A decision space algorithm for multiobjective convex quadratic integer optimization. - In: Computers & operations research : an international journal.. - Amsterdam [u.a.] : Elsevier, ISSN 0305-0548, Bd. 134 (2021)
We present a branch-and-bound algorithm for minimizing multiple convex quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm stops after finitely many fixings of variables with detecting both the full efficient and the nondominated set of multiobjective strictly convex quadratic integer problems. A major advantage of the approach is that the expensive calculations are done in a preprocessing phase so that the nodes in the branch-and-bound tree can be enumerated fast. We show numerical experiments on biobjective instances and on instances with three and four objectives.
A general branch-and-bound framework for continuous global multiobjective optimization. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, Bd. 80 (2021), 1, S. 195-227
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.
Optimality conditions in discrete-continuous nonlinear optimization. - In: Minimax theory and its applications. - [Lemgo] : Heldermann Verlag, ISSN 2199-1413, Bd. 6 (2021), 1, S. 127-144
Proximity measures based on KKT points for constrained multi-objective optimization. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, Bd. 80 (2021), 1, S. 63-86
An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush-Kuhn-Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems.
Nonconvex constrained optimization by a filtering branch and bound. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, Bd. 80 (2021), 1, S. 31-61
A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the [alpha]BB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained optimization problem. Moreover, the multiobjective reformulation enables to identify the trade-off between constraint satisfaction and objective value which is also reflected in the quality guarantee. Numerical tests validate that we indeed can find feasible and often optimal solutions where the classical single-objective [alpha]BB method fails, i.e., it terminates without ever finding a feasible solution.
Coordinate transformation and its uncertainty under consideration of a non-orthogonal coordinate base. - In: Measurement science and technology : devoted to the theory, practice and application of measurement in physics, chemistry, engineering and the environmental and life sciences from inception to commercial exploitation.. - Bristol : IOP Publ., ISSN 1361-6501, Volume 32 (2021), number 4, 045001, Seite 1-6
Nanopositioning and nanomeasuring machines are 3D coordinate measuring systems with nanometer precision at measurement volumes in the cubic centimeter range. The coordinate base is formed by an interferometer system with a common mirror corner. The orthogonality deviations of the mirror corner require a coordinate transformation of the measuring axes. The uncertainty of the coordinate transformation must be taken into account in the overall measurement uncertainty budget. Starting from a complete transformation model, the result of model simplications on the transformation behaviour is analysed and discussed.
Expensive multi-objective optimization of electromagnetic mixing in a liquid metal. - In: Optimization and engineering. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2924, (2020), insges. 25 S.
- Published: 07 October 2020
This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet located underneath the rectangular vessel. The rectangular magnet provides a spatially non-uniform magnetic field that is known analytically. The magnet oscillation creates a steady mean flow (steady streaming) similar to those observed from oscillating rigid bodies. In the optimization problem, randomly distributed mass-less particles are advected by the flow to achieve a homogeneous distribution (objective function 1) while keeping the work done to move the permanent magnet minimal (objective function 2). A single evaluation of these two objective functions may take more than two hours. For that reason, to save computational time, the proposed method uses interpolation models on trust-regions for finding descent directions. We show that, even for our significantly simplified model problem, the mixing patterns vary significantly with the control parameters, which justifies the use of improved optimization techniques and their further development.
Solving multiobjective mixed integer convex optimization problems. - In: SIAM journal on optimization. - Philadelphia, Pa. : SIAM, ISSN 1095-7189, Bd. 30 (2020), 4, S. 3122-3145
Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we build linear outer approximations of the image set in an adaptive way. We are able to guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. As far as we know, the procedure we present is the first non-scalarization-based deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances.
Methods for multiobjective bilevel optimization. - In: Bilevel optimization. - Cham, Switzerland : Springer, (2020), S. 423-449
This chapter is on multiobjective bilevel optimization, i.e. on bilevel optimization problems with multiple objectives on the lower or on the upper level, or even on both levels. We give an overview on the major optimality notions used in multiobjective optimization. We provide characterization results for the set of optimal solutions of multiobjective optimization problems by means of scalarization functionals and optimality conditions. These can be used in theoretical and numerical approaches to multiobjective bilevel optimization.As multiple objectives arise in multiobjective optimization as well as in bilevel optimization problems, we also point out the results on the connection between these two classes of optimization problems. Finally, we give reference to numerical approaches which have been followed in the literature to solve these kind of problems. We concentrate in this chapter on nonlinear problems, while the results and statements naturally also hold for the linear case.
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. - In: Optimization. - London [u.a.] : Taylor & Francis, ISSN 1029-4945, (2020), insges. 42 S.
- Published online: 14 Sep 2020
Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set-valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference for the evaluation of the numerator in the difference quotient. We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as anembedding of convex sets in this space using Steiner points.In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts.