Certificates of infeasibility via nonsmooth optimization

An important aspect in the solution process of constraint satisfaction problems is to identify exclusion boxes which are boxes that do not contain feasible points. This paper presents a certificate of infeasibility for finding such boxes by solving a linearly constrained nonsmooth optimization problem. Furthermore, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem.Comment: arXiv admin note: substantial text overlap with arXiv:1506.0802

Verified global optimization for estimating the parameters of nonlinear models

Nonlinear parameter estimation is usually achieved via the minimization of some possibly non-convex cost function. Interval analysis allows one to derive algorithms for the guaranteed characterization of the set of all global minimizers of such a cost function when an explicit expression for the output of the model is available or when this output is obtained via the numerical solution of a set of ordinary differential equations. However, cost functions involved in parameter estimation are usually challenging for interval techniques, if only because of multi-occurrences of the parameters in the formal expression of the cost. This paper addresses parameter estimation via the verified global optimization of quadratic cost functions. It introduces tools for the minimization of generic cost functions. When an explicit expression of the output of the parametric model is available, significant improvements may be obtained by a new box exclusion test and by careful manipulations of the quadratic cost function. When the model is described by ODEs, some of the techniques available in the previous case may still be employed, provided that sensitivity functions of the model output with respect to the parameters are available

Intertemporal and Spatial Location of Disposal Facilities

Optimal capacity and location of a sequence of land.lls are studied, and the interactions between both decisions are pointed out.The decision capacity has some spatial implications, because it a.ects the feasible region for the rest of land.lls, and some temporal implications, because the capacity determines the lifetime of the land.ll and hence the instant of time where next land.lls will need to be constructed.Some general mathematical properties of the solution are provided and interpreted from an economic point of view.The resulting problem turns out to be no convex and therefore it can not be solved by conventional optimization techniques.Some global optimization methods are used to solve the problem in a particular case, in order to illustrate the behavior of the solution depending on parameter values.Landfilling;Optimal Capacity;Optimal Location;Global Optimization

Branch-and-Prune Search Strategies for Numerical Constraint Solving

When solving numerical constraints such as nonlinear equations and inequalities, solvers often exploit pruning techniques, which remove redundant value combinations from the domains of variables, at pruning steps. To find the complete solution set, most of these solvers alternate the pruning steps with branching steps, which split each problem into subproblems. This forms the so-called branch-and-prune framework, well known among the approaches for solving numerical constraints. The basic branch-and-prune search strategy that uses domain bisections in place of the branching steps is called the bisection search. In general, the bisection search works well in case (i) the solutions are isolated, but it can be improved further in case (ii) there are continuums of solutions (this often occurs when inequalities are involved). In this paper, we propose a new branch-and-prune search strategy along with several variants, which not only allow yielding better branching decisions in the latter case, but also work as well as the bisection search does in the former case. These new search algorithms enable us to employ various pruning techniques in the construction of inner and outer approximations of the solution set. Our experiments show that these algorithms speed up the solving process often by one order of magnitude or more when solving problems with continuums of solutions, while keeping the same performance as the bisection search when the solutions are isolated.Comment: 43 pages, 11 figure

Intertemporal and Spacial Location of Disposal Facilities

Optimal capacity and location of a sequence of landfills are studied, and the interactions between both decisions are pointed out. The decision capacity has some spatial implications, because it affects the feasible region for the rest of landfills, and some temporal implications, because the capacity determines the lifetime of the landfill and hence the instant of time where next landfills will need to be constructed. Some general mathematical properties of the solution are provided and interpreted from an economic point of view. The resulting problem turns out to be no convex and therefore it can not be solved by conventional optimization techniques. Some global optimization methods are used to solve the problem in a particular case, in order to illustrate the behavior of the solution depending on parameter value

Intertemporal and Spatial Location of Disposal Facilities

Optimal capacity and location of a sequence of land.lls are studied, and the interactions between both decisions are pointed out.The decision capacity has some spatial implications, because it a.ects the feasible region for the rest of land.lls, and some temporal implications, because the capacity determines the lifetime of the land.ll and hence the instant of time where next land.lls will need to be constructed.Some general mathematical properties of the solution are provided and interpreted from an economic point of view.The resulting problem turns out to be no convex and therefore it can not be solved by conventional optimization techniques.Some global optimization methods are used to solve the problem in a particular case, in order to illustrate the behavior of the solution depending on parameter values.

Rigorous techniques for continuous constraint satisfaction problems

Diese Arbeit beschäftigt sich mit rigorosen Techniken für das Lösen kontinuierlicher Zulässigkeitsprobleme. Das heißt, wir suchen nach einem oder allen Punkte, genannt zulässige Punkte, die eine Familie von Gleichungen und/oder Ungleichungen erfüllen, die wir im Weiteren Nebenbedingungen nennen werden. Zahlreiche Anwendungen führen auf kontinuierliche Zulässigkeitsprobleme. Neue und bereits existierende moderne Methoden werden präsentiert und integriert in GloptLab, eine neue, leicht bedienbare Test- und Entwicklungsplattform zum Lösen quadratischer Zulässigkeitsprobleme. Der Lösungsalgorithmus beruht auf dem Grundprinzip von Branch-and-Prune und auf Filterung. Filterungsmethoden dienen zur Verkleinerung/Reduktion einer Box, definiert als das kartesische Produkt der Intervalle, die die Schranken an die Variablen festlegen. Um den Verlust zulässiger Punkte zu vermeiden, werden alle Fehlerabschätzungen rigoros mittels Intervallarithmetik und gerichteter Rundung durchgeführt. Das stellt sicher, dass alle Rechnungen auch in Gleitkommaarithmetik gültig sind. In der Doktorarbeit werden die folgenden Themen diskutiert: der mathematische Hintergrund, Algorithmen und Tests für Constraint-Propagation, strikt konvexe Einschließungen, lineare Relaxationen, das Berechnen, korrekte Benutzen und Verifizieren approximativ zulässiger Punkte, optimale Skalierung und diverse Hilfsmethoden. Insbesondere: - Constraint-Propagation basiert auf einer Folge von Schritten, die jeweils eine einzelne Nebenbedingung verwenden. Traditionelle Techniken werden durch eine spezielle quadratische Methode erweitert, die neue Verfahren für die Eliminierung bilinearer Einträge und für das Berechnen optimaler Einschließungen für separable quadratische Ausdrücke verwendet. - Eine quadratische Ungleichungsnebenbedingung, die eine positiv definite Hesse-Matrix besitzt, definiert ein Ellipsoid. Eine spezielle rundungsfehlerkontrollierte Version der Cholesky-Zerlegung wird verwendet, um die strikt konvexe quadratische Nebenbedingungen in Norm-Ungleichungen zu transformieren. Für diese ist es dann einfach, die Intervall-Hülle analytisch zu bestimmen. - Diverse Methoden für die Erzeugung linearer Relaxationen werden diskutiert, kombiniert und erweitert. Teilweise verbesserte, existierende und neue Verfahren für das rigorose Einschließen der Lösungsmenge linearer Systeme werden präsentiert. - Eine Vielzahl von Beispielen demonstrieren, dass die präsentierten Verfahren einander ergänzen. Außerdem zeigen sie, wie man Lösungsstrategien entwickelt, die Zulässigkeitsprobleme global und effizient lösen.This thesis contributes rigorous techniques for solving continuous constraint satisfaction problems, i.e., finding one or all points (called feasible points) satisfying a given family of equations and/or inequalities (called constraints). Many real word problems are continuous constraint satisfaction problems. New and old state of the art methods are presented, integrated in GloptLab, a new easy-to-use testing and development platform for solving quadratic constraint satisfaction problems. The basic solution principle is branch and prune and filtering. Filtering techniques tighten a box -- the Cartesian product of intervals defined by the bounds on the variables. In order to avoid a loss of feasible points, rigorous error estimation using interval arithmetic and directed rounding are used, to take care that all calculations are valid even though the calculations are performed in floating-point arithmetic only. Discussed are the mathematical background, algorithms and tests of constraint propagation, strictly convex enclosures, linear relaxations, finding, using and verifying approximately feasible points, optimal scaling and other auxiliary techniques. In particular: - Constraint propagation is based on a sequence of steps, each using a single constraint only. Traditional techniques are extended by special quadratic constraint propagation methods using new techniques for eliminating bilinear entries and finding optimal enclosures for separable quadratic expressions. - A quadratic inequality constraint with a positive definite Hessian defines an ellipsoid. A rounding error controlled version of the Cholesky factorization is used to transform a strictly convex quadratic constraint into a norm inequality, for which the interval hull is easy to compute analytically. - Different methods for computing linear relaxations are discussed, combined and extended. Partially improved and existing methods, as well as new approaches for rigorously enclosing the solution set of linear systems of inequalities are presented. - Numerous examples show that the above methods complement each other and how to create solution strategies that solve constraint satisfaction problems globally and efficiently

A Software Platform for Global Optimization, Journal of Telecommunications and Information Technology, 2010, nr 3

This paper addresses issues associated with the global optimization algorithms, which are methods to find optimal solutions for given problems. It focuses on an integrated software environment – global optimization object-oriented library (GOOL), which provides the graphical user interface together with the library of solvers for convex and nonconvex, unconstrained and constrained problems. We describe the design, performance and possible applications of the GOOL system. The practical example – price management problem – is provided to illustrate the effectiveness and range of applications of our software tool