Package com.polytechnik.kgo
Class KGOIterationalLagrangeMultipliersInDenominatorU
java.lang.Object
com.polytechnik.kgo.KGOIterationalLagrangeMultipliersInDenominatorU
A Knowledge Generalized Operator iteration optimization,
a try with Lagrange multipliers in the denominator;
calculations directly in u_orig basis.
Since the variation of Rayleigh quotients
$$
\mathcal{F}=\frac{
\sum\limits_{j,j^{\prime}=0}^{nC-1}\sum\limits_{k,k^{\prime}=0}^{nX-1}
u_{jk}
S_{jk;j^{\prime}k^{\prime}}
u_{j^{\prime}k^{\prime}}}
{
\sum\limits_{j,j^{\prime}=0}^{nC-1}\sum\limits_{k=0}^{nX-1}
u_{jk}Q_{jj^{\prime}}u_{j^{\prime}k}
}
\xrightarrow[{u}]{\quad }\max
$$
gives
$$
0=\sum\limits_{j^{\prime}=0}^{nC-1}\sum\limits_{k^{\prime}=0}^{nX-1}
S_{sq;j^{\prime}k^{\prime}}u_{j^{\prime}k^{\prime}}
- \mathcal{F} \sum\limits_{j^{\prime}=0}^{nC-1} Q_{sj^{\prime}} u_{j^{\prime}q}
$$
one can use Lagrange multipliers \(\lambda_{ij}\)
in denomination in place of \(Q_{ij}\)
instead of considering regular
$$
\mathcal{L}=\sum\limits_{j,j^{\prime}=0}^{nC-1}\sum\limits_{k,k^{\prime}=0}^{nX-1}
u_{jk}
S_{jk;j^{\prime}k^{\prime}}
u_{j^{\prime}k^{\prime}}
+ \sum\limits_{j,j^{\prime}=0}^{nC-1}
\lambda_{jj^{\prime}}\left[\delta_{jj^{\prime}}-\sum\limits_{k^{\prime}=0}^{nX-1}u_{jk^{\prime}} u_{j^{\prime}k^{\prime}} \right]
$$
The result is almost the same as with
KGOIterationalSimpleOptimizationU
wich uses regular Lagrangian approach.- See Also:
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Field Summary
FieldsModifier and TypeFieldDescriptionfinal UAdjustment
final KGOEVSelection
private static final boolean
private static final int
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Constructor Summary
ConstructorsConstructorDescriptionKGOIterationalLagrangeMultipliersInDenominatorU
(int nC, int nX, double[] SK, double eps) -
Method Summary
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Field Details
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N_iterations
private static final int N_iterations- See Also:
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FORCE_DIAG_DUMP
private static final boolean FORCE_DIAG_DUMP- See Also:
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evSelected
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aep
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Constructor Details
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KGOIterationalLagrangeMultipliersInDenominatorU
public KGOIterationalLagrangeMultipliersInDenominatorU(int nC, int nX, double[] SK, double eps)
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