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package org.apache.commons.math3.optim.nonlinear.scalar;
import org.apache.commons.math3.analysis.MultivariateFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathUtils;
Adapter extending bounded MultivariateFunction to an unbouded domain using a penalty function.
This adapter can be used to wrap functions subject to simple bounds on
parameters so they can be used by optimizers that do not directly
support simple bounds.
The principle is that the user function that will be wrapped will see its parameters bounded as required, i.e when its value method is called with argument array point, the elements array will fulfill requirement lower[i] <= point[i] <= upper[i] for all i. Some of the components may be unbounded or bounded only on one side if the corresponding bound is set to an infinite value. The optimizer will not manage the user function by itself, but it will handle this adapter and it is this adapter that will take care the bounds are fulfilled. The adapter value(double[]) method will be called by the optimizer with unbound parameters, and the adapter will check if the parameters is within range or not. If it is in range, then the underlying user function will be called, and if it is not the value of a penalty function will be returned instead.
This adapter is only a poor-man's solution to simple bounds optimization constraints that can be used with simple optimizers like
SimplexOptimizer. A better solution is to use an optimizer that directly supports simple bounds like
CMAESOptimizer or
BOBYQAOptimizer. One caveat of this poor-man's solution is that if start point or start simplex is completely outside of the allowed range, only the penalty function is used, and the optimizer may converge without ever entering the range.
See Also: Since: 3.0
/**
* <p>Adapter extending bounded {@link MultivariateFunction} to an unbouded
* domain using a penalty function.</p>
*
* <p>
* This adapter can be used to wrap functions subject to simple bounds on
* parameters so they can be used by optimizers that do <em>not</em> directly
* support simple bounds.
* </p>
* <p>
* The principle is that the user function that will be wrapped will see its
* parameters bounded as required, i.e when its {@code value} method is called
* with argument array {@code point}, the elements array will fulfill requirement
* {@code lower[i] <= point[i] <= upper[i]} for all i. Some of the components
* may be unbounded or bounded only on one side if the corresponding bound is
* set to an infinite value. The optimizer will not manage the user function by
* itself, but it will handle this adapter and it is this adapter that will take
* care the bounds are fulfilled. The adapter {@link #value(double[])} method will
* be called by the optimizer with unbound parameters, and the adapter will check
* if the parameters is within range or not. If it is in range, then the underlying
* user function will be called, and if it is not the value of a penalty function
* will be returned instead.
* </p>
* <p>
* This adapter is only a poor-man's solution to simple bounds optimization
* constraints that can be used with simple optimizers like
* {@link org.apache.commons.math3.optim.nonlinear.scalar.noderiv.SimplexOptimizer
* SimplexOptimizer}.
* A better solution is to use an optimizer that directly supports simple bounds like
* {@link org.apache.commons.math3.optim.nonlinear.scalar.noderiv.CMAESOptimizer
* CMAESOptimizer} or
* {@link org.apache.commons.math3.optim.nonlinear.scalar.noderiv.BOBYQAOptimizer
* BOBYQAOptimizer}.
* One caveat of this poor-man's solution is that if start point or start simplex
* is completely outside of the allowed range, only the penalty function is used,
* and the optimizer may converge without ever entering the range.
* </p>
*
* @see MultivariateFunctionMappingAdapter
*
* @since 3.0
*/
public class MultivariateFunctionPenaltyAdapter
implements MultivariateFunction {
Underlying bounded function. /** Underlying bounded function. */
private final MultivariateFunction bounded;
Lower bounds. /** Lower bounds. */
private final double[] lower;
Upper bounds. /** Upper bounds. */
private final double[] upper;
Penalty offset. /** Penalty offset. */
private final double offset;
Penalty scales. /** Penalty scales. */
private final double[] scale;
Simple constructor.
When the optimizer provided points are out of range, the value of the
penalty function will be used instead of the value of the underlying
function. In order for this penalty to be effective in rejecting this
point during the optimization process, the penalty function value should
be defined with care. This value is computed as:
penalty(point) = offset + ∑i[scale[i] * √|point[i]-boundary[i]|]
where indices i correspond to all the components that violates their boundaries.
So when attempting a function minimization, offset should be larger than
the maximum expected value of the underlying function and scale components
should all be positive. When attempting a function maximization, offset
should be lesser than the minimum expected value of the underlying function
and scale components should all be negative.
minimization, and lesser than the minimum expected value of the underlying
function when attempting maximization.
These choices for the penalty function have two properties. First, all out
of range points will return a function value that is worse than the value
returned by any in range point. Second, the penalty is worse for large
boundaries violation than for small violations, so the optimizer has an hint
about the direction in which it should search for acceptable points.
Params: - bounded – bounded function
- lower – lower bounds for each element of the input parameters array (some elements may be set to
Double.NEGATIVE_INFINITY for unbounded values) - upper – upper bounds for each element of the input parameters array (some elements may be set to
Double.POSITIVE_INFINITY for unbounded values) - offset – base offset of the penalty function
- scale – scale of the penalty function
Throws: - DimensionMismatchException – if lower bounds, upper bounds and
scales are not consistent, either according to dimension or to bounadary
values
/**
* Simple constructor.
* <p>
* When the optimizer provided points are out of range, the value of the
* penalty function will be used instead of the value of the underlying
* function. In order for this penalty to be effective in rejecting this
* point during the optimization process, the penalty function value should
* be defined with care. This value is computed as:
* <pre>
* penalty(point) = offset + ∑<sub>i</sub>[scale[i] * √|point[i]-boundary[i]|]
* </pre>
* where indices i correspond to all the components that violates their boundaries.
* </p>
* <p>
* So when attempting a function minimization, offset should be larger than
* the maximum expected value of the underlying function and scale components
* should all be positive. When attempting a function maximization, offset
* should be lesser than the minimum expected value of the underlying function
* and scale components should all be negative.
* minimization, and lesser than the minimum expected value of the underlying
* function when attempting maximization.
* </p>
* <p>
* These choices for the penalty function have two properties. First, all out
* of range points will return a function value that is worse than the value
* returned by any in range point. Second, the penalty is worse for large
* boundaries violation than for small violations, so the optimizer has an hint
* about the direction in which it should search for acceptable points.
* </p>
* @param bounded bounded function
* @param lower lower bounds for each element of the input parameters array
* (some elements may be set to {@code Double.NEGATIVE_INFINITY} for
* unbounded values)
* @param upper upper bounds for each element of the input parameters array
* (some elements may be set to {@code Double.POSITIVE_INFINITY} for
* unbounded values)
* @param offset base offset of the penalty function
* @param scale scale of the penalty function
* @exception DimensionMismatchException if lower bounds, upper bounds and
* scales are not consistent, either according to dimension or to bounadary
* values
*/
public MultivariateFunctionPenaltyAdapter(final MultivariateFunction bounded,
final double[] lower, final double[] upper,
final double offset, final double[] scale) {
// safety checks
MathUtils.checkNotNull(lower);
MathUtils.checkNotNull(upper);
MathUtils.checkNotNull(scale);
if (lower.length != upper.length) {
throw new DimensionMismatchException(lower.length, upper.length);
}
if (lower.length != scale.length) {
throw new DimensionMismatchException(lower.length, scale.length);
}
for (int i = 0; i < lower.length; ++i) {
// note the following test is written in such a way it also fails for NaN
if (!(upper[i] >= lower[i])) {
throw new NumberIsTooSmallException(upper[i], lower[i], true);
}
}
this.bounded = bounded;
this.lower = lower.clone();
this.upper = upper.clone();
this.offset = offset;
this.scale = scale.clone();
}
Computes the underlying function value from an unbounded point.
This method simply returns the value of the underlying function
if the unbounded point already fulfills the bounds, and compute
a replacement value using the offset and scale if bounds are
violated, without calling the function at all.
Params: - point – unbounded point
Returns: either underlying function value or penalty function value
/**
* Computes the underlying function value from an unbounded point.
* <p>
* This method simply returns the value of the underlying function
* if the unbounded point already fulfills the bounds, and compute
* a replacement value using the offset and scale if bounds are
* violated, without calling the function at all.
* </p>
* @param point unbounded point
* @return either underlying function value or penalty function value
*/
public double value(double[] point) {
for (int i = 0; i < scale.length; ++i) {
if ((point[i] < lower[i]) || (point[i] > upper[i])) {
// bound violation starting at this component
double sum = 0;
for (int j = i; j < scale.length; ++j) {
final double overshoot;
if (point[j] < lower[j]) {
overshoot = scale[j] * (lower[j] - point[j]);
} else if (point[j] > upper[j]) {
overshoot = scale[j] * (point[j] - upper[j]);
} else {
overshoot = 0;
}
sum += FastMath.sqrt(overshoot);
}
return offset + sum;
}
}
// all boundaries are fulfilled, we are in the expected
// domain of the underlying function
return bounded.value(point);
}
}