(10c) New Mccormick-Style Convex Relaxations of Implicit Functions in Global Optimization

Several chemical engineering applications demand global optimization of nonconvex process models, including safety verification and determination of thermodynamic equilibria. Deterministic methods for global optimization typically proceed by generating upper and lower bounds on the unknown objective value at a global minimum, and progressively improving these bounds. Lower bounds in global minimization are typically obtained by minimizing convex relaxations of the original objective function. When this function is known in closed form, then useful convex relaxations can be automatically generated using the well-known alphaBB or McCormick relaxations and their later variants. However, these methods do not extend directly to implicit functions defined as the solutions of nonlinear equation systems, such as thermodynamic equations of state and numerical ODE solutions computed by implicit solvers.

This presentation introduces a new general formula for automatically generating convex relaxations for implicit functions, where these implicit functions are defined by an equation system that involves a known residual function with known relaxations. This formula is compatible with the multivariate McCormick relaxations of composite functions by Tsoukalas and Mitsos [2], and is readily extended to handle inverse functions, to handle equality constraints in nonlinear programming, and to tighten a priori interval bounds for implicit functions. Unlike previous approaches for handling implicit functions, our approach does not place any structural assumptions on the supplied relaxations of the residual functions; to our knowledge, it is the first approach that can use alphaBB relaxations of the residual function. For example, the attached figure shows our implicit function relaxations based on alphaBB relaxations of a particular residual function. Moreover, our approach does not proceed by essentially relaxing the first iterations of a nonlinear equation solver; instead, it generates its relaxations as the solutions of auxiliary convex NLPs that may be solved or further relaxed by any valid method. Several numerical examples are presented for illustration, demonstrating the tightness and versatility of our new relaxations.

References

[1] H. Cao and K.A. Khan, Optimization-based convex relaxations of implicit functions, under review.

[2] A. Tsoukalas and A. Mitsos, Multivariate McCormick relaxations, J Glob Optim, 59:633-662, 2014.