On the area of feasible solutions and its reduction by the complementarity theorem

• We show how the rotational ambiguity can drastically be reduced by means of the complementarity theorem.
• The results directly apply to the area of feasible solutions (AFS).
• A reduction of the rotational ambiguity up to uniqueness is possible.
• The new methods are tested for FT-IR spectroscopic data.
• A software implementation is made available in FAC-PACK.

Multivariate curve resolution techniques in chemometrics allow to uncover the pure component information of mixed spectroscopic data. However, the so-called rotational ambiguity is a difficult hurdle in solving this factorization problem. The aim of this paper is to combine two powerful methodological approaches in order to solve the factorization problem successfully. The first approach is the simultaneous representation of all feasible nonnegative solutions in the area of feasible solutions (AFS) and the second approach is the complementarity theorem. This theorem allows to formulate serious restrictions on the factors under partial knowledge of certain pure component spectra or pure component concentration profiles.In this paper the mathematical background of the AFS and of the complementarity theorem is introduced, their mathematical connection is analyzed and the results are applied to spectroscopic data. We consider a three-component reaction subsystem of the Rhodium-catalyzed hydroformylation process and a four-component model problem.

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