Matthew D. Stuber

Generalized McCormick relaxations

Convex and concave relaxations are used extensively in global optimization algorithms. Among the various techniques available for generating relaxations of a given function, McCormick’s relaxations are attractive due to the recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures. This article leverages the recursive nature of McCormick’s relaxations to define a generalized form which both affords a new framework within which to analyze the properties of McCormick’s relaxations, and extends the applicability of McCormick’s technique to challenging open problems in global optimization. Specifically, relaxations of the parametric solutions of ordinary differential equations are considered in detail, and prospects for relaxations of the parametric solutions of nonlinear algebraic equations are discussed. For the case of ODEs, a complete computational procedure for evaluating convex and concave relaxations of the parametric solutions is described. Through McCormick’s composition rule, these relaxations may be used to construct relaxations for very general optimal control problems.

Convex and concave relaxations of implicit functions

A deterministic algorithm for solving nonconvex NLPs globally using a reduced-space approach is presented. These problems are encountered when real-world models are involved as nonlinear equality constraints and the decision variables include the state variables of the system. By solving the model equations for the dependent (state) variables as implicit functions of the independent (decision) variables, a significant reduction in dimensionality can be obtained. As a result, the inequality constraints and objective function are implicit functions of the independent variables, which can be estimated via a fixed-point iteration. Relying on the recently developed ideas of generalized McCormick relaxations and McCormick-based relaxations of algorithms and subgradient propagation, the development of McCormick relaxations of implicit functions is presented. Using these ideas, the reduced space, implicit optimization formulation can be relaxed. When applied within a branch-and-bound framework, finite convergence to ε-optimal global solutions is guaranteed.

Robust simulation and design using semi-infinite programs with implicit functions

A method is presented for guaranteeing robust steady-state operation of chemical processes using a model-based approach, taking into account uncertainty in the model parameters and disturbances in the process inputs. Intractable constrained max–min optimisation formulations have been proposed for this problem in the past. A new approach is presented in which the equality constraints (process model equations) are solved numerically for the process variables as implicit functions of the uncertain parameters and controls. The problem is then formulated as a semi-infinite program (SIP) constrained only by the performance specifications as semi-infinite inequality constraints. A rigorous, finite e-optimal convergent algorithm for solving such SIPs is proposed, making no assumptions on convexity, which makes use of the novel developments of parametric interval-Newton methods for bounding implicit functions, and novel developments in McCormick relaxations of algorithms.

Robust Simulation and Design using Parametric Interval Methods

Taking into account disturbance uncertainty is required to guarantee robust operation of chemical processes. Likewise, a model-based approach must be taken, inherently introducing model uncertainty, to allow engineers to make robustness guarantees for physical systems at the design stage. One such application of interest is in remote deep sea oil recovery technologies, where the hazards and costs associated with repairs to subsea processing units are extraordinarily high. Swaney and Grossman (1985) and Floudas et al. (2001) considered such robustness problems with steady-state process models expressed as equality constraints and performance specifications as inequality constraints in a bilevel optimization formulation. Due to problem complexity, the bilevel formulation is difficult to solve. A new approach is proposed in which the equality constraints are numerically solved for the process variables as implicit functions of the uncertainty parameters and controls and then formulated as a semi-infinite program (SIP) constrained only by the performance specifications as inequality constraints. Bhattacharjee et al. (2004) developed an interval approach to SIPs with explicit constraints. Stuber and Barton (2009) have applied interval Newton-type methods to parametric nonlinear problems to calculate valid bounds on the range of the implicit function solutions. These two ideas are coupled, resulting in an effective way to solve the implicitly constrained SIP formulation and give robust performance guarantees under uncertainty.

Corrections to: Differentiable McCormick relaxations

These errata correct various errors in the closed-form relaxations provided by Khan, Watson, and Barton in the article “Differentiable McCormick Relaxations” (J Glob Optim, 67:687–729, 2017). Without these corrections, the provided closed-form relaxations may fail to be convex or concave and may fail to be valid relaxations.