Theory and implementation for the global optimization of a wide class of algorithms is presented via convex/affine relaxations. The basis for the proposed relaxations is the systematic construction of subgradients for the convex relaxations of factorable functions by McCormick [Math. Prog., 10 (1976), pp. 147–175]. Similar to the convex relaxation, the subgradient propagation relies on the recursive application of a few rules, namely, the calculation of subgradients for addition, multiplication, and composition operations. Subgradients at interior points can be calculated for any factorable function for which a McCormick relaxation exists, provided that subgradients are known for the relaxations of the univariate intrinsic functions. For boundary points, additional assumptions are necessary. An automated implementation based on operator overloading is presented, and the calculation of bounds based on affine relaxation is demonstrated for illustrative examples. Two numerical examples for the global optimization of algorithms are presented. In both examples a parameter estimation problem with embedded differential equations is considered. The solution of the differential equations is approximated by algorithms with a fixed number of iterations.

This paper is concerned with tube-based model predictive control (MPC) for both linear and nonlinear, input-affine continuous-time dynamic systems that are affected by time-varying disturbances. We derive a min-max differential inequality describing the support function of positive robust forward invariant tubes, which can be used to construct a variety of tube-based model predictive controllers. These constructions are conservative, but computationally tractable and their complexity scales linearly with the length of the prediction horizon. In contrast to many existing tube-based MPC implementations, the proposed framework does not involve discretizing the control policy and, therefore, the conservatism of the predicted tube depends solely on the accuracy of the set parameterization. The proposed approach is then used to construct a robustMPCscheme based on tubes with ellipsoidal cross-sections. This ellipsoidal MPC scheme is based on solving an optimal control problem under linear matrix inequality constraints. We illustrate these results with the numerical case study of a spring-mass-damper system

Ionic liquids have found their way into many applications where they show a high potential to replace traditional chemicals. But concerns over their ecological impacts (toxicity and biodegradability) and high cost have limited their use so far. The outcome of existing techno-economic and life-cycle assessments comparing ionic liquids with existing solvents has proven hard to interpret due to the many metrics used and trade-offs between them. For the first time, this paper couples the concept of monetization with detailed process simulation and life-cycle assessment to estimate the true cost of ionic liquids. A comparative case study on four solvents used in lignocellulosic biomass pretreatment is conducted: triethylammonium hydrogen sulfate [TEA][HSO₄], 1-methylimidazolium hydrogen sulfate [HMIM][HSO₄], acetone from fossil sources, and glycerol from renewable sources. The results show that the total monetized cost of production accounting for externalities can be more than double the direct costs estimated using conventional economic assessment methods. [TEA][HSO₄] is found to have the lowest total cost, while glycerol presents the highest total cost. We expect this methodology to provide a starting point for future research and development in sustainable ionic liquids.