Convex relaxations of componentwise convex functions

Abstract

Deterministic global optimization is widely spread in the field of process engineering, e.g., for the solution of chemical equilibrium problems, heat exchanger networks, or process synthesis. The performance of state-of-the-art deterministic global optimization algorithms largely depends on the tightness of convex and concave relaxations. We propose results on relaxations of componentwise convex functions, i.e., functions that are convex with respect to each component when all other components are fixed. We show that if the partial derivatives of the original function satisfy a specific monotonicity condition, we can construct a valid underestimator of the original componentwise convex function. Concave relaxations of componentwise concave functions can be derived analogously. Finally, we show numerical examples and applications in property models, for example, the enthalpy of the subcooled liquid found in the IAPWS-IF97 model. The numerical result show drastic improvement in relaxation tightness when the proposed result is used compared to standard methods.