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## weka.core
Class Optimization

    java.lang.Object
      weka.core.Optimization

    public abstract class Optimization

extends java.lang.Object

Implementation of Active-sets method with BFGS update to solve optimization problem with only bounds constraints in multi-dimensions. In this implementation we consider both the lower and higher bound constraints.

Here is the sketch of our searching strategy, and the detailed description of the algorithm can be found in the Appendix of Xin Xu's MSc thesis:

Initialize everything, incl. initial value, direction, etc.

LOOP (main algorithm):
1. Perform the line search using the directions for free variables
1.1 Check all the bounds that are not "active" (i.e. binding variables) and compute the feasible step length to the bound for each of them
1.2 Pick up the least feasible step length, say \alpha, and set it as the upper bound of the current step length, i.e. 0<\lambda<=\alpha
1.3 Search for any possible step length<=\alpha that can result the "sufficient function decrease" (\alpha condition) AND "positive definite inverse Hessian" (\beta condition), if possible, using SAFEGUARDED polynomial interpolation. This step length is "safe" and thus is used to compute the next value of the free variables .
1.4 Fix the variable(s) that are newly bound to its constraint(s).

2. Check whether there is convergence of all variables or their gradients. If there is, check the possibilities to release any current bindings of the fixed variables to their bounds based on the "reliable" second-order Lagarange multipliers if available. If it's available and negative for one variable, then release it. If not available, use first-order Lagarange multiplier to test release. If there is any released variables, STOP the loop. Otherwise update the inverse of Hessian matrix and gradient for the newly released variables and CONTINUE LOOP.

3. Use BFGS formula to update the inverse of Hessian matrix. Note the already-fixed variables must have zeros in the corresponding entries in the inverse Hessian.

4. Compute the new (newton) search direction d=H^{-1}*g, where H^{-1} is the inverse Hessian and g is the Jacobian. Note that again, the already- fixed variables will have zero direction.

ENDLOOP

A typical usage of this class is to create your own subclass of this class and provide the objective function and gradients as follows:

`class MyOpt extends Optimization{
// Provide the objective function
protected double objectiveFunction(double[] x){
// How to calculate your objective function...
// ...
}`

// Provide the first derivatives
protected double[] evaluateGradient(double[] x){
// How to calculate the gradient of the objective function...
// ...
}

// If possible, provide the index^{th} row of the Hessian matrix
protected double[] evaluateHessian(double[] x, int index){
// How to calculate the index^th variable's second derivative
// ...
}
}

// When it's the time to use it, in some routine(s) of other class...
MyOpt opt = new MyOpt();

// Set up initial variable values and bound constraints
double[] x = new double[numVariables];
// Lower and upper bounds: 1st row is lower bounds, 2nd is upper
double[] constraints = new double[2][numVariables];
...

// Find the minimum, 200 iterations as default
x = opt.findArgmin(x, constraints);
while(x == null){ // 200 iterations are not enough
x = opt.getVarbValues(); // Try another 200 iterations
x = opt.findArgmin(x, constraints);
}

// The minimal function value
double minFunction = opt.getMinFunction();
...

It is recommended that Hessian values be provided so that the second-order Lagrangian multiplier estimate can be calcluated. However, if it is not provided, there is no need to override the `evaluateHessian()` function.

REFERENCES:
The whole model algorithm is adapted from Chapter 5 and other related chapters in Gill, Murray and Wright(1981) "Practical Optimization", Academic Press. and Gill and Murray(1976) "Minimization Subject to Bounds on the Variables", NPL Report NAC72, while Chong and Zak(1996) "An Introduction to Optimization", John Wiley & Sons, Inc. provides us a brief but helpful introduction to the method.

Dennis and Schnabel(1983) "Numerical Methods for Unconstrained Optimization and Nonlinear Equations", Prentice-Hall Inc. and Press et al.(1992) "Numeric Recipe in C", Second Edition, Cambridge University Press. are consulted for the polynomial interpolation used in the line search implementation.

The Hessian modification in BFGS update uses Cholesky factorization and two rank-one modifications:
Bk+1 = Bk + (Gk*Gk')/(Gk'Dk) + (dGk*(dGk)'))/[alpha*(dGk)'*Dk].
where Gk is the gradient vector, Dk is the direction vector and alpha is the step rate.
This method is due to Gill, Golub, Murray and Saunders(1974) ``Methods for Modifying Matrix Factorizations'', Mathematics of Computation, Vol.28, No.126, pp 505-535.

Version:
  $Revision: 1.6 $
Author:
  Xin Xu (xx5@cs.waikato.ac.nz)

Constructor Summary
`Optimization()`

Method Summary
`double[]` | `findArgmin(double[] initX, double[][] constraints)` Main algorithm.
`double` | `getMinFunction()` Get the minimal function value
`double[]` | `getVarbValues()` Get the variable values.
`double[]` | `lnsrch(double[] xold, double[] gradient, double[] direct, double stpmax, boolean[] isFixed, double[][] nwsBounds, weka.core.Optimization.DynamicIntArray wsBdsIndx)` Find a new point x in the direction p from a point xold at which the value of the function has decreased sufficiently, the positive definiteness of B matrix (approximation of the inverse of the Hessian) is preserved and no bound constraints are violated.
`void` | `setDebug(boolean db)` Set whether in debug mode
`void` | `setMaxIteration(int it)` Set the maximal number of iterations in searching (Default 200)
`static double[]` | `solveTriangle(Matrix t, double[] b, boolean isLower, boolean[] isZero)` Solve the linear equation of TX=B where T is a triangle matrix It can be solved using back/forward substitution, with O(N^2) complexity

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

Constructor Detail

### Optimization

    public Optimization()

Method Detail

### getMinFunction

    public double getMinFunction()

  Get the minimal function value

  Returns:
    minimal function value found

### setMaxIteration

    public void setMaxIteration(int it)

  Set the maximal number of iterations in searching (Default 200)

  Parameters:
    `it` - the maximal number of iterations

### setDebug

    public void setDebug(boolean db)

  Set whether in debug mode

  Parameters:
    `db` - use debug or not

### getVarbValues

    public double[] getVarbValues()

  Get the variable values. Only needed when iterations exceeds the max threshold.

  Returns:
    the current variable values

### lnsrch

    public double[] lnsrch(double[] xold,
                           double[] gradient,
                           double[] direct,
                           double stpmax,
                           boolean[] isFixed,
                           double[][] nwsBounds,
                           weka.core.Optimization.DynamicIntArray wsBdsIndx)
                    throws java.lang.Exception

  Find a new point x in the direction p from a point xold at which the value of the function has decreased sufficiently, the positive definiteness of B matrix (approximation of the inverse of the Hessian) is preserved and no bound constraints are violated. Details see "Numerical Methods for Unconstrained Optimization and Nonlinear Equations". "Numeric Recipes in C" was also consulted.

  Parameters:
    `xold` - old x value
    `gradient` - gradient at that point
    `direct` - direction vector
    `stpmax` - maximum step length
    `isFixed` - indicating whether a variable has been fixed
    `nwsBounds` - non-working set bounds. Means these variables are free and subject to the bound constraints in this step
    `wsBdsIndx` - index of variables that has working-set bounds. Means these variables are already fixed and no longer subject to the constraints
  Returns:
    new value along direction p from xold, null if no step was taken
  Throws:
    `java.lang.Exception` - if an error occurs

### findArgmin

    public double[] findArgmin(double[] initX,
                               double[][] constraints)
                        throws java.lang.Exception

  Main algorithm. Descriptions see "Practical Optimization"

  Parameters:
    `initX` - initial point of x, assuming no value's on the bound!
    `constraints` - the bound constraints of each variable constraints[0] is the lower bounds and constraints[1] is the upper bounds
  Returns:
    the solution of x, null if number of iterations not enough
  Throws:
    `java.lang.Exception` - if an error occurs

### solveTriangle

    public static double[] solveTriangle(Matrix t,
                                         double[] b,
                                         boolean isLower,
                                         boolean[] isZero)

  Solve the linear equation of TX=B where T is a triangle matrix It can be solved using back/forward substitution, with O(N^2) complexity

  Parameters:
    `t` - the matrix T
    `b` - the vector B
    `isLower` - whether T is a lower or higher triangle matrix
    `isZero` - which row(s) of T are not used when solving the equation. If it's null or all 'false', then every row is used.
  Returns:
    the solution of X

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