OPTIMIZATION with MATLAB. DIRECT SEARCH ALGORITMS and SURROGATE OPTIMIZATION

By J Lopez

Global Optimization Toolbox provides functions that search for global solutions to problems that contain multiple maxima or minima. Toolbox solvers include surrogate, pattern search, genetic algorithm, particle swarm, simulated annealing, multi start, and global search. Direct search is a method for solving optimization problems that does not require any information about the gradient of the objective function. Unlike more traditional optimization methods that use information about the gradient or higher derivatives to search for an optimal point, a direct search algorithm searches a set of points around the current point, looking for one where the value of the objective function is lower than the value at the current point. You can use direct search to solve problems for which the objective function is not differentiable, or is not even continuous.Global Optimization Toolbox functions include three direct search algorithms called the generalized pattern search (GPS) algorithm, the generating set search (GSS) algorithm, and the mesh adaptive search (MADS) algorithm. All are pattern search algorithms that compute a sequence of points that approach an optimal point. At each step, the algorithm searches a set of points, called a mesh, around the current point-the point computed at the previous step of the algorithm. The mesh is formed by adding the current point to a scalar multiple of a set of vectors called a pattern. If the pattern search algorithm finds a point in the mesh that improves the objective function at the current point, the new point becomes the current point at the next step of the algorithm.The GPS algorithm uses fixed direction vectors. The GSS algorithm is identical to the GPS algorithm, except when there are linear constraints, and when the current point is near a linear constraint boundary. The MADS algorithm uses a random selection of vectors to define the mesh.A surrogate is a function that approximates an objective function. The surrogate is useful because it takes little time to evaluate. So, for example, to search for a point that minimizes an objective function, simply evaluate the surrogate on thousands of points, and take the best value as an approximation to the minimizer of the objective function. Surrogate optimization is best suited to time-consuming objective functions. The objective function need not be smooth, but the algorithm works best when the objective function is continuous. Surrogate optimization attempts to fin a global minimum of an objective function using few objective function evaluations. Simulated annealing is a method for solving unconstrained and bound-constrained optimization problems. The method models the physical process of heating a material and then slowly lowering the temperature to decrease defects, thus minimizing the system energy. At each iteration of the simulated annealing algorithm, a new point is randomly generated. The distance of the new point from the current point, or the extent of the search, is based on a probability distribution with a scale proportional to the temperature. The algorithm accepts all new points that lower the objective, but also, with a certain probability, points that raise the objective. By accepting points that raise the objective, the algorithm avoids being trapped in local minima, and is able to explore globally for more possible solutions. An annealing schedule is selected to systematically decrease the temperature as the algorithm proceeds.

• Format: Paperback
• Language: English
• ISBN: 1075818346
• ISBN13: 9781075818349
• Release Date: January 1
• Publisher: Independently Published
• Length: 360 Pages
• Weight: 1.16 lbs.
• Dimensions: 0.8" x 6.0" x 9.0"