Class KGOIterationalLagrangeMultipliersInDenominatorU

java.lang.Object
com.polytechnik.kgo.KGOIterationalLagrangeMultipliersInDenominatorU

public class KGOIterationalLagrangeMultipliersInDenominatorU extends Object
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.
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