Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems

We address nonconvex mixed-integer bilinear problems where the main challenge is the computation of a tight upper bound for the objective function to be maximized. This can be obtained by using the recently developed concept of multiparametric disaggregation following the solution of a mixed-integer linear relaxation of the bilinear problem. Besides showing that it can provide tighter bounds than a commercial global optimization solver within a given computational time, we propose to also take advantage of the relaxed formulation for contracting the variables domain and further reduce the optimality gap. Through the solution of a real-life case study from a hydroelectric power system, we show that this can be an efficient approach depending on the problem size. The relaxed formulation from multiparametric formulation is provided for a generic numeric representation system featuring a base between 2 (binary) and 10 (decimal)

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

GALINI: an extensible solver for mixed-integer quadratically-constrained problems

Many industrial relevant optimization problems can be formulated as Mixed-Integer Quadratically Constrained Problems. This class of problems are difficult to solve because of 1) the non-convex bilinear terms 2) integer variables. This thesis develops the Python library \suspect{} for detecting special structure (monotonicity and convexity) of Pyomo models. This library can be extended to provide specialized detection for complex expressions. As a motivating example, we show how the library can be used to detect the convexity of the reciprocal of the log mean temperature difference. This thesis introduces GALINI: a novel solver that is easy to extend at runtime with new 1) cutting planes, 2) primal heuristics, 3) branching strategies, 4) node selection strategies, and 5) relaxations.GALINI uses Pyomo to represent optimization problems, this decision makes it possible to integrate with the existing Pyomo ecosystem to provide, for example, building blocks for relaxations. We test the solver on two large datasets and show that the performance is comparable to existing open source solvers. Finally, we present a library to formulate pooling problems, a class of network flow problems, using Pyomo. The library provides a mechanism to automatically generate the PQ-formulation for pooling problems. Since the library keeps the knowledge of the original network, it can 1) use a mixed-integer programming primal heuristic specialized for the pooling problem to find a feasible solution, and 2) generate valid cuts for the pooling problem. We use this library to develop an extension for GALINI that uses the mixed-integer programming primal heuristic to find a feasible solution and that generates cuts at every node of the branch & cut algorithm. We test GALINI with the pooling extensions on large scale instances of the pooling problem and show that we obtain results that are comparable to or better than the best available commercial solver on dense instances.Open Acces

Integration of process design and control: A review

There is a large variety of methods in literature for process design and control, which can be classified into two main categories. The methods in the first category have a sequential approach in which, the control system is designed, only after the details of process design are decided. However, when process design is fixed, there is little room left for improving the control performance. Recognizing the interactions between process design and control, the methods in the second category integrate some control aspects into process design. With the aim of providing an exploration map and identifying the potential areas of further contributions, this paper presents a thematic review of the methods for integration of process design and control. The evolution paths of these methods are described and the advantages and disadvantages of each method are explained. The paper concludes with suggestions for future research activities

Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

Branching strategies for mixed-integer programs containing logical constraints and decomposable structure

Decision-making optimisation problems can include discrete selections, e.g. selecting a route, arranging non-overlapping items or designing a network of items. Branch-and-bound (B&B), a widely applied divide-and-conquer framework, often solves such problems by considering a continuous approximation, e.g. replacing discrete variable domains by a continuous superset. Such approximations weaken the logical relations, e.g. for discrete variables corresponding to Boolean variables. Branching in B&B reintroduces logical relations by dividing the search space. This thesis studies designing B&B branching strategies, i.e. how to divide the search space, for optimisation problems that contain both a logical and a continuous structure. We begin our study with a large-scale, industrially-relevant optimisation problem where the objective consists of machine-learnt gradient-boosted trees (GBTs) and convex penalty functions. GBT functions contain if-then queries which introduces a logical structure to this problem. We propose decomposition-based rigorous bounding strategies and an iterative heuristic that can be embedded into a B&B algorithm. We approach branching with two strategies: a pseudocost initialisation and strong branching that target the structure of GBT and convex penalty aspects of the optimisation objective, respectively. Computational tests show that our B&B approach outperforms state-of-the-art solvers in deriving rigorous bounds on optimality. Our second project investigates how satisfiability modulo theories (SMT) derived unsatisfiable cores may be utilised in a B&B context. Unsatisfiable cores are subsets of constraints that explain an infeasible result. We study two-dimensional bin packing (2BP) and develop a B&B algorithm that branches on SMT unsatisfiable cores. We use the unsatisfiable cores to derive cuts that break 2BP symmetries. Computational results show that our B&B algorithm solves 20% more instances when compared with commercial solvers on the tested instances. Finally, we study convex generalized disjunctive programming (GDP), a framework that supports logical variables and operators. Convex GDP includes disjunctions of mathematical constraints, which motivate branching by partitioning the disjunctions. We investigate separation by branching, i.e. eliminating solutions that prevent rigorous bound improvement, and propose a greedy algorithm for building the branches. We propose three scoring methods for selecting the next branching disjunction. We also analyse how to leverage infeasibility to expedite the B&B search. Computational results show that our scoring methods can reduce the number of explored B&B nodes by an order of magnitude when compared with scoring methods proposed in literature. Our infeasibility analysis further reduces the number of explored nodes.Open Acces

Simultaneous minimisation of water and energy within a water and membrane network superstructure

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2015The scarcity of water and strict environmental regulations have made sustainable engineering a prime concern in the process and manufacturing industries. Water minimisation involves the reduction of freshwater use and effluent discharge in chemical plants. This is achieved through water reuse, water recycle and water regeneration. Optimisation of the water network (WN) superstructure considers all possible interconnections between water sources, water sinks and regenerator units (membrane systems). In most published works, membrane systems have been represented using the “black-box” approach, which uses a simplified linear model to represent the membrane systems. This approach does not give an accurate representation of the energy consumption and associated costs of the membrane systems. The work presented in this dissertation therefore looks at the incorporation of a detailed reverse osmosis network (RON) superstructure within a water network superstructure in order to simultaneously minimise water, energy, operating and capital costs. The WN consists of water sources, water sinks and reverse osmosis (RO) units for the partial treatment of the contaminated water. An overall mixed-integer nonlinear programming (MINLP) framework is developed, that simultaneously evaluates both water recycle/reuse and regeneration reuse/recycle opportunities. The solution obtained from optimisation provides the optimal connections between various units in the network arrangement, size and number of RO units, booster pumps as well as energy recovery turbines. The work looks at four cases in order to highlight the importance of including a detailed regeneration network within the water network instead of the traditional “black-box’’ model. The importance of using a variable removal ratio in the model is also highlighted by applying the work to a literature case study, which leads to a 28% reduction in freshwater consumption and 80% reduction in wastewater generation.GR201

Superstructure optimisation of a water minimisation network with a embedded multicontaminant electrodialysis model

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering, 2016The water-energy nexus considers the relationship between water and energy resources. Increases in environmental degradation and social pressures in recent years have necessitated the development of manufacturing processes that are conservative with respect to both these resources, while maintaining financial viability. This can be achieved by process integration (PI); a holistic approach to design which emphasises the unity of processes. Within the realm of PI, water network synthesis (WNS) explores avenues for reuse, recycle and regeneration of effluent in order to minimise freshwater consumption and wastewater production. When regeneration is required, membrane-based treatment processes may be employed. These processes are energy intensive and result in a trade-off between water and energy minimisation, thus creating an avenue for optimisation. Previous work in WNS employed a black box approach to represent regenerators in water minimisation problems. However, this misrepresents the cost of regeneration and underestimates the energy requirements of a system. The aim of the research presented in this dissertation is to develop an integrated water regeneration network synthesis model to simultaneously minimise water and energy in a water network. A novel MINLP model for the design of an electrodialysis (ED) unit that is capable of treating a binary mixture of simple salts was developed from first principles. This ED model was embedded into a water network superstructure optimisation model, where the objective was to minimise freshwater and energy consumption, wastewater productions, and associated costs. The model was applied to a pulp and paper case study, considering several scenarios. Global optimisation of the integrated water network and ED design model, with variable contaminant removal ratios, was found to yield the best results. A total of 38% savings in freshwater, 68% reduction in wastewater production and 55% overall cost reduction were observed when compared with the original design. This model also led to a 80% reduction in regeneration (energy) cost.GS201