Commons Math User Guide

##### User Guide

- Contents
- Overview
- Statistics
- Data Generation
- Linear Algebra
- Numerical Analysis
- Special Functions
- Utilities
- Complex Numbers
- Distributions
- Fractions
- Transform Methods
- 3D Geometry
- Optimization
- Ordinary Differential Equations
- Genetic Algorithms

## 12 Optimization

### 12.1 Overview

The optimization package provides algorithms to optimize (i.e. either minimize or maximize) some objective or cost function. The package is split in several sub-packages dedicated to different kind of functions or algorithms.

- the univariate package handles univariate scalar functions,
- the linear package handles multivariate vector linear functions with linear constraints,
- the direct package handles multivariate scalar functions using direct search methods (i.e. not using derivatives),
- the general package handles multivariate scalar or vector functions using derivatives.
- the fitting package handles curve fitting by univariate real functions

The top level optimization package provides common interfaces for the optimization algorithms provided in sub-packages. The main interfaces defines defines optimizers and convergence checkers. The functions that are optimized by the algorithms provided by this package and its sub-packages are a subset of the one defined in the `analysis` package, namely the real and vector valued functions. These functions are called objective function here. When the goal is to minimize, the functions are often called cost function, this name is not used in this package.

The type of goal, i.e. minimization or maximization, is defined by the enumerated GoalType which has only two values: `MAXIMIZE` and `MINIMIZE`.

Optimizers are the algorithms that will either minimize or maximize, the objective function by changing its input variables set until an optimal set is found. There are only four interfaces defining the common behavior of optimizers, one for each supported type of objective function:

- UnivariateRealOptimizer for univariate real functions
- MultivariateRealOptimizer for multivariate real functions
- DifferentiableMultivariateRealOptimizer for differentiable multivariate real functions
- DifferentiableMultivariateVectorialOptimizer for differentiable multivariate vectorial functions

Despite there are only four types of supported optimizers, it is possible to optimize a transform a non-differentiable multivariate vectorial function by converting it to a non-differentiable multivariate real function thanks to the LeastSquaresConverter helper class. The transformed function can be optimized using any implementation of the MultivariateRealOptimizer interface.

For each of the four types of supported optimizers, there is a special implementation which wraps a classical optimizer in order to add it a multi-start feature. This feature call the underlying optimizer several times in sequence with different starting points and returns the best optimum found or all optima if desired. This is a classical way to prevent being trapped into a local extremum when looking for a global one.

### 12.2 Univariate Functions

A UnivariateRealOptimizer is used to find the minimal values of a univariate real-valued function `f`.

These algorithms usage is very similar to root-finding algorithms usage explained in the analysis package. The main difference is that the `solve` methods in root finding algorithms is replaced by `optimize` methods.

### 12.3 Linear Programming

This package provides an implementation of George Dantzig's simplex algorithm for solving linear optimization problems with linear equality and inequality constraints.

### 12.4 Direct Methods

Direct search methods only use cost function values, they don't need derivatives and don't either try to compute approximation of the derivatives. According to a 1996 paper by Margaret H. Wright (Direct Search Methods: Once Scorned, Now Respectable), they are used when either the computation of the derivative is impossible (noisy functions, unpredictable discontinuities) or difficult (complexity, computation cost). In the first cases, rather than an optimum, a not too bad point is desired. In the latter cases, an optimum is desired but cannot be reasonably found. In all cases direct search methods can be useful.

Simplex-based direct search methods are based on comparison of the cost function values at the vertices of a simplex (which is a set of n+1 points in dimension n) that is updated by the algorithms steps.

The instances can be built either in single-start or in multi-start mode. Multi-start is a traditional way to try to avoid being trapped in a local minimum and miss the global minimum of a function. It can also be used to verify the convergence of an algorithm. In multi-start mode, the `minimizes`method returns the best minimum found after all starts, and the `etMinima` method can be used to retrieve all minima from all starts (including the one already provided by the `minimizes` method).

The `direct` package provides two solvers. The first one is the classical Nelder-Mead method. The second one is Virginia Torczon's multi-directional method.

### 12.5 General Case

The general package deals with non-linear vectorial optimization problems when the partial derivatives of the objective function are available.

One important class of estimation problems is weighted least squares problems. They basically consist in finding the values for some parameters pk such that a cost function J = sum(wi(mesi - modi)2) is minimized. The various (targeti - modeli(pk)) terms are called residuals. They represent the deviation between a set of target values targeti and theoretical values computed from models modeli depending on free parameters pk. The wi factors are weights. One classical use case is when the target values are experimental observations or measurements.

Solving a least-squares problem is finding the free parameters pk of the theoretical models such that they are close to the target values, i.e. when the residual are small.

Two optimizers are available in the general package, both devoted to least-squares problems. The first one is based on the Gauss-Newton method. The second one is the Levenberg-Marquardt method.

In order to solve a vectorial optimization problem, the user must provide it as an object implementing the DifferentiableMultivariateVectorialFunction interface. The object will be provided to the `estimate` method of the optimizer, along with the target and weight arrays, thus allowing the optimizer to compute the residuals at will. The last parameter to the `estimate` method is the point from which the optimizer will start its search for the optimal point.

In addition to least squares solving, the NonLinearConjugateGradientOptimizer class provides a non-linear conjugate gradient algorithm to optimize DifferentiableMultivariateRealFunction. Both the Fletcher-Reeves and the Polak-Ribière search direction update methods are supported. It is also possible to set up a preconditioner or to change the line-search algorithm of the inner loop if desired (the default one is a Brent solver).

The PowellOptimizer provides an optimization method for non-differentiable functions.

### 12.6 Curve Fitting

The fitting package deals with curve fitting for univariate real functions. When a univariate real function y = f(x) does depend on some unknown parameters p0, p1 ... pn-1, curve fitting can be used to find these parameters. It does this by fitting the curve so it remains very close to a set of observed points (x0, y0), (x1, y1) ... (xk-1, yk-1). This fitting is done by finding the parameters values that minimizes the objective function sum(yi-f(xi))2. This is really a least squares problem.

For all provided curve fitters, the operating principle is the same. Users must first create an instance of the fitter, then add the observed points and once the complete sample of observed points has been added they must call the `fit` method which will compute the parameters that best fit the sample. A weight is associated with each observed point, this allows to take into account uncertainty on some points when they come from loosy measurements for example. If no such information exist and all points should be treated the same, it is safe to put 1.0 as the weight for all points.

The CurveFitter class provides curve fitting for general curves. Users must provide their own implementation of the curve template as a class implementing the ParametricRealFunction interface and they must provide the initial guess of the parameters. The more specialized PolynomialFitter and HarmonicFitter classes require neither an implementation of the parametric real function not an initial guess as they are able to compute them by themselves.

An example of fitting a polynomial is given here:

    PolynomialFitter fitter = new PolynomialFitter(degree, new LevenbergMarquardtOptimizer());
    fitter.addObservedPoint(-1.00,  2.021170021833143);
    fitter.addObservedPoint(-0.99   2.221135431136975);
    fitter.addObservedPoint(-0.98   2.09985277659314);
    fitter.addObservedPoint(-0.97   2.0211192647627025);
    // lots of lines ommitted
    fitter.addObservedPoint( 0.99, -2.4345814727089854);
    PolynomialFunction fitted = fitter.fit();