799 research outputs found
A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization
This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures
The Hybridization of Branch and Bound with Metaheuristics for Nonconvex Multiobjective Optimization
A hybrid framework combining the branch and bound method with multiobjective
evolutionary algorithms is proposed for nonconvex multiobjective optimization.
The hybridization exploits the complementary character of the two optimization
strategies. A multiobjective evolutionary algorithm is intended for inducing
tight lower and upper bounds during the branch and bound procedure. Tight
bounds such as the ones derived in this way can reduce the number of
subproblems that have to be solved. The branch and bound method guarantees the
global convergence of the framework and improves the search capability of the
multiobjective evolutionary algorithm. An implementation of the hybrid
framework considering NSGA-II and MOEA/D-DE as multiobjective evolutionary
algorithms is presented. Numerical experiments verify the hybrid algorithms
benefit from synergy of the branch and bound method and multiobjective
evolutionary algorithms
Nonconvex and mixed integer multiobjective optimization with an application to decision uncertainty
Multiobjective optimization problems commonly arise in different fields like economics or engineering. In general, when dealing with several conflicting objective functions, there is an infinite number of optimal solutions which cannot usually be determined analytically.
This thesis presents new branch-and-bound-based approaches for computing the globally optimal solutions of multiobjective optimization problems of various types. New algorithms are proposed for smooth multiobjective nonconvex optimization problems with convex constraints as well as for multiobjective mixed integer convex optimization problems. Both algorithms guarantee a certain accuracy of the computed solutions, and belong to the first deterministic algorithms within their class of optimization problems. Additionally, a new approach to compute a covering of the optimal solution set of multiobjective optimization problems with decision uncertainty is presented. The three new algorithms are tested numerically. The results are evaluated in this thesis as well.
The branch-and-bound based algorithms deal with box partitions and use selection rules, discarding tests and termination criteria. The discarding tests are the most important aspect, as they give criteria whether a box can be discarded as it does not contain any optimal solution. We present discarding tests which combine techniques from global single objective optimization with outer approximation techniques from multiobjective convex optimization and with the concept of local upper bounds from multiobjective combinatorial optimization. The new discarding tests aim to find appropriate lower bounds of subsets of the image set in order to compare them with known upper bounds numerically.Multikriterielle Optimierungprobleme sind in diversen Anwendungsgebieten wie beispielsweise in den Wirtschafts- oder Ingenieurwissenschaften zu finden. Da hierbei mehrere konkurrierende Zielfunktionen auftreten, ist die Lösungsmenge eines derartigen Optimierungsproblems im Allgemeinen unendlich groß und kann meist nicht in analytischer Form berechnet werden.
In dieser Dissertation werden neue Branch-and-Bound basierte Algorithmen zur Lösung verschiedener Klassen von multikriteriellen Optimierungsproblemen entwickelt und vorgestellt. Der Branch-and-Bound Ansatz ist eine typische Methode der globalen Optimierung. Einer der neuen Algorithmen löst glatte multikriterielle nichtkonvexe Optimierungsprobleme mit konvexen Nebenbedingungen, während ein zweiter zur Lösung multikriterieller gemischt-ganzzahliger konvexer Optimierungsprobleme dient. Beide Algorithmen garantieren eine gewisse Genauigkeit der berechneten Lösungen und gehören damit zu den ersten deterministischen Algorithmen ihrer Art. Zusätzlich wird ein Algorithmus zur Berechnung einer Überdeckung der Lösungsmenge multikriterieller Optimierungsprobleme mit Entscheidungsunsicherheit vorgestellt. Alle drei Algorithmen wurden numerisch getestet. Die Ergebnisse werden ebenfalls in dieser Arbeit ausgewertet.
Die neuen Algorithmen arbeiten alle mit Boxunterteilungen und nutzen Auswahlregeln, sowie Verwerfungs- und Terminierungskriterien. Dabei spielen gute Verwerfungskriterien eine zentrale Rolle. Diese entscheiden, ob eine Box verworfen werden kann, da diese sicher keine Optimallösung enthält. Die neuen Verwerfungskriterien nutzen Methoden aus der globalen skalarwertigen Optimierung, Approximationstechniken aus der multikriteriellen konvexen Optimierung sowie ein Konzept aus der kombinatorischen Optimierung. Dabei werden stets untere Schranken der Bildmengen konstruiert, die mit bisher berechneten oberen Schranken numerisch verglichen werden können
A general branch-and-bound framework for continuous global multiobjective optimization
Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples
Differential Evolution for Multiobjective Portfolio Optimization
Financial portfolio optimization is a challenging problem. First, the problem is multiobjective (i.e.: minimize risk and maximize profit) and the objective functions are often multimodal and non smooth (e.g.: value at risk). Second, managers have often to face real-world constraints, which are typically non-linear. Hence, conventional optimization techniques, such as quadratic programming, cannot be used. Stochastic search heuristic can be an attractive alternative. In this paper, we propose a new multiobjective algorithm for portfolio optimization: DEMPO - Differential Evolution for Multiobjective Portfolio Optimization. The main advantage of this new algorithm is its generality, i.e., the ability to tackle a portfolio optimization task as it is, without simplifications. Our empirical results show the capability of our approach of obtaining highly accurate results in very reasonable runtime, in comparison with quadratic programming and another state-of-art search heuristic, the so-called NSGA II.Portfolio Optimization, Multiobjective, Real-world Constraints, Value at Risk, Expected Shortfall, Differential Evolution
Multiobjective optimization to a TB-HIV/AIDS coinfection optimal control problem
We consider a recent coinfection model for Tuberculosis (TB), Human
Immunodeficiency Virus (HIV) infection and Acquired Immunodeficiency Syndrome
(AIDS) proposed in [Discrete Contin. Dyn. Syst. 35 (2015), no. 9, 4639--4663].
We introduce and analyze a multiobjective formulation of an optimal control
problem, where the two conflicting objectives are: minimization of the number
of HIV infected individuals with AIDS clinical symptoms and coinfected with
AIDS and active TB; and costs related to prevention and treatment of HIV and/or
TB measures. The proposed approach eliminates some limitations of previous
works. The results of the numerical study provide comprehensive insights about
the optimal treatment policies and the population dynamics resulting from their
implementation. Some nonintuitive conclusions are drawn. Overall, the
simulation results demonstrate the usefulness and validity of the proposed
approach.Comment: This is a preprint of a paper whose final and definite form is with
'Computational and Applied Mathematics', ISSN 0101-8205 (print), ISSN
1807-0302 (electronic). Submitted 04-Feb-2016; revised 11-June-2016 and
02-Sept-2016; accepted for publication 15-March-201
An approximation algorithm for multi-objective optimization problems using a box-coverage
For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin
Obtaining properly Pareto optimal solutions of multiobjective optimization problems via the branch and bound method
In multiobjective optimization, most branch and bound algorithms provide the
decision maker with the whole Pareto front, and then decision maker could
select a single solution finally. However, if the number of objectives is
large, the number of candidate solutions may be also large, and it may be
difficult for the decision maker to select the most interesting solution. As we
argue in this paper, the most interesting solutions are the ones whose
trade-offs are bounded. These solutions are usually known as the properly
Pareto optimal solutions. We propose a branch-and-bound-based algorithm to
provide the decision maker with so-called -properly Pareto optimal
solutions. The discarding test of the algorithm adopts a dominance relation
induced by a convex polyhedral cone instead of the common used Pareto dominance
relation. In this way, the proposed algorithm excludes the subboxes which do
not contain -properly Pareto optimal solution from further
exploration. We establish the global convergence results of the proposed
algorithm. Finally, the algorithm is applied to benchmark problems as well as
to two real-world optimization problems
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