Soft constraints for reducing the intrinsic rotational ambiguity of the area of feasible solutions

• The reduction of the rotational ambiguity is demonstrated within the Area of Feasible Solutions (AFS).
• Constraints on unimodality, monotonicity and on window restrictions are used to illustrate the concepts.
• Numerical results are presented for a model reaction and for experimental FT-IR data from the hydroformylation process.
• The paper proposes a path from the AFS to MCR methods by taking into consideration additional soft constraints.

The reduction of the rotational ambiguity in multivariate curve resolution problems is a central challenge in order to construct an effective chemometric method. Soft modeling is a method of choice to solve this problem.The aim of this paper is to demonstrate the impact of soft constraints on the full set of all feasible, nonnegative solutions. To this end the starting point is the Area of Feasible Solutions (AFS) for a three-component system. Then soft constraints, namely constraints on the unimodality, monotonicity and windowing for certain concentration profiles, are used in order to reduce the AFS. This process extracts chemically meaningful solutions from the set of all feasible nonnegative factors and demonstrates the mode of action of soft constraints. Results are presented for a model problem as well as for FT-IR data for a catalytic subsystem of the rhodium-catalyzed hydroformylation process. Typically, the AFS can significantly be reduced by adding soft constraints.