THE FOUNDATIONS OF SET-THEORETIC ESTIMATION

The conventional approach to estimation problems has been to optimize an objective function with or without constraints. The solvability of the resulting optimization problem is definitely a central issue and m ay lead to the selection of an unrealistic objective function and seve re limitations in the incorporation of available information. Conseque ntly, the reliability of the solutions becomes questionable, as they m ay violate known constraints about the problem. Set theoretic estimati on is governed by the notion of feasibility and produces solutions who se sole property is to be consistent with all information arising from the observed data and a priori knowledge. Each piece of information i s associated with a set in the solution space and the intersection of these sets, the feasibility set, represents the acceptable solutions. The practical use of the set theoretic framework stems from the existe nce of efficient techniques for finding these solutions. Many scattere d problems in systems science and signal processing have been approach ed in set theoretic terms over the past three decades. The purpose of this paper is to synthesize these various approaches into a single, ge neral framework, to examine its fundamental philosophy, goals, and ana lytical techniques, and to relate it to conventional methods. Better u nderstanding of the set theoretic approach will result in more applica tions in sciences and engineering and will stimulate further theoretic al research.