decomposition (GBD), Outer-Approximation (OA), and backtracking, among others [323; 324; 325; 326; 327]. The most basic optimization techniques perform an explicit enumeration of all admissible solutions to obtain the optimal solution. However, for most problems this is computationally expensive. Local search calculus-based algorithms, such as the gradient descent method or Newton-based methods cannot solve non-convex problems to optimality, since they don't have the required capacity to overcome local extreme points. To overcome this issue, procedures based on global search [328; 329] and multi-start procedures have been proposed in the literature to improve its performance. These approaches perform a local search from multiple feasible starting points and sample multiple basins of attraction (_i.e._, sets of initial starting points that converge to the same local extreme point) to assess if an optimal solution is obtained. The limitations of such approaches are that the objective function must be continuous, differentiable, bounded, and must not be too complex, otherwise a significant amount of starting points will be required to properly sample the search space. Different approaches in the literature have addressed the solution of small and simplified versions of the WFDO problem using gradient-based algorithms [27; 180; 219]. However, it is recognized that the capabilities of such algorithms to find optimal solutions to WFDO problem are limited [219]. Improved exact techniques, such as tree search-based procedures (e.g., branch and bound algorithms), perform an implicit enumeration of all the admissible solutions (_i.e._, all the admissible solutions are considered and implicitly evaluated but are not explicitly constructed) [324; 326; 328; 329]. Although tree-search based procedures have shown improved computational efficiency to find optimal solutions, these methods perform poor when highly constrained models and non-convex search spaces are considered. To overcome this limitation new improvements based on relaxation procedures are being developed in recent research [330; 331; 332]. Relaxation consists of transforming the original problem into another equivalent problem of reduced complexity (e.g., having less and/or different restrictions, different type of variables, objective functions with reduced order, _etc._), which can be solved with other commonly used solution procedures. Relaxation denominations depend on the kind of relaxation performed (e.g., the Linear Programming (LP) relaxation consist of transforming an integer-type problem into a problem with continuous variables by disregarding the constraint that the variables must be integer). The solution of the relaxed problem can be used to gain information about the solution to the original problem. However, relaxation procedures can significantly affect the original problem and thus its optimal solution (_i.e._, depending on the problem solved and the adopted relaxation procedure, both the shape of the solution and the solution quality of the relaxed problem can be significantly different that the optimal solution shape and quality of original problem).