Genetic Algorithm and Direct Search Toolbox™ 2.3

Product Description

• Introduction and Key Features
• Graphical User Interface and Command-Line Functions
• Genetic Algorithm Tools
• Direct Search and Simulated Annealing Tools
• Solving Constrained Optimization Problems and Using Other Solvers
• Displaying, Monitoring, and Outputting Results

Direct Search Tools

Genetic Algorithm and Direct Search Toolbox contains two direct search algorithms: the generalized pattern search algorithm (GPS) and mesh adaptive search algorithm (MADS). While more traditional optimization algorithms use information about the gradient or higher derivatives to search for an optimal point, these toolbox algorithms use a pattern search method that implements a minimal and maximal positive basis pattern. The pattern search method handles optimization problems with nonlinear, linear, and bound constraints, and does not require functions to be differentiable or continuous.

The pattern search algorithm includes the following options:

• Polling methods let you decide how to generate and evaluate the points in a pattern and the maximum number of points generated at each step. You can also control the polling order of the points to improve efficiency.
• Search methods let you choose a search method that is more efficient than a poll step. You can perform a search in a pattern or in the entire search space. Global search methods, like the genetic algorithm, can be used to obtain a good starting point.
• Mesh lets you control how the pattern changes over iterations and adjusts the mesh for problems that vary in scale across dimensions. You can choose the initial mesh size, mesh refining factor, or mesh contrac­tion factor. The mesh accelerator speeds up convergence when it is near a minimum.
• Cache lets you store points evaluated during optimization of expensive objective functions. You can specify the size and tolerance of the cache that the pattern search algorithm uses and vary the cache tolerance as the algorithm proceeds, improving opti­mization speed and efficiency.
• Nonlinear constraint algorithm settings let you specify a penalty parameter for the nonlinear constraints as well as a penalty update factor.

You can change any of the options from the command line or the graphical user interface.

The pattern search GUI helps you quickly set up and solve your optimization problems (far left) and is used here to used to find the peak, or global optima, of the white mountains (left). Click on image to see enlarged view.

Simulated Annealing and Threshold Acceptance Tools

Simulated annealing solves optimization problems using a probabilistic search algorithm that mimics the physical process of annealing. Annealing is the process of heating a material and then slowly lowering the temperature to decrease defects, thus minimizing the system energy. By analogy, each iteration of a simulated annealing algorithm seeks to find the global minimum by slowly reducing the extent of the search.

The simulated annealing 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 better solutions.

Threshold acceptance uses an approach similar to simulated annealing, but instead of accepting new points that raise the objective with a certain probability, it accepts only the points that fall below a fixed threshold. Because threshold acceptance avoids the probabilistic acceptance calculations of simulated annealing, it may locate an optimal value faster than simulated annealing.

Simulated annealing and threshold acceptance allow you to solve unconstrained or bound-constrained optimization problems and do not require that the functions be differentiable or continuous. Toolbox functions are accessible from the command line for

• Solving problems using adaptive simulated annealing, Boltzmann annealing, or fast annealing algorithms
• Controlling acceptance criteria for simulated annealing problems at predetermined temperature levels using the threshold acceptance algorithm
• Creating custom functions to define the annealing process, acceptance criteria, temperature schedule, plotting functions, or simulation output
• Performing hybrid optimization by specifying another optimization method to run at defined intervals or at normal solver termination