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:

• UnivariateOptimizer for univariate real functions
• MultivariateOptimizer for multivariate real functions
• DifferentiableMultivariateOptimizer for differentiable multivariate real functions
• DifferentiableMultivariateVectorOptimizer 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 MultivariateOptimizer 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 UnivariateOptimizer 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 minimizesmethod 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 four solvers:

• the classical Nelder-Mead method,
• Virginia Torczon's multi-directional method,
• Nikolaus Hansen's Covariance Matrix Adaptation Evolution Strategy (CMA-ES),
• Mike Powell's BOBYQA method.

The first two simplex-based methods do not handle simple bounds constraints by themselves. However there are two adapters( MultivariateFunctionMappingAdapter and MultivariateFunctionPenaltyAdapter) that can be used to wrap the user function in such a way the wrapped function is unbounded and can be used with these optimizers, despite the fact the underlying function is still bounded and will be called only with feasible points that fulfill the constraints. Note however that using these adapters are only a poor man solutions to simple bounds optimization constraints. Better solutions are to use an optimizer that directly supports simple bounds. Some caveats of the mapping adapter solution are that

• behavior near the bounds may be numerically unstable as bounds are mapped from infinite values,
• start value is evaluated by the optimizer as an unbounded variable, so it must be converted from bounded to unbounded by user,
• optimum result is evaluated by the optimizer as an unbounded variable, so it must be converted from unbounded to bounded by user,
• convergence values are evaluated by the optimizer as unbounded variables, so there will be scales differences when converted to bounded variables,
• in the case of simplex based solvers, the initial simplex should be set up as delta in unbounded variables.

The last methods do handle simple bounds constraints directly, so the adapters are not needed with them.

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 p_k such that a cost function J = sum(w_i(mes_i - mod_i)²) is minimized. The various (target_i - model_i(p_k)) terms are called residuals. They represent the deviation between a set of target values target_i and theoretical values computed from models model_i depending on free parameters p_k. The w_i 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 p_k 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 DifferentiableMultivariateVectorFunction 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.

Quadratic Problem Example

We are looking to find the best parameters [a, b, c] for the quadratic function f(x) = a x2 + b x + c. The data set below was generated using [a = 8, b = 10, c = 16]. A random number between zero and one was added to each y value calculated.

X	Y
1	34.234064369
2	68.2681162306108
3	118.615899084602
4	184.138197238557
5	266.599877916276
6	364.147735251579
7	478.019226091914
8	608.140949270688
9	754.598868667148
10	916.128818085883

First we need to implement the interface DifferentiableMultivariateVectorFunction. This requires the implementation of the method signatures:

• MultivariateMatrixFunction jacobian()
• double[] value(double[] point)

We'll tackle the implementation of the MultivariateMatrixFunction jacobian() method first. You may wish to familiarize yourself with what a Jacobian Matrix is. In this case the Jacobian is the partial derivative of the function with respect to the parameters a, b and c. These derivatives are computed as follows:

• d(ax² + bx + c)/da = x²
• d(ax² + bx + c)/db = x
• d(ax² + bx + c)/dc = 1

For a quadratic which has three variables the Jacobian Matrix will have three columns, one for each variable, and the number of rows will equal the number of rows in our data set, which in this case is ten. So for example for [a = 1, b = 1, c = 1], the Jacobian Matrix is (excluding the first column which shows the value of x):

x	d(ax2 + bx + c)/da	d(ax2 + bx + c)/db	d(ax2 + bx + c)/dc
1	1	1	1
2	4	2	1
3	9	3	1
4	16	4	1
5	25	5	1
6	36	6	1
7	49	7	1
8	64	8	1
9	81	9	1
10	100	10	1

The implementation of the MultivariateMatrixFunction jacobian() for this problem looks like this (The x parameter is an ArrayList containing the independent values of the data set):

 private double[][] jacobian(double[] variables) {
     double[][] jacobian = new double[x.size()][3];
     for (int i = 0; i < jacobian.length; ++i) {
         jacobian[i][0] = x.get(i) * x.get(i);
         jacobian[i][1] = x.get(i);
         jacobian[i][2] = 1.0;
     }
     return jacobian;
 }

 public MultivariateMatrixFunction jacobian() {
     return new MultivariateMatrixFunction() {
         private static final long serialVersionUID = -8673650298627399464L;
         public double[][] value(double[] point) {
             return jacobian(point);
         }
     };
 }

Note that if for some reason the derivative of the objective function with respect to its variables is difficult to obtain, Numerical differentiation can be used.

The implementation of the double[] value(double[] point) method, which returns a double array containing the values the objective function returns per given independent value and the current set of variables or parameters, can be seen below:

    public double[] value(double[] variables) {
        double[] values = new double[x.size()];
        for (int i = 0; i < values.length; ++i) {
            values[i] = (variables[0] * x.get(i) + variables[1]) * x.get(i) + variables[2];
        }
        return values;
    }

Below is the the class containing all the implementation details (Taken from the Apache Commons Math org.apache.commons.math4.optimization.general.LevenbergMarquardtOptimizerTest):

private static class QuadraticProblem
    implements DifferentiableMultivariateVectorFunction, Serializable {

    private static final long serialVersionUID = 7072187082052755854L;
    private List<Double> x;
    private List<Double> y;

    public QuadraticProblem() {
        x = new ArrayList<Double>();
        y = new ArrayList<Double>();
    }

    public void addPoint(double x, double y) {
        this.x.add(x);
        this.y.add(y);
    }

    public double[] calculateTarget() {
        double[] target = new double[y.size()];
        for (int i = 0; i < y.size(); i++) {
            target[i] = y.get(i).doubleValue();
        }
        return target;
    }

    private double[][] jacobian(double[] variables) {
        double[][] jacobian = new double[x.size()][3];
        for (int i = 0; i < jacobian.length; ++i) {
            jacobian[i][0] = x.get(i) * x.get(i);
            jacobian[i][1] = x.get(i);
            jacobian[i][2] = 1.0;
        }
        return jacobian;
    }

    public double[] value(double[] variables) {
        double[] values = new double[x.size()];
        for (int i = 0; i < values.length; ++i) {
            values[i] = (variables[0] * x.get(i) + variables[1]) * x.get(i) + variables[2];
        }
        return values;
    }

    public MultivariateMatrixFunction jacobian() {
        return new MultivariateMatrixFunction() {
            private static final long serialVersionUID = -8673650298627399464L;
            public double[][] value(double[] point) {
                return jacobian(point);
            }
        };
    }
}

The below code shows how to go about using the above class and a LevenbergMarquardtOptimizer instance to produce an optimal set of quadratic curve fitting parameters:

 QuadraticProblem problem = new QuadraticProblem();

 problem.addPoint(1, 34.234064369);
 problem.addPoint(2, 68.2681162306);
 problem.addPoint(3, 118.6158990846);
 problem.addPoint(4, 184.1381972386);
 problem.addPoint(5, 266.5998779163);
 problem.addPoint(6, 364.1477352516);
 problem.addPoint(7, 478.0192260919);
 problem.addPoint(8, 608.1409492707);
 problem.addPoint(9, 754.5988686671);
 problem.addPoint(10, 916.1288180859);

 LevenbergMarquardtOptimizer optimizer = new LevenbergMarquardtOptimizer();

 final double[] weights = { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };

 final double[] initialSolution = {1, 1, 1};

 PointVectorValuePair optimum = optimizer.optimize(100,
                                                   problem,
                                                   problem.calculateTarget(),
                                                   weights,
                                                   initialSolution);

 final double[] optimalValues = optimum.getPoint();

 System.out.println("A: " + optimalValues[0]);
 System.out.println("B: " + optimalValues[1]);
 System.out.println("C: " + optimalValues[2]);

    

If you run the above sample you will see the following printed by the console:

A: 7.998832172372726
B: 10.001841530162448
C: 16.324008168386605

In addition to least squares solving, the NonLinearConjugateGradientOptimizer class provides a non-linear conjugate gradient algorithm to optimize DifferentiableMultivariateFunction. 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.