An interval-matrix branch-and-bound algorithm for bounding eigenvalues

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the k-th largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn’s method (an interval extension of Weyl’s theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision > 0 (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4,000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated

Designing optimal mixtures using generalized disjunctive programming: Hull relaxations

A general modeling framework for mixture design problems, which integrates Generalized Disjunctive Programming (GDP) into the Computer-Aided Mixture/blend Design (CAMbD) framework, was recently proposed (S. Jonuzaj, P.T. Akula, P.-M. Kleniati, C.S. Adjiman, 2016. AIChE Journal 62, 1616–1633). In this paper we derive Hull Relaxations (HR) of GDP mixture design problems as an alternative to the big-M (BM) approach presented in this earlier work. We show that in restricted mixture design problems, where the number of components is fixed and their identities and compositions are optimized, BM and HR formulations are identical. For general mixture design problems, where the optimal number of mixture components is also determined, a generic approach is employed to enable the derivation and solution of the HR formulation for problems involving functions that are not defined at zero (e.g., logarithms). The design methodology is applied successfully to two solvent design case studies: the maximization of the solubility of a drug and the separation of acetic acid from water in a liquid-liquid extraction process. Promising solvent mixtures are identified in both case studies. The HR and BM approaches are found to be effective for the formulation and solution of mixture design problems, especially via the general design problem

Arbitrarily tight aBB underestimators of general non-linear functions over sub-optimal domains

In this paper we explore the construction of arbitrarily tight αBB relaxations of C2 general non-linear non-convex functions. We illustrate the theoretical challenges of building such relaxations by deriving conditions under which it is possible for an αBB underestimator to provide exact bounds. We subsequently propose a methodology to build αBB underestimators which may be arbitrarily tight (i.e., the maximum separation distance between the original function and its underestimator is arbitrarily close to 0) in some domains that do not include the global solution (defined in the text as “sub-optimal”), assuming exact eigenvalue calculations are possible. This is achieved using a transformation of the original function into a μ-subenergy function and the derivation of αBB underestimators for the new function. We prove that this transformation results in a number of desirable bounding properties in certain domains. These theoretical results are validated in computational test cases where approximations of the tightest possible μ-subenergy underestimators, derived using sampling, are compared to similarly derived approximations of the tightest possible classical αBB underestimators. Our tests show that μ-subenergy underestimators produce much tighter bounds, and succeed in fathoming nodes which are impossible to fathom using classical αBB

A QM-CAMD approach to solvent design for optimal reaction rates

The choice of solvent in which to carry out liquid-phase organic reactions often has a large impact on reaction rates and selectivity and is thus a key decision in process design. A systematic methodology for solvent design that does not require any experimental data on the effect of solvents on reaction kinetics is presented. It combines quantum mechanical computations for the reaction rate constant in various solvents with a computer-aided molecular design (CAMD) formulation. A surrogate model is used to derive an integrated design formulation that combines kinetics and other considerations such as phase equilibria, as predicted by group contribution methods. The derivation of the mixed-integer nonlinear formulation is presented step-by-step. In the application of the methodology to a classic SN2 reaction, the Menschutkin reaction, the reaction rate is used as the key performance objective. The results highlight the tradeoffs between different chemical and physical properties such as reaction rate constant, solvent density and solid reactant solubility and lead to the identification of several promising solvents to enhance reaction performance

A library of nonconvex bilevel test problems with the corresponding AMPL input files

This library provides a description of nonconvex bilevel test problems in AMPL format, compatible with the Branch-And-Sandwich BiLevel solver - BASBL: http://basblsolver.github.io/home/ Kindly note, that this is a growing collection of nonconvex bilevel problems meant as a resource for researchers in the field, including problem statement, analysis, solution(s) and input file(s). For more details on these problems please visit wiki-page: https://github.com/basblsolver/test-problems/wikiThis library provides a description of nonconvex bilevel test problems in AMPL format, compatible with the Branch-And-Sandwich BiLevel solver - BASBL: http://basblsolver.github.io/home/ Kindly note, that this is a growing collection of nonconvex bilevel problems meant as a resource for researchers in the field, including problem statement, analysis, solution(s) and input file(s). For more details on these problems please visit wiki-page: https://github.com/basblsolver/test-problems/wiki

The formulation of optimal mixtures with generalized disjunctive programming: a solvent design case study

Systematic approaches for the design of mixtures, based on a computer‐aided mixture/blend design (CAMbD) 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 CAMbD 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 CAMbD 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

Repulsion-dispersion parameters for the modelling of organic molecular crystals containing N, O, S and Cl

In lattice energy models that combine ab initio and empirical components, it is important to ensure consistency between these components so that meaningful quantitative results are obtained. A method for deriving parameters of atom-atom repulsion dispersion potentials for crystals, tailored to different ab initio models is presented. It is based on minimization of the sum of squared de- viations between experimental and calculated structures and energies. The solution algorithm is designed to avoid convergence to local minima in the parameter space by combining a deter- ministic low-discrepancy sequence for the generation of multiple initial parameter guesses with an efficient local minimization algorithm. The proposed approach is applied to derive transferable exp-6 potential parameters suitable for use in conjunction with a distributed multipole electrostatics model derived from isolated molecule charge densities calculated at the M06/6-31G(d,p) level of theory. Data for hydrocarbons, azahydrocarbons, oxohydrocarbons, organosulphur compounds and chlorohydrocarbons are used for the estimation. A good fit is achieved for the new set of parameters with a mean absolute error in sublimation enthalpies of 4.1 kJ/mol and an average rmsd 15 of 0.31 Å. The parameters are found to perform well on a separate cross-validation set of 39 compounds

The design of optimal mixtures from atom groups using Generalized Disjunctive Programming

A comprehensive computer-aided mixture/blend design methodology for formulating a gen- eral mixture design problem where the number, identity and composition of mixture constituents are optimized simultaneously is presented in this work. Within this approach, Generalized Dis- junctive Programming (GDP) is employed to model the discrete decisions (number and identities of mixture ingredients) in the problems. The identities of the components are determined by designing molecules from UNIFAC groups. The sequential design of pure compounds and blends, and the arbitrary pre-selection of possible mixture ingredients can thus be avoided, making it possible to consider large design spaces with a broad variety of molecules and mixtures. The proposed methodology is first applied to the design of solvents and solvent mixtures for max- imising the solubility of ibuprofen, often sought in crystallization processes; next, antisolvents and antisolvent mixtures are generated for minimising the solubility of the drug in drowning out crystallization; and finally, solvent and solvent mixtures are designed for liquid-liquid extraction. The GDP problems are converted into mixed-integer form using the big-M approach. Integer cuts are included in the general models leading to lists of optimal solutions which often contain a combination of pure and mixed solvents