Global Optimization Toolbox

MAJOR UPDATE

 

Global Optimization Toolbox

Solve multiple maxima, multiple minima, and nonsmooth optimization problems

 

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, multistart, and global search. You can use these solvers for optimization problems where the objective or constraint function is continuous, discontinuous, stochastic, does not possess derivatives, or includes simulations or black-box functions. For problems with multiple objectives, you can identify a Pareto front using genetic algorithm or pattern search solvers.

You can improve solver effectiveness by adjusting options and, for applicable solvers, customizing creation, update, and search functions. You can use custom data types with the genetic algorithm and simulated annealing solvers to represent problems not easily expressed with standard data types. The hybrid function option lets you improve a solution by applying a second solver after the first.

Solving Optimization Problems

Choose a solver, define your optimization problem, and set options for algorithm behavior, tolerances, stopping criteria, visualizations, and customizations.

Specify Solver and Problem

Decide on the solver based on problem characteristics and desired results. Write functions to specify nonlinear objectives and constraints. 

Smooth and nonsmooth problems.

Set Common Options

Set the stopping criteria applicable to the selected solver. Set tolerances for optimality and constraints. Accelerate with parallel computing.

Speed-ups from parallel computing.

Assess Intermediate Results

Use plotting functions to get live feedback about optimization progress. Write your own or use those provided. Use output functions to create your own stopping criteria, write results to files, or write your own apps to run the solvers.

Custom plot function for pattern search.

GlobalSearch and MultiStart

Apply gradient-based solvers to find local minima from multiple starting points in search of global minima. Other local or global minima are returned. Solve unconstrained and constrained problems that are smooth.

Compare Solvers

Use GlobalSearch to generate multiple starting points and filter them before starting the nonlinear solver, often resulting in high-quality solutions. MultiStart lets you choose local solvers and a variety of ways to create starting points.

GlobalSearch and MultiStart results.

Select GlobalSearch Options

Specify the number of trial points and tune the search. 

Select MultiStart Options

Specify the nonlinear solver. Choose a method to generate starting points or use a user-defined set. Accelerate with parallel computing.

Surrogate Optimization

Search for global minima on problems with time-consuming objective functions. The solver builds an approximation to the function that can be quickly evaluated and minimized.

Specify the Problem

Apply to problems with finite bound constraints. The objective function does not need to be differentiable or continuous.

Select Options

Provide a set of initial points and optional objective values for constructing the initial surrogate. Set the number of points to use for the surrogate and a minimal sample distance. Accelerate with parallel computing.

Built-in plot of sample, adaptive, and best points.

Pattern Search

Solve optimization problems by using one of three direct search algorithms: generalized pattern search (GPS), generating set search (GSS), and mesh adaptive search (MADS). At each step, a mesh pattern of points is generated and evaluated.

Specify the Problem

Apply to problems that are unconstrained or have bound, linear, or nonlinear constraints. The objective and constraint functions do not need to be differentiable or continuous.

Climbing Mount Washington in the White Mountains.

Select Options

Choose among polling options and set the number of points to evaluate at each step. Use an optional search step to improve efficiency. Control how the mesh changes, including refinement and contraction. Accelerate with parallel computing.

Built-in plots for function value and evaluations.

Genetic Algorithm

Search for global minima by mimicking the principles of biological evolution, repeatedly modifying a population of individual points using rules modeled on gene combinations in biological reproduction.

Specify the Problem

Apply to problems that are unconstrained or have bound, linear, nonlinear, or integer constraints. The objective and constraint functions do not need to be differentiable or continuous.

Select Options

Choose among options for creation, fitness scaling, selection, crossover, and mutation. Specify population size, number of elite children, and crossover fraction. Accelerate with parallel computing.

Function with several local minima.

Customize

Provide your own functions for creation, selection, and mutation. Use custom data types to more easily express your problem. Apply a second optimizer to refine solutions.

Solution to the traveling salesman problem.

Particle Swarm

Search for global minima using an algorithm inspired by the behavior of insects swarming. Each particle moves with a velocity and direction influenced by the best location it has found so far and the best location the swarm has found.

Specify the Problem

Apply to unconstrained problems or problems with bound constraints. The objective function does not need to be differentiable or continuous.

Showing five-move path per particle.

Select Options

Tune velocity computation through setting of inertia and self- and social adjustment weights. Set the neighborhood size. Accelerate with parallel computing.

Built-in plot functions.

Customize

Provide your own function for creating the initial swarm. Apply a second optimizer to refine solutions.

Particle swarm on a stochastic function.

Simulated Annealing

Search for global minima with a probabilistic search algorithm that mimics the physical process of annealing, in which a material is heated and then the temperature is slowly lowered to decrease defects, thus minimizing the system energy.

Specify the Problem

Apply to unconstrained problems or problems with bound constraints. The objective function does not need to be differentiable or continuous

Function with many local minima.

Select Options

Choose among options for adaptive simulated annealing, Boltzmann annealing, or fast annealing algorithms.

Simulated annealing visualization.

Customize

Create functions to define the annealing process, acceptance criteria, and temperature schedule. Use custom data types to more easily express your problem. Apply a second optimizer to refine solutions.

Multiprocessor schedule.

Multiobjective Optimization

Identify the Pareto front—the set of nondominated solutions—for problems with multiple objectives and bound, linear, or nonlinear constraints. Use either the pattern search or genetic algorithm solvers.

Compare Solvers

Use the multiobjective pattern search algorithm to generate a Pareto front in fewer function evaluations than with the multiobjective genetic algorithm. The genetic algorithm may generate more widely spaced points. 

Select Pattern Search Options

Provide a set of initial points. Specify the desired Pareto set size, minimum polling fraction, and volume change tolerance. Automatically plot 2D and 3D Pareto fronts. Accelerate with parallel computing.

Pareto surface of the three objectives.

Set Genetic Algorithm Options

Specify the fraction of individuals to keep on the top-ranked Pareto front. Automatically plot 2D Pareto fronts. Accelerate with parallel computing.

Pareto front of two objectives.

Latest Features

surrogateopt Solver

Solve time-consuming, bound-constrained optimization problems using fewer objective function evaluations

paretosearch Multiobjective Solver

Find Pareto sets quickly and accurately for problems with bound, linear, or smooth nonlinear constraints

Parallel Computation

Accelerate surrogateopt and paretosearch functions (using Parallel Computing Toolbox™)

See the release notes for details on any of these features and corresponding functions.

Global Optimization Toolbox™ は、複数の最大値または最小値をもつ問題の大域的解を探索する機能を提供します。このツールボックスには、大域的探索法、マルチスタート法、パターン探索法、遺伝的アルゴリズム法、多目的遺伝的アルゴリズム法、シミュレーテッド アニーリング法、粒子群法のソルバーが含まれます。これらのソルバーを使用して、次のような性質の目的関数または制約関数をもつ最適化問題を解くことができます。 1)目的関数または制約関数が連続関数、不連続関数または確率関数である、2)目的関数または制約関数が導関数をもたない、3)目的関数または制約関数にシミュレーションやブラックボックス関数が含まれる。

オプションを設定し、作成、更新、探索関数をカスタマイズすることで、ソルバーの有効性を改善できます。汎用アルゴリズムおよびシミュレーテッド アニーリング法のソルバーでカスタム データ型を使用して、標準のデータ型では表現するのが容易ではなかった問題を表現することができます。ハイブリッド関数オプションにより、最初のソルバーの後に 2 番目のソルバーを適用することで、ソリューションを改善できます。

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