Accurate determination of the aerosol mixing state is indispensable in reducing uncertainties in the assessment of aerosol direct and indirect effects. However, the characterization of mixing state is often limited by the scarcity of direct measurements of aerosol chemical composition and their morphology. Thus, previous studies have largely relied on heuristic approaches to infer mixing states from optical measurements, which are generally deemed only as probable mixing states due to a lack of uncertainty quantification. To address this gap, this study formulates an optical inversion framework to infer aerosol mixing states and composition, explicitly characterizing the inherently ill-posed and ill-conditioned nature of the inversion problem. This formulation leverages the scalability of the AeroMix model in simulating optical properties of complex aerosol mixtures constituted by an arbitrary number of externally mixed and core-shell mixed aerosol components. Recognizing that standard optimization techniques cannot resolve this ill-posed system deterministically, the inversion framework is reframed as a system of linear inequalities. This approach geometrically bounds the feasible solution space as the boundary of a convex polytope. This theoretical confinement is further evaluated and empirically illustrated using simulated aerosol mixtures, confirming that all physically valid solutions strictly reside on the boundary facets of the formulated convex polytope. By transitioning from heuristic methods to an analytical geometry framework using AeroMix, this study establishes a rigorous foundation for the probabilistic determination of aerosol mixing states from widely available optical measurements.