Global Optimality Conditions and their Geometric Interpretation for the Chemical and Phase Equilibrium Problem

Abstract

A general class of nonlinear optimization problems motivated by the chemical and phase equilibrium problem in chemical thermodynamics is discussed. The relationships between Kuhn–Tucker points and the global minimum are investigated. The relationships are interpreted in terms of common tangent planes to a function with domain in $\mathcal{R}^{N - 1} $ associated with the objective function, whose domain is in $\mathcal{R}^N $. Necessary and sufficient conditions for a global minimum are established, which we call the reaction tangent-plane criterion. The conditions related to the common tangent planes may be considered separately from the feasibility conditions, which allows a novel geometric interpretation of the overall optimality conditions. Illustrative examples are provided of systems involving up to three chemical species.