Ten years of feasibility pump, and counting

The Feasibility Pump (fp) is probably the best-known primal heuristic for mixed-integer programming. The original work by Fischetti et al. (Math Program 104(1):91\u2013104, 2005), which introduced the heuristic for 0\u20131 mixed-integer linear programs, has been succeeded by more than twenty follow-up publications which improve the performance of the fp and extend it to other problem classes. Year 2015 was the tenth anniversary of the first fp publication. The present paper provides an overview of the diverse Feasibility Pump literature that has been presented over the last decade

Rounding-based heuristics for nonconvex MINLPs

We propose two primal heuristics for nonconvex mixed-integer nonlinear programs. Both are based on the idea of rounding the solution of a continuous nonlinear program subject to linear constraints. Each rounding step is accomplished through the solution of a mixed-integer linear program. Our heuristics use the same algorithmic scheme, but they differ in the choice of the point to be rounded (which is feasible for nonlinear constraints but possibly fractional) and in the linear constraints. We propose a feasibility heuristic, that aims at finding an initial feasible solution, and an improvement heuristic, whose purpose is to search for an improved solution within the neighborhood of a given point. The neighborhood is defined through local branching cuts or box constraints. Computational results show the effectiveness in practice of these simple ideas, implemented within an open-source solver for nonconvex mixed-integer nonlinear programs

Pump scheduling in drinking water distribution networks with an LP/NLP-based branch and bound

This paper offers a novel approach for computing globally optimal solutions to the pump scheduling problem in drinking water distribution networks. A tight integer linear relaxation of the original non-convex formulation is devised and solved by branch and bound where integer nodes are investigated through non-linear programming to check the satisfaction of the non-convex constraints and compute the actual cost. This generic method can tackle a large variety of networks , e.g. with variable-speed pumps. We also propose to specialize it for a common subclass of networks with several improving techniques, including a new primal heuristic to repair near-feasible integer relaxed solutions. Our approach is numerically assessed on various case studies of the literature and compared with recently reported results

Newly proposed strategies to increase the energy efficiency of water systems

One of the main challenges in the water industry consists of the reduction of environmental impacts, as well as the containment of energy use. In this research work, new solutions to achieve a sustainable management of water networks have been developed and organized in three lines of research. The main line of research is based on the optimal location of hydraulic devices within a water distribution network in order to maximize the energy production and water savings, as well as to minimize the investment cost. Firstly, the installation of only Pumps As Turbines (PATs) has been analyzed within a literature synthetic network and a new Mixed Integer Non-Linear Programming (MINLP) model has been developed to perform the optimization. Such an optimization model has been defined by a thorough mathematical formulation in order to deal with the extremely hard technical and computational complexities affecting the optimization procedure. In this research, only deterministic solvers have been employed to search the optima, and a comparison of their performance has been also carried out. Most of the computations have been performed by a global optimization solver, which potentially finds the global optimum in both convex and non-convex problems, but is also used to find good quality local optima in very complex problems, where the achievement of the exact solution may require infinite computational time. Compared to other studies in literature on the same network, the proposed study accounts for crucial hydraulic aspects, such as the phenomenon of flow reversion during the day affecting the installation and the operation of the devices, as well as the need for installing machines generating a power above a minimum fixed value. A comparison with such previous literature works has been carried out in order to highlight the effectiveness of the newly proposed optimization procedure. Moreover, to develop a more realistic and comprehensive mathematical model, the simultaneous installation of PATs and Pressure Reducing Valves (PRVs) has been also modeled by the introduction of new variables and mathematical constraints. Indeed, in presence of large water savings but small energy recovery, a PRV might be a more viable solution than a PAT. Compared to other studies in literature optimizing the only location of PATs within the same synthetic network, the simultaneous installation of valves and turbines, as well as the formulation of new hydraulic constraints, has significantly increased the value of the optimization model. In addition, the optimization has been extended to a real water distribution network serving the Blackstairs region (IE), with the aim of testing the robustness of the model and of the optimization procedure in more complex and larger problems. Indeed, the computational complexity affecting the optimization procedure increases according to the size of the network and the mathematical formulation proposed for the synthetic network might be not suitable for such a more complex case study. Compared to the synthetic network where the pressure reduction up to defined minimum requirements has not compromised the hydraulic operation of the system, in the analyzed real water network the exploitation of the available excess pressure to save both water and energy raises the need for employing also pumping systems to supply the most remote nodes of the network. The installation of pumping systems within the network has been therefore included within the optimization procedure and the outcome has been a new model for a Global Optimization of Hydraulic Devices Location (GOHyDeL), suitable for any water distribution network. Such a new model has been the result of progressive findings and hard attempts to deal with the enormous complexities arising during the computation. In all the performed optimization, the maximized water and energy savings and the minimized installation costs have been assessed according to a cost model used by previous authors in literature, in order to make a more straightforward comparison with such literature works. However, more recent cost models available in literature have been also employed to achieve more reasonable and realistic values of the results. According to the comparison between results obtained by using different cost models, despite the employment of more recent models leading to significantly larger investment costs and, thus, smaller values of NPV, the solutions are quite similar in terms of location of installed devices, and the achieved savings are comparable as well. However, among all the devices, the PRVs have resulted to be more affected by the choice of the cost model, due to the strong dependency of the valve costs on the pipe diameter. On the whole, beyond the large feasibility of the model within the optimal location field, a remarkable value of the proposed research also results from the new formulation of mathematical constraints and variables, which requires less computational effort and could find application also in more general optimization problems. The second line of research defines and compares two alternative strategies to supply a real water distribution network. The first solution consists of an elevated reservoir, which is located upstream of a water distribution network and is supplied from the water source by a pumping system. In this scheme, the excess pressure is not dissipated by a traditional valve, but rather a pump as turbine is installed to contain the pressure, thus water leakage, and also recover energy. The second hydraulic scheme instead consists of a pump supplying the downstream network directly from the source. In this scheme there is not an excess pressure to convert in energy, since the elevated reservoir is bypassed and the flow is pumped to the network with lower head. Such new schemes represent two different strategies to increase the energy efficiency of a supply system, as alternatives to the use of elevated reservoir with dissipation of the excess pressure by means of pressure reducing valves. The two schemes have been properly designed in order to find the devices, in terms of diameter and rotational speed, minimizing the energy requirements, thus maximizing the energy efficiency of the whole system. Given the water network supplying a small village in Ballacolla area (IE), the direct supply of the network has resulted a more efficient strategy than the indirect supply scheme with energy recovery. Moreover, the two schemes have been compared by varying the operating conditions, thus considering different combinations of distance and elevation of the source from the water distribution network. The energy audit of the two schemes has been assessed by new energy efficiency indices and also by literature indices. The comparison has showed that the convenience of a scheme over the other significantly depends on the operating conditions. However, with equal values of pumping head in both the schemes, the indirect scheme with energy recovery is up to 5 % more convenient than the direct pumping scheme, which is instead more efficient if the pumping head could be reduced up to 6 %. In the third line of research a new strategy to save energy in the urban water management is presented. The proposed solution consists of a mixed PAT-pump turbocharger, that is a PAT-equipped turbopump exploiting an excess pressure within the fresh water network to produce energy, which is entirely used to carry a wastewater stream towards a treatment plant. In this system, the excess pressure is converted by the PAT in a mechanical torque, which in turn supplies the pump mounted on the same shaft. Such a plant arises whenever wastewater pumping station and excess pressure point could be co-located, thus in low ground areas where high clean-water pressures occur and sewage networks are equipped with pumping systems due to the need to treat the wastewater. In this application, the water distribution network serving Ballacolla area (IE) has been assumed as case study, since it is suitable for the installation of this kind of plant. A preliminary geometric selection of the devices has been performed by a new selection method based on the maximum daily averaged values of fresh and wastewater discharge. Then, the behavior of the plant has been simulated for several wastewater hydrographs by a new mathematical model. The benefits of the plants have been assessed and compared with a conventional wastewater pumping system working in ON/OFF mode. According to the comparison, the higher Net Present Value (NPV) of the MPP plant proves the advantage of this scheme over the conventional system, at least until the useful life of the plant is reached

Trajectory-based methods for solving nonlinear and mixed integer nonlinear programming problems

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2015.I would like to acknowledge a number of people who contributed towards the completion of this thesis. Firstly, I thank my supervisor Professor Montaz Ali for his patience, enthusiasm, guidance and teachings. The skills I have acquired during this process have infiltrated every aspect of my life. I remain forever grateful. Secondly, I would like to say a special thank you to Professor Jan Snyman for his assistance, which contributed immensely towards this thesis. I would also like to thank Professor Dominque Orban for his willingness to assist me for countless hours with the installation of CUTEr, as well as Professor Jose Mario Martinez for his email correspondence. A heartfelt thanks goes out to my family and friends at large, for their prayers, support and faith in me when I had little faith in myself. Thank you also to my colleagues who kept me sane and motivated, as well as all the support staff who played a pivotal roll in this process. Above all, I would like to thank God, without whom none of this would have been possible

Multi-objective optimisation: algorithms and application to computer-aided molecular and process design

Computer-Aided Molecular Design (CAMD) has been put forward as a powerful and systematic technique that can accelerate the identification of new candidate molecules. Given the benefits of CAMD, the concept has been extended to integrated molecular and process design, usually referred to as Computer-Aided Molecular and Process Design (CAMPD). In CAMPD approaches, not only is the interdependence between the properties of the molecules and the process performance captured, but it is also possible to assess the optimal overall performance of a given fluid using an objective function that may be based on process economics, energy efficiency, or environmental criteria. Despite the significant advances made in the field of CAM(P)D, there are remaining challenges in handling the complexities arising from the large mixed-integer nonlinear structure-property and process models and the presence of conflicting performance criteria that cannot be easily merged into a single metric. Many of the algorithms proposed to date, however, resort to single-objective decomposition-based approaches. To overcome these challenges, a novel CAMPD optimisation framework is proposed, in the first part of thesis, in the context of identifying optimal amine solvents for carbon dioxide (CO2) chemical absorption. This requires development and validation of a model that enables the prediction of process performance metrics for a wide range of solvents for which no experimental data exist. An equilibrium-stage model that incorporates the SAFT-γ Mie group contribution approach is proposed to provide an appropriate balance between accuracy and predictive capability with varying molecular design spaces. In order to facilitate the convergence behaviour of the process-molecular model, a tailored initialisation strategy is established based on the inside-out algorithm. Novel feasibility tests that are capable of recognising infeasible regions of molecular and process domains are developed and incorporated into an outer-approximation framework to increase solution robustness. The efficiency of the proposed algorithm is demonstrated by applying it to the design of CO2 chemical absorption processes. The algorithm is found to converge successfully in all 150 runs carried out. To derive greater insights into the interplay between solvent and process performance, it is desirable to consider multiple objectives. In the second part of the thesis, we thus explore the relative performance of five multi-objective optimisations (MOO) solution techniques, modified from the literature to address nonconvex MINLPs, on CAM(P)D problems to gain a better understanding of the performance of different algorithms in identifying the Pareto front efficiently. The combination of the sandwich algorithm with a multi-level single-linkage algorithm to solve nonconvex subproblems is found to perform best on average. Next, a robust algorithm for bi-objective optimisation (BOO), the SDNBI algorithm, is designed to address the theoretical and numerical challenges associated with the solution of general nonconvex and discrete BOO problems. The main improvements in the development of the algorithm are focused on the effective exploration of the nonconvex regions of the Pareto front and the early identification of regions where no additional Pareto solutions exist. The performance of the algorithm is compared to that of the sandwich algorithm and the modified normal boundary intersection method (mNBI) over a set of literature benchmark problems and molecular design problems. The SDNBI found to provide the most evenly distributed approximation of the Pareto front as well as useful information on regions of the objective space that do not contain a nondominated point. The advances in this thesis can accelerate the discovery of novel solvents for CO2 capture that can achieve improved process performance. More broadly, the modelling and algorithmic development presented extend the applicability of CAMPD and MOO based CAMD/CAMPD to a wider range of applications.Open Acces

Developing an Enhanced Algorithms to Solve Mixed Integer Non-Linear Programming Problems Based on a Feasible Neighborhood Search Strategy

Engineering optimization problems often involve nonlinear objective functions, which can capture complex relationships and dependencies between variables. This study focuses on a unique nonlinear mathematics programming problem characterized by a subset of variables that can only take discrete values and are linearly separable from the continuous variables. The combination of integer variables and non-linearities makes this problem much more complex than traditional nonlinear programming problems with only continuous variables. Furthermore, the presence of integer variables can result in a combinatorial explosion of potential solutions, significantly enlarging the search space and making it challenging to explore effectively. This issue becomes especially challenging for larger problems, leading to long computation times or even infeasibility. To address these challenges, we propose a method that employs the "active constraint" approach in conjunction with the release of nonbasic variables from their boundaries. This technique compels suitable non-integer fundamental variables to migrate to their neighboring integer positions. Additionally, we have researched selection criteria for choosing a nonbasic variable to use in the integerizing technique. Through implementation and testing on various problems, these techniques have proven to be successful

Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity