MinConGenLin Class

MinConGenLin Class

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

Inheritance Hierarchy

System.Object
Imsl.Math.MinConGenLin

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class MinConGenLin

<SerializableAttribute>
Public Class MinConGenLin

[SerializableAttribute]
public ref class MinConGenLin

[<SerializableAttribute>]
type MinConGenLin =  class end

The MinConGenLin type exposes the following members.

Constructors

	Name	Description
	MinConGenLin	Constructor for MinConGenLin.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetFinalActiveConstraints	Returns the indices of the final active constraints.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetLagrangeMultiplierEstimate	Returns the Lagrange multiplier estimates of the final active constraints.
	GetSolution	Returns the computed solution.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetGuess	Sets an initial guess of the solution.
	Solve	Minimizes a general objective function subject to linear equality/inequality constraints.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	FinalActiveConstraintsNum	Returns the final number of active constraints.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	ObjectiveValue	Returns the value of the objective function.
	Parallel	Enable or disable performing MinConGenLin.IFunctionn.F in parallel.
	Tolerance	The nonnegative tolerance on the first order conditions at the calculated solution.

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Remarks

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

$\min f (x)$

subject to

$A_2x\le b_2$

$x_l \le x\le x_u$

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

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

subject to

$a_jd = 0, \,j \in I_k$

$a_jd \le 0, \,j \in J_k$

is solved to get $(d^k, \lambda^k)$ 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 $x_i \le (x_u)_i$ and for the constraint $-x_i \le (x_l)_i$ . Here, is a vector with 1 as the i-th component, and zeros elsewhere. Variables $\lambda^k$ are the Lagrange multipliers, and is a positive definite approximation to the second derivative $\nabla ^2 f(x^k)$ .

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

$f(x^k + \alpha^kd^k) \le f(x^k) + 0.1 \alpha^k (d^k)^T\nabla f(x^k)$

and

$(d^k)^T\nabla f(x^k +\alpha ^kd^k) \ge 0.7 (d^k)^T\nabla f(x^k)$

The main idea in forming the set is that, if any of the equality constraints restricts the step-length $\alpha^k$ , 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

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

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 $\tau$ is the supplied tolerance. For more details, see Powell (1988, 1989).

Reference

Imsl.Math Namespace

Other Resources

Example 1

Example 2