MinConGenLin (JMSL Numerical Library)

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JMSL^TM Numerical Library 4.0

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com.imsl.math
Class MinConGenLin

java.lang.Object
  com.imsl.math.MinConGenLin

All Implemented Interfaces:: Cloneable, Serializable

public class MinConGenLin
extends Object
implements Serializable, Cloneable

Minimizes a general objective function subject to linear equality/inequality constraints.

The class MinConGenLin is based on M.J.D. Powell's TOLMIN, which solves linearly constrained optimization problems, i.e., problems of the form

subject to

given the vectors , , , and and the matrices and .

The algorithm starts by checking the equality constraints for inconsistency and redundancy. If the equality constraints are consistent, the method will revise , the initial guess, to satisfy

Next, is adjusted to satisfy the simple bounds and inequality constraints. This is done by solving a sequence of quadratic programming subproblems to minimize the sum of the constraint or bound violations.

Now, for each iteration with a feasible , let be the set of indices of inequality constraints that have small residuals. Here, the simple bounds are treated as inequality constraints. Let be the set of indices of active constraints. The following quadratic programming problem

$min fleft( {x^k } right) + d^T nabla ,fleft( {x^k } right) + {1 over 2}d^T B^k d$

subject to

is solved to get where is a row vector representing either a constraint in or or a bound constraint on x. In the latter case, the for the bound constraint and for the constraint . Here, is a vector with 1 as the i-th component, and zeros elsewhere. Variables are the Lagrange multipliers, and is a positive definite approximation to the second derivative .

After the search direction is obtained, a line search is performed to locate a better point. The new point $x^{k+1}= x^k +alpha^kd^k$ has to satisfy the conditions

and

The main idea in forming the set is that, if any of the equality constraints restricts the step-length , then its index is not in . Therefore, small steps are likely to be avoided.

Finally, the second derivative approximation , is updated by the BFGS formula, if the condition

$left( {d^K } right)^T nabla fleft( {x^k + alpha ^k d^k } right) - nabla fleft( {x^k } right) > 0$

holds. Let $x^k leftarrow x^{k + 1}$ , and start another iteration.

The iteration repeats until the stopping criterion

$left| {nabla f(x^k ) - A^k lambda ^K } right|_2 le tau$

is satisfied. Here is the supplied tolerance. For more details, see Powell (1988, 1989).

See Also:: Example 1, Example 2, Serialized Form

Nested Class Summary
`static class`	`MinConGenLin.ConstraintsInconsistentException` The equality constraints are inconsistent.
`static class`	`MinConGenLin.ConstraintsNotSatisfiedException` No vector x satisfies all of the constraints.
`static class`	`MinConGenLin.EqualityConstraintsException` the variables are determined by the equality constraints.
`static interface`	`MinConGenLin.Function` Public interface for the user-supplied function to evaluate the function to be minimized.
`static interface`	`MinConGenLin.Gradient` Public interface for the user-supplied function to compute the gradient.
`static class`	`MinConGenLin.VarBoundsInconsistentException` The equality constraints and the bounds on the variables are found to be inconsistent.

Constructor Summary
`MinConGenLin(MinConGenLin.Function fcn, int nvar, int ncon, int neq, double[] a, double[] b, double[] lowerBound, double[] upperBound)` Constructor for `MinConGenLin`.

Method Summary
`int[]`	`getFinalActiveConstraints()` Returns the indices of the final active constraints.
`int`	`getFinalActiveConstraintsNum()` Returns the final number of active constraints.
`double[]`	`getLagrangeMultiplierEst()` Returns the Lagrange multiplier estimates of the final active constraints.
`double`	`getObjectiveValue()` Returns the value of the objective function.
`double[]`	`getSolution()` Returns the computed solution.
`void`	`setGuess(double[] guess)` Sets an initial guess of the solution.
`void`	`setTolerance(double tolerance)` Sets the nonnegative tolerance on the first order conditions at the calculated solution.
`void`	`solve()` Minimizes a general objective function subject to linear equality/inequality constraints.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

MinConGenLin

public MinConGenLin(MinConGenLin.Function fcn,
                    int nvar,
                    int ncon,
                    int neq,
                    double[] a,
                    double[] b,
                    double[] lowerBound,
                    double[] upperBound)

Constructor for MinConGenLin.
Parameters:: fcn - A Function object, user-supplied function to evaluate the function to be minimized.; nvar - A int scalar containing the number of variables.; ncon - A int scalar containing the number of linear constraints (excluding simple bounds).; neq - A int scalar containing the number of linear equality constraints.; a - A double array containing the equality constraint gradients in the first neq rows followed by the inequality constraint gradients. a.length = ncon * nvar; b - A double array containing the right-hand sides of the linear constraints.; lowerBound - A double array containing the lower bounds on the variables. Choose a very large negative value if a component should be unbounded below or set lowerBound[i] = upperBound[i] to freeze the i-th variable. lowerBound.length = nvar; upperBound - A double array containing the upper bounds on the variables. Choose a very large positive value if a component should be unbounded above. upperBound.length = nvar
Throws:: IllegalArgumentException - is thrown if the dimensions of nvar, ncon, neq, a.length , b.length, lowerBound.length and upperBound.length are not consistent.

Method Detail

getFinalActiveConstraints

public int[] getFinalActiveConstraints()

Returns the indices of the final active constraints.

Returns:: a int array containing the indices of the final active constraints.

getFinalActiveConstraintsNum

public int getFinalActiveConstraintsNum()

Returns the final number of active constraints.

Returns:: a int scalar containing the final number of active constraints.

getLagrangeMultiplierEst

public double[] getLagrangeMultiplierEst()

Returns the Lagrange multiplier estimates of the final active constraints.

Returns:: a double array containing the Lagrange multiplier estimates of the final active constraints.

getObjectiveValue

public double getObjectiveValue()

Returns the value of the objective function.

Returns:: a double scalar containing the value of the objective function.

getSolution

public double[] getSolution()

Returns the computed solution.

Returns:: a double array containing the computed solution.

setGuess

public void setGuess(double[] guess)

Sets an initial guess of the solution.

Parameters:: guess - a double array containing an initial guess.

setTolerance

public void setTolerance(double tolerance)

Sets the nonnegative tolerance on the first order conditions at the calculated solution.

Parameters:: tolerance - a double scalar containing the tolerance.

solve

public final void solve()
                 throws MinConGenLin.ConstraintsInconsistentException,
                        MinConGenLin.VarBoundsInconsistentException,
                        MinConGenLin.ConstraintsNotSatisfiedException,
                        MinConGenLin.EqualityConstraintsException