We study a variant of the pessimistic bi-level optimization problem, which comprises constraints that must be satisfied for any optimal solution of a subordinate (lower-level) optimization problem. We present conditions that guarantee the existence of optimal solutions in such a problem, and we characterize the computational complexity of various subclasses of the problem. We then focus on problem instances that may lack convexity, but that satisfy a certain independence property. We develop convergent approximations for these instances, and we derive an iterative solution scheme that is reminiscent of the discretization techniques used in semi-infinite programming. We also present a computational study that illustrates the numerical behavior of our algorithm on standard benchmark instances.

We present a global optimization algorithm, Branch-and-Sandwich, for optimistic bilevel programming problems that satisfy a regularity condition in the inner problem. The functions involved are assumed to be nonconvex and twice continuously differentiable. The proposed approach can be interpreted as the exploration of two solution spaces (corresponding to the inner and the outer problems) using a single branch-and-bound tree. A novel branching scheme is developed such that classical branch-and-bound is applied to both spaces without violating the hierarchy in the decisions and the requirement for (global) optimality in the inner problem. To achieve this, the well-known features of branch-and-bound algorithms are customized appropriately. For instance, two pairs of lower and upper bounds are computed: one for the outer optimal objective value and the other for the inner value function. The proposed bounding problems do not grow in size during the algorithm and are obtained from the corresponding problems at the parent node.Keywords Bilevel programming · Nonconvex inner problem · Branch and bound

Systematic approaches for the design of mixtures, based on a computer-aided mixture/blend design (CAM b D) framework, have the potential to deliver better products and processes. In most existing methodologies the number of mixture ingredients is fixed (usually a binary mixture) and the identity of at least one compound is chosen from a given set of candidate molecules. A novel CAM b D methodology is presented for formulating the general mixture design problem where the number, identity and composition of mixture constituents are optimized simultaneously. To this end, generalized disjunctive programming is integrated into the CAM b D framework to formulate the discrete choices. This generic methodology is applied to a case study to find an optimal solvent mixture that maximizes the solubility of ibuprofen. The best performance in this case study is obtained with a solvent mixture, showing the benefit of using mixtures instead of pure solvents to attain enhanced behavior. V C 2016 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 62: [1616][1617][1618][1619][1620][1621][1622][1623][1624][1625][1626][1627][1628][1629][1630][1631][1632][1633] 2016 Keywords: computer-aided mixture design, generalized disjunctive programming, optimal mixture design, ibuprofen, solubility IntroductionMixtures play an important role in the process industries and blends of refrigerants, 1-3 polymers, 4,5 and solvents 6 are used in a wide range of applications. Solvent mixtures, for example, are used in separation processes, such as extraction, 7,8 absorption, 9 and crystallisation, 10,11 and in chemical reactions. 12,13 In product design, the desired performance can often only be achieved with a mixture or formulation (e.g., pesticide formulation, crude oils blended into a single product).14,15 The current regulatory environment is making mixtures increasingly relevant as restrictions are placed on the use of a growing number of compounds. Some common compounds are thus being removed from use as a result of changing regulations (e.g., REACH regulations 16 ). Given this context, the formulation of mixtures offers a potential route to enhanced performance, because mixtures can exhibit properties that equate or even surpass those of pure compounds. Some of the benefits of using mixtures have been demonstrated by Granberg and Rasmuson 17 who studied the solubility of paracetamol in a binary mixture of water and acetone. The results of their study are shown in Figure 1, where it can be seen that, at 308C, mixtures containing up to 75% water by mass achieve at least as high a solubility as pure acetone, with a 30:70 mixture of water and acetone achieving the highest solubility. A fivefold increase in solubility relative to pure acetone is observed, although paracetamol is poorly water-soluble. This nonlinear behavior, which is commonly observed in solubility experiments (e.g., the work of Pacheco et al. 18 ), arises from the nonideal thermodynamics of...

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