The semi-continous quadratic mixture design problem

The semi-continuous quadratic mixture design problem (SCQMDP) is described as a problem with linear, quadratic and semi-continuity con- straints. Moreover, a linear cost objective and an integer valued objective are introduced. The research question is to deal with the SCQMD prob- lem from a Branch-and-Bound perspective generating robust solutions. Therefore, an algorithm is outlined which is rigorous in the sense it iden- ti¯es instances where decision makers tighten requirements such that no ²-robust solution exists. The algorithm is tested on several cases derived from industry

Robust solutions of bi-blend recipe optimization with quadratic constraints

Production companies use raw materials to compose end-products. They often make different products with the same raw materials. In this research, the focus lies on the production of two end-products consisting of (partly) the same raw materials as cheap as possible. Each of the products has its own demand and quality requirements consisting of quadratic constraints. The minimization of the costs, given the quadratic constraints is a global optimization problem, which can be difficult because of possible local optima. Therefore, the multi modal character of the (bi-) blend problem is investigated. Standard optimization packages (solvers) in Matlab and GAMS were tested on their ability to solve the problem. In total 20 test cases were generated and taken from literature to test solvers on their effectiveness and efficiency to solve the problem. The research also gives insight in adjusting the quadratic constraints of the problem in order to make a robust problem formulation of the bi-blend problem

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

Locating a competitive facility in the plane with a robustness criterion

A new continuous location model is presented and embedded in the literature on robustness in facility location. The multimodality of the model is investigated, and a branch and bound method based on dc optimization is described. Numerical experience is reported, showing that the developed method allows one to solve in a few seconds problems with thousands of demand points.Ministerio de Ciencia e InnovaciónJunta de AndalucíaEuropean Regional Development Fun

On a branch-and-bound approach for a Huff-like Stackelberg location problem

Modelling the location decision of two competing firms that intend to build a new facility in a planar market can be done by a Huff-like Stackelberg location problem. In a Huff-like model, the market share captured by a firm is given by a gravity model determined by distance calculations to facilities. In a Stackelberg model, the leader is the firm that locates first and takes into account the actions of the competing chain (follower) locating a new facility after the leader. The follower problem is known to be a hard global optimisation problem. The leader problem is even harder, since the leader has to decide on location given the optimal action of the follower. So far, in literature only heuristic approaches have been tested to solve the leader problem. Our research question is to solve the leader problem rigorously in the sense of having a guarantee on the reached accuracy. To answer this question, we develop a branch-and-bound approach. Essentially, the bounding is based on the zero sum concept: what is gain for one chain is loss for the other. We also discuss several ways of creating bounds for the underlying (follower) sub-problems, and show their performance for numerical cases

On the minimum number of simplex shapes in longest edge bisection refinement of a regular n-simplex

In several areas like Global Optimization using branch-and-bound methods, the unit n-simplex is refined by bisecting the longest edge such that a binary search tree appears. This process generates simplices belonging to different shape classes. Having less simplex shapes facilitates the prediction of the further workload from a node in the binary tree, because the same shape leads to the same sub-tree. Irregular sub-simplices generated in the refinement process may have more than one longest edge when n\geqslant 3. The question is how to choose the longest edge to be bisected such that the number of shape classes is as small as possible. We develop a Branch-and-Bound (B&B) algorithm to find the minimum number of classes in the refinement process. The developed B&B algorithm provides a minimum number of eight classes for a regular 3-simplex. Due to the high computational cost of solving this combinatorial problem, future research focuses on using high performance computing to derive the minimum number of shapes in higher dimensions

Global optimization at work

In many research situations where mathematical models are used, researchers try to find parameter values such that a given performance criterion is at an optimum. If the parameters can be varied in a continuous way, this in general defines a so-called Nonlinear Programming Problem. Methods for Nonlinear Programming usually result in local optima. A local optimum is a solution (parameter values) which is the best with respect to values in the neighbourhood of that solution, not necessarily the best over the total admissible, feasible set of all possible parameter values, solutions.For mathematicians this results in the research question: How to find the best, global optimum in situations where several local optima exist?, the field of Global Optimization (GLOP). Literature, books and a specific journal, has appeared during the last decades on the field. Main focus has been on the mathematical side, i.e. given assumptions on the structure of the problems to be solved and specific global optimization methods and properties are derived. Cooperation between mathematicians and researchers (in this book called 'the modeller' or 'the potential user'), who saw global optimization problems in practical problems has lead to application of GLOP algorithms to practical optimization problems. Some of those can be found in this book. In this book we started with the question:Given a potential user with an arbitrary global optimization problem, what route can be taken in the GLOP forest to find solutions of the problem?From this first question we proceed by raising new questions. In Chapter 1 we outline the target group of users we have in mind, i.e. agricultural and environmental engineers, designers and OR workers in agricultural science. These groups are not clearly defined, nor mutually exclusive, but have in common that mathematical modelling is used and there is knowledge of linear programming and possibly of combinatorial optimization.In general, when modellers are confronted with optimization aspects, the first approach is to develop heuristics or to look for standard nonlinear programming codes to generate solutions of the optimization problem. During the search for solutions, multiple local optima may appear. We distinguished two major tracks for the path to be taken from there by the potential user to solve the problem. One track is called the deterministic track and is discussed in Chapters 2, 3 and 4. The other track is called the stochastic track and is discussed in Chapters 5 and 6. The two approaches are intended to reach a different goal.The deterministic track aims at:The global optimum is approximated (found) with certainty in a finite number of steps.The stochastic track is understood to contain some stochastic elements and aims at:Approaching the optimum in a probabilistic sense as effort grows to infinity.Both tracks are investigated in this book from the viewpoint of a potential user corresponding to the way of thinking in Popperian science. The final results are new challenging problems, questions for further research. A side question along the way is:How can the user influence the search process given the knowledge of the underlying problem and the information that becomes available during the search?The deterministic approachWhen one starts looking into the deterministic track for a given problem, one runs into the requirements which determine a major difference in applicability of the two approaches.Deterministic methods require the availability of explicit mathematical expressions of the functions to be optimized.In many practical situations which are also discussed in this book, these expressions are not available and deterministic methods cannot be applied. The operations in deterministic methods are based on concepts such as Branch-and-Bound and Cutting which require bounding of functions and parameters based on so-called mathematical structures.In Chapter 2 we describe these structures and distinguish between those which can be derived directly from the expressions, such as quadratic, bilinear and fractional functions and other structures which require analysis of the expressions such as concave and Lipschitz continuous functions. Examples are given of optimization problems revealing their structure. Moreover, we show that symmetry in the model formulation may cause models to have more than one extreme.In Chapter 3 the relationship between GLOP and Integer Programming (IP) is highlighted for several reasons.Sometimes practical GLOP problems can be approximated by IP variants and solved by standard Mixed Integer Linear Programming (MILP) techniques.The algorithms of GLOP and IP can similarly be classified.The transformability of GLOP problems to IP problems and vice versa shows that difficult problems in one class will not become easier to solve in the other.Analysis of problems, which is common in Global Optimization, can be used to better understand the complexity of some IP problems.In Chapter 4 we analyze the use of deterministic methods, demonstrating the application of the Branch-and-Bound concept. The following can be stated from the point of view of the potential user:Analysis of the expressions is required to find useful mathematical structures (Chapter 2). It should be noted that also interval arithmetic techniques can be applied directly on the expressions.The elegance of the techniques is the guarantee that we are certain about the global optimality of the optimum, when it has been discovered and verified.The methods are hard to implement. Thorough use should be made of special data structures to store the necessary information in memory.Two cases are elaborated. The quadratic product design problem illustrates how the level of Decision Support Systems can be reached for low dimensional problems, i.e. the number of variables, components or ingredients, is less than 10. The other case, the nutrient problem, shows how by analysis of the problem many useful properties can be derived which help to cut away large areas of the feasible space where the optimum cannot be situated. However, it also demonstrates the so-called Curse of Dimensionality; the problem has so many variables in a realistic situation that it is impossible to traverse the complete Branch-and-Bound tree. Therefore it is good to see the relativity of the use of deterministic methods:No global optimization method can guarantee to find and verify the global optimum for every practical situation, within a humans lifetime.The stochastic approachThe stochastic approach is followed in practice for many optimization problems by combining the generation of random points with standard nonlinear optimization algorithms. The following can be said from the point of view of the potential user.The methods require no mathematical structure of the problem and are therefore more generally applicable.The methods are relatively easy to implement.The user is never completely certain that the global optimum has been reached.The optimum is approximated in a probabilistic sense when effort increases to infinity.In Chapter 5 much attention is paid to the question what happens when a user wants to spend a limited (not infinite) amount of time to the search for the optimum, preferably less than a humans lifetime:What to do when the time for solving the problem is finite?First we looked at the information which becomes available during the search and the instruments with which the user can influence the search. It appeared that besides classical instruments which are also available in traditional nonlinear programming, the main instrument is to influence the trade-off between global (random) and local search (looking for a local optimum). This lead to a new question:Is there a best way to rule the choice between global and local search, given the information which becomes available?Analyzing in a mathematical way with extreme cases lead to the comfortable conclusion that a best method of choosing between global and local search -thus a best global optimization method- does not exist. This is valid for cases where further information (more than the information which becomes available during the search) on the function to be optimized is not available, called in literature the black-box case. The conclusion again shows that mathematical analysis with extreme cases is a powerful tool to demonstrate that so-called magic algorithms -algorithms which are said in scientific journals to be very promising, because they perform well on some test cases- can be analyzed and 'falsified' in the way of Popperian thinking. This leads to the conclusion that:Magic algorithms which are going to solve all of your problems do not exist.Several side questions derived from the main problem are investigated in this book.In Chapter 6 we place the optimization problem in the context of parameter estimation. One practical question is raised by the phenomenonEvery local search leads to a new local optimum.We know from parameter estimation that this is a symptom in so called non-identifiable systems. The minimum is obtained at a lower dimensional surface or curve. Some (non-magic) heuristics are discussed to overcome this problem.There are two side questions of users derived from the general remark:"I am not interested in the best (GLOP) solution, but in good points".The first question is that of Robust Solutions, introduced in Chapter 4, and the other is called Uniform Covering, concerning the generation of points which are nearly as good as the optimum, discussed in Chapter 6.Robust solutions are discussed in the context of product design. The robustness is defined as a measure of the error one can make from the solution so that the solution (product) is still acceptable. Looking for the most robust product is looking for that point which is as far away as possible from the boundaries of the feasible (acceptable) area. For the solution procedures, we had a look at the appearance of the problem in practice, where boundaries are given by linear and quadratic surfaces, properties of the product.For linear boundaries, finding the most robust solution is an LP problem and thus rather easy.For quadratic properties the development of specific algorithms is required.The question of Uniform Covering concerns the desire to have a set of "suboptimal" points, i.e. points with low function value (given an upper level of the function value); the points are in a so-called level set. To generate "low" points, one could run a local search many times. However, we want the points not to be concentrated in one of the compartments or one sub-area of the level set, we want them to be equally, uniformly spread over the region. This is a very difficult problem for which we test and analyze several approaches in Chapter 6. The analysis taught us that:It is unlikely that stochastic methods will be proposed which solve problems in an expected calculation time, which is polynomial in the number of variables of the problem.Final resultWhether an arbitrary problem of a user can be solved by GLOP requires analysis. There are many optimization problems which can be solved satisfactorily. Besides the selection of algorithms the user has various instruments to steer the process. For stochastic methods it mainly concerns the trade-off between local and global search. For deterministic methods it includes setting bounds and influencing the selection rule in Branch-and-Bound. We hope with this book to have given a tool and a guidance to solution procedures. Moreover, it is an introduction to further literature on the subject of Global Optimization.</p

Global optimisation in process design

This thesis concerns the development of rigorous global optimisation techniques and their application to process engineering problems. Many Process Engineering optimisation problems are nonlinear. Local optimisation approaches may not provide global solutions to these problems if they are nonconvex. The global optimisation approach utilised in this work is based on interval branch and bound algorithms. The interval global optimisation approach is extended to take advantage of information about the structure of the problem and facilitate efficient solution of constrained NLPs using interval analysis. This is achieved by reformulating the interval lower bounding procedure as a convex programming problem which allows inclusion of convex constraints in the lower bounding problem. The approach is applied to a number of standard constrained test problems indicating that this algorithm retains the wide applicability of the interval methods while allowing efficient solution of constrained problems. A new approach to the construction of modular flowsheets is developed. This approach allows construction of flowsheets from linked unit models which enable the application of a number of global optimisation algorithms. The modular flowsheets are constructed with 'generic' unit operations which provide interval bounds, linear bounds, derivatives and derivative bounds using extended numerical types. The genericity means that new 'extended types' can be devised and used without rewriting the unit operations models. The new interval global optimisation algorithm is applied to the generic modular flowsheet. Using interval analysis and automatic differentiation as the arithmetic types, lower bounding linear programs are constructed and used in a branch and bound framework to globally optimise the modular flowsheet