János D. Pintér

Solving circle packing problems by global optimization: Numerical results and industrial applications

A (general) circle packing is an optimized arrangement of N arbitrary sized circles inside a container (e.g., a rectangle or a circle) such that no two circles overlap. In this paper, we present several circle packing problems, review their industrial applications, and some exact and heuristic strategies for their solution. We also present illustrative numerical results using generic global optimization software packages. Our work highlights the relevance of global optimization in solving circle packing problems, and points towards the necessary advancements in both theory and numerical practice.

Nonlinear optimization with GAMS /LGO

The Lipschitz Global Optimizer (LGO) software integrates global and local scope search methods, to handle a very general class of nonlinear optimization models. Here we discuss the LGO implementation linked to the General Algebraic Modeling System (GAMS). First we review the key features and basic usage of the GAMS /LGO solver option, then present reproducible numerical results to illustrate its performance.

The impact of accelerating tools on the interval subdivision algorithm for global optimization

Abstract The interval subdivision method of Moore and Skelboe is considered for global optimization. The new implementation allows the use of a long list in the procedure, which is limited only by the available computer memory. Standard global optimization test problems are used to measure the efficiency of different versions (with or without the monotonicity and the cut-off tests) of the algorithm. The new implementation and the inclusion of the monotonicity test made it possible, for the first time, for all standard test problems to be solved completely by an interval method. Moreover, in some problems the efficiency of the algorithm was better than that of the best traditional techniques, which do not give guaranteed reliability results.

Global Optimization Toolbox for Maple: an introduction with illustrative applications

This article presents a concise review of the scientific–technical computing system Maple and its application potentials in Operations Research. The primary emphasis is placed on non-linear optimization models that may be formulated using a broad range of components, including e.g. special functions, integrals, systems of differential equations, deterministic or stochastic simulation, external function calls, and other computational procedures. Such models may have a complicated structure, with possibly a non-convex feasible set and multiple, global and local, optima. We introduce the Global Optimization Toolbox to solve such models in Maple, and illustrate its usage by numerical examples.

Automatic model calibration applying global optimization techniques

This paper summarizes a study aimed at the application of global optimization techniques for the purpose of quantitative characterization of the Wolfville formation located in Nova Scotia, Canada. Aquifer parameters (transmissivity, storativity, areal recharge and boundary flux) are calibrated in order to yield the best possible match with the available field observations. The calibration is accomplished using a global approach to the inverse procedure in “black box” systems optimization which makes possible the simultaneous fitting of several tens of parameters. This study indicates that, even for a limited number of optimized parameters, a global search procedure should be considered. Numerical results are presented and discussed to show the validity of the approach.

Set partition by globally optimized cluster seed points

Abstract A new method is proposed for solving set partition (clustering) problems: the suggested approach is based on the application of global optimization methodology for selecting ‘best’ cluster seed points. The applicability of the method is illustrated by numerical examples.

Globally optimized calibration of nonlinear models: Techniques, software, and applications

The search for a better understanding of complex systems calls for quantitative model development. Within this process, model fitting to observation data often plays an important role. Traditionally, standard (convex) nonlinear optimization techniques have been applied to solve model calibration problems: the limitations of such approaches - due to their local search scope - are well known. By contrast, global optimization (GO) strategies are able to find (i.e., to numerically approximate) the absolutely best model parameterization. This point is illustrated by numerical examples and case studies which have been analyzed and solved by our various software implementations.

Groundwater remediation design using physics-based flow, transport, and optimization technologies

BackgroundThe purpose of this work was to demonstrate an approach to groundwater remedial design that is automated, cost-effective, and broadly applicable to contaminated aquifers in different geologic settings. The approach integrates modeling and optimization for use as a decision support framework for the optimal design of groundwater remediation systems employing pump and treat and re-injection technologies. The technology resulting from the implementation of the methodology, which we call Physics-Based Management Optimization (PBMO), integrates physics-based groundwater flow and transport models, management science, and nonlinear optimization tools to provide stakeholders with practical, optimized well placement locations and flow rates for remediating contaminated groundwater at complex sites.ResultsThe algorithm implementation, verification, and effectiveness testing was conducted using groundwater conditions at the Umatilla Chemical Depot in Umatilla, Oregon, as a case study. This site was the subject of a government-sponsored remedial optimization study. Our methodology identified the optimal solution 40 times faster than other methods, did not fail to perform when the physics-based models failed to converge, and did not require human intervention during the solution search, in contrast to the other methods. The integration of the PBMO and Lipschitz Global Optimization (LGO) methods with standalone physically based models provides an approach that is applicable to a wide range of hydrogeological flow and transport settings.ConclusionsThe global optimization based solutions obtained from this study were similar to those found by others, providing method verification. Automation of the optimal search strategy combined with the reliability to overcome inherent difficulties of non-convergence when using physics models in optimization promotes its usefulness. The application of our methodology to the Umatilla case study site represents a rigorous testing of our optimization methodology for handling groundwater remediation problems.

A new interval method for locating the boundary of level sets

An interval method for sensitivity analysis is presented that locates the boundary of level sets. Convergence conditions and the rate of convergence of the procedure are studied. The algorithm is applied to a real life parameter estimation problem, and its results are compared with those obtained with a traditional method based on the approximation of the Hessian matrix.

Ibrahim H. Osman and James P. Kelly (eds.),Meta-Heuristics: Theory and Applications.

This book is based on a reviewed collection of papers presented at the MetaHeuristics International Conference (Breckenridge, Colorado, USA, 22–26 July 1995). Meta-heuristics – as the name suggests – is a general search principle or algorithm framework that can be applied in a context-dependent manner to find good approximate solutions to (complicated) optimization problems. In this sense, the concept has specific relevance with respect to mathematical programming models which are notoriously difficult to solve: combinatorial optimization, (continuous or mixed integer) global optimization, stochastic programming, and other areas provide an abundance of examples. The book discusses several types of meta-heuristics that have found a variety of applications in the area of combinatorial optimization. Similar to many global optimization problems in character, discrete programming models are often very simple to state, but may prove extremely difficult (NP-hard) to solve. This fact directly calls for good heuristic approaches, for model-sizes arising in practice. In Chapter 1, Osman and Kelly – the editors of the volume – provide a concise overview of the currently most popular ideas in meta-heuristics. These include the following broad classes of approaches (that can be tailored to specific problems, and may also direct the operation of respective subordinate algorithms):