Motivated by applications to open program reasoning such as maximal specification inference, this paper studies \emph{optimal CHC solving}, a problem to compute maximal and/or minimal solutions of constrained Horn clauses (CHCs). This problem and its subproblems have been studied in the literature, and a major approach is to iteratively improve a solution of CHCs until it becomes optimal. So a key ingredient of optimization methods is the optimality checking of a given solution.
We propose a novel optimality checking method, as well as an optimization method using the proposed optimality checker, based on a computational theoretical analysis of the optimality checking problem. The key observation is that the optimality checking problem is closely related to the termination analysis of programs, and this observation is useful both theoretically and practically. From a theoretical perspective, it clarifies a limitation of an existing method and incorrectness of another method in the literature. From a practical perspective, it allows us to apply techniques of termination analysis to the optimality checking of a solution of CHCs. We present an optimality checking method based on constraint-based synthesis of termination arguments, implemented our method, evaluated it on CHCs that encode maximal specification synthesis problems, and obtained promising results.