The problem of approximating the Pareto front of a multiobjective optimization problem can be reformulated as the problem of finding a set that maximizes the hypervolume indicator. This paper...Show moreThe problem of approximating the Pareto front of a multiobjective optimization problem can be reformulated as the problem of finding a set that maximizes the hypervolume indicator. This paper establishes the analytical expression of the Hessian matrix of the mapping from a (fixed size) collection of n points in the d-dimensional decision space (or m dimensional objective space) to the scalar hypervolume indicator value. To define the Hessian matrix, the input set is vectorized, and the matrix is derived by analytical differentiation of the mapping from a vectorized set to the hypervolume indicator. The Hessian matrix plays a crucial role in second-order methods, such as the Newton-Raphson optimization method, and it can be used for the verification of local optimal sets. So far, the full analytical expression was only established and analyzed for the relatively simple bi-objective case. This paper will derive the full expression for arbitrary dimensions (m ≥ 2 objective functions). For the practically important three-dimensional case, we also provide an asymptotically efficient algorithm with time complexity in O(n log n) for the exact computation of the Hessian Matrix’ non-zero entries. We establish a sharp bound of 12m−6 for the number of non-zero entries. Also, for the general m-dimensional case, a compact recursive analytical expression is established, and its algorithmic implementation is discussed. Also, for the general case, some sparsity results can be established; these results are implied by the recursive expression. To validate and illustrate the analytically derived algorithms and results, we provide a few numerical examples using Python and Mathematica implementations. Open-source implementations of the algorithms and testing data are made available as a supplement to this paper.Show less
Over several decades, a variety of computational methods for drug discovery have been proposed and applied in practice. With the accumulation of data and the development of machine learning methods... Show moreOver several decades, a variety of computational methods for drug discovery have been proposed and applied in practice. With the accumulation of data and the development of machine learning methods, computational drug design methods have gradually shifted to a new paradigm, i.e. deep learning methods have attracted particular interest in drug design. In this study, a new deep learning-based method (DrugEx) was proposed to design de novo drug-like molecules. It was proven that candidate molecules designed by DrugEx had a larger chemical diversity, and better covered the chemical space of known ligands. In order to address the issue of polypharmacology, the DrugEx algorithm was updated with multi-objective optimization towards multiple targets. The results of its application demonstrated the generation of compounds with a diverse predicted selectivity profile toward multiple targets, offering the potential of high efficacy and lower toxicity. In order to improve its generality, DrugEx was further updated to have the capability of designing molecules based on given scaffolds. We extended the architecture of Transformer to deal with each molecule as a graph. As a proof, its effectiveness in that 100% valid molecules are generated and most of them had predicted high affinity towards A2AAR with given scaffolds. Moreover, GenUI was developed as a visualizion software platform that makes it possible to integrate molecular generators within a feature-rich graphical user interface to facilitate collaboration in the disparate communities interested in computer-aided drug discovery.These studies highlight the overwhelming power of AI methods in drug discovery. Show less
In anaerobic co-digestion plants a mix of organic materials is converted to biogas using the anaerobic digestion process. These organic materials, called substrates, can be crops, sludge, manure,... Show moreIn anaerobic co-digestion plants a mix of organic materials is converted to biogas using the anaerobic digestion process. These organic materials, called substrates, can be crops, sludge, manure, organic wastes and many more. They are fed on a daily basis and significantly affect the biogas production process. In this thesis dynamic real-time optimization of the substrate feed for anaerobic co-digestion plants is developed. In dynamic real-time optimization a dynamic simulation model is used to predict the future performance of the controlled plant. Therefore, a complex simulation model for biogas plants is developed, which uses the famous Anaerobic Digestion Model No. 1 (ADM1). With this model the future economics as well as stability can be calculated resulting in a multi-objective performance criterion. Using multi-objective nonlinear model predictive control (NMPC) the model predictions are used to find the optimal substrate feed for the biogas plant. Therefore, NMPC solves an optimization problem over a moving horizon and applies the optimal substrate feed to the plant for a short while before recalculating the new optimal solution. The multi-objective optimization problem is solved using state-of-the-art methods such as SMS-EMOA and SMS-EGO. The performance of the proposed approach is validated in a detailed simulation study Show less
