Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition

This is the author accepted manuscript. The final version is available from Springer via the DOI in this record Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig–Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a “price-and-verify” algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.Federal Ministry of Education and Researc

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

Productivity enhancement through process integration

A hierarchical procedure is developed to determine maximum overall yield of a process and optimize process changes to achieve such a yield. First, a targeting procedure is developed to identify an upper bound of the overall yield ahead of detailed design. Several mass integration strategies are proposed to attain maximum yield. These strategies include rerouting of raw materials, optimization of reaction yield, rerouting of product from undesirable outlets to desirable outlets, and recycling of unreacted raw materials. Path equations are tailored to provide the appropriate level of detail for modeling process performance as a function of the optimization variables pertaining to design and operating variables. Interval analysis is used as an inclusion technique that provides rigorous bounds regardless of the process nonlinearities and without enumeration. Then, a new approach for identification of cost-effective implementation of maximum attainable targets for yield is presented. In this approach, a mathematical program was developed to identify the maximum feasible yield using a combination of iterative additions of constraints and problem reformulation. Next, cost objectives were employed to identify a cost-effective solution with the details of design and operating variables. Constraint convexification was used to improve the quality of the solution towards globability. A trade-off procedure between the saving and expenses for yield maximization problem is presented. The proposed procedure is systematic, rigorous, and computationally efficient. A case study was solved to demonstrate the applicability and usefulness of the developed procedure

MINLP OPTIMIZATION OF STEEL FRAMES

ABSTRACT: The paper presents the discrete dimension optimization of unbraced rigid steel plane frames. The optimization of steel frames was carried out by the Mixed-Integer Non-linear Programming (MINLP) approach. The MINLP is a combined discrete-continuous optimization technique. It performs the discrete optimization of discrete decisions simultaneously with the continuous optimization of continuous parameters. The task of the optimization is to minimize the mass of the frame structure and to find the optimal discrete sizes of standard steel sections for frame members. The finite element equations are defined as the equality constraints for the second-order elastic structural analysis. The design constraints for the steel members were formulated according to Eurocode 3. The Modified Outer-Approximation/ Equality-Relaxation algorithm and a two-phase MINLP optimization approach were applied for the optimization. The latter starts with the continuous optimization of the frame, while the standard dimensions are temporarily relaxed into continuous parameters. When the optimal continuous solution is found, standard sizes of cross-sections are re-established and the simultaneous continuous and discrete dimension optimization by MINLP is then continued until the optimal solution is found. A numerical example of the optimization of a steel frame is presented at the end of the paper to show the suitability of the proposed approach

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

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

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Synthesis and design of integrated reaction-separation systems with complex configurations and rigorous models

Chemical engineering, and specially process design, synthesis and intensification, are well positioned to support both society and industry in overcoming present global challenges of environment degradation, energy supply, water scarcity and food supply. These challenges have been translated into industrial problems that involve the design of chemical processes with decreased water and energy consumption, and improved efficiencies. In this context the present study focuses on the simultaneous synthesis and design of reaction-separation systems including complex configuration distillation columns and using rigorous models. The study is considered a further step in this research area, as previous works have usually focused on the synthesis of sub-networks and have used shortcut models. Additionally, among complex configuration, thermally coupled distillation columns are reported to present significant savings in terms of the total annualised cost of the system. Among the available approaches to synthesis and design, a superstructure optimisation approach is used. The procedure involves the construction of a superstructure that includes a reaction superstructure, taken from Ma et al. (Ma et al. 2019) and a separation superstructure, proposed by Sargent and Gaminibandara (Sargent and K. Gaminibandara 1976). The modelling is performed using generalised disjunctive programming (GDP) to produce a logic-based model. This model is then reformulated into a mixed-integer nonlinear programming (MINLP) optimisation problem, where the objective is to minimise the total annualised cost of the process. For the reformulation convex hull and bypass efficiency methods are used. A modified version of the solving strategy presented by Ma et al. (Ma et al. 2019) is used, which involves using the solver SBB in General Algebraic Modelling System (GAMS). The proposed framework is applied to a case study previously addressed by Zhang et al. (Zhang et al. 2018) and Ma et al. (Ma et al. 2019). Economic models and assumptions made in those studies are maintained in order to evaluate the benefits of including complex configuration columns in the design possibilities. Results present a flowsheet with one PFR reactor and complex configuration distillation columns that are partially thermally coupled. The total annualised cost of the process is 5.85x105 $/yr, which is 6.3% and 4.7% less than the value achieved by Zhang et al. (Zhang et al. 2018)and Ma et al., respectively. Results show that it is both possible and beneficial to consider complex configuration distillation columns, including thermally coupled ones, in the simultaneous synthesis and design of reaction-separation systems using rigorous models.Chevening AwardsAgencia Nacional de Investigación e Innovació