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Results: 159
Created on: Sat, 28 Jan 2023 23:09:36 +0100 in 0.1022 sec

Eichfelder, Gabriele; Gerlach, Tobias; Rocktäschel, Stefan
Convexity and continuity of specific set-valued maps and their extremal value functions. - In: Journal of applied and numerical optimization, ISSN 2562-5535, Bd. 5 (2023), 1, S. 71-92

In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals over the image sets of the set-valued maps. Such extremal value functions play an important role for instance for derivative concepts for set-valued maps or for algorithmic approaches in set-valued optimization. We formulate conditions under which the set-valued maps and their extremal value functions inherit properties like (Lipschitz-)continuity and convexity.
Eichfelder, Gabriele; Stein, Oliver
Limit sets in global multiobjective optimization. - In: Optimization, ISSN 1029-4945, Bd. 0 (2022), 0, S. 1-27

Inspired by the recently introduced branch-and-bound method for continuous multiobjective optimization problems from G. Eichfelder, P. Kirst, L. Meng, O. Stein [A general branch-and-bound framework for continuous global multiobjective optimization. J Glob Optim. 2021;80:195-227], we study for a general class of branch-and-bound methods in which sense the generated terminal enclosure and the terminal provisional nondominated set approximate the nondominated set when the termination accuracy is driven to zero. Our convergence analysis of the enclosures relies on constructions from the above paper, but is self-contained and also covers the mixed-integer case. The analysis for the provisional nondominated set is based on general convergence properties of the epsilon-nondominated set, and hence it is also applicable to other algorithms which generate such points. Furthermore, we discuss post-processing steps for the terminal enclosure and provide numerical illustrations for the cases of two and three objective functions.
Eichfelder, Gabriele; Grüne, Lars; Krügel, Lisa; Schießl, Jonas
Relaxed dissipativity assumptions and a simplified algorithm for multiobjective MPC. - In: Computational optimization and applications, ISSN 1573-2894, (2022), insges. 36 S.

We consider nonlinear model predictive control (MPC) with multiple competing cost functions. In each step of the scheme, a multiobjective optimal control problem with a nonlinear system and terminal conditions is solved. We propose an algorithm and give performance guarantees for the resulting MPC closed loop system. Thereby, we significantly simplify the assumptions made in the literature so far by assuming strict dissipativity and the existence of a compatible terminal cost for one of the competing objective functions only. We give conditions which ensure asymptotic stability of the closed loop and, what is more, obtain performance estimates for all cost criteria. Numerical simulations on various instances illustrate our findings. The proposed algorithm requires the selection of an efficient solution in each iteration, thus we examine several selection rules and their impact on the results. and we also examine numerically how different selection rules impact the results
Eichfelder, Gabriele; Quintana, Ernest; Rocktäschel, Stefan
A vectorization scheme for nonconvex set optimization problems. - In: SIAM journal on optimization, ISSN 1095-7189, Bd. 32 (2022), 2, S. 1184-1209

In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective optimization. Our strategy consists of deriving a parametric family of multiobjective optimization problems whose optimal solution sets approximate, in a specific sense, that of the set-valued problem with arbitrary accuracy. We also examine particular classes of set-valued mappings for which the corresponding set optimization problem is equivalent to a multiobjective optimization problem in the generated family. Surprisingly, this includes set-valued mappings with a convex graph.
De Santis, Marianna; Eichfelder, Gabriele; Patria, Daniele
On the exactness of the ε-constraint method for biobjective nonlinear integer programming. - In: Operations research letters, ISSN 0167-6377, Bd. 50 (2022), 3, S. 356-361

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.
Eichfelder, Gabriele; Groetzner, Patrick
A note on completely positive relaxations of quadratic problems in a multiobjective framework. - In: Journal of global optimization, ISSN 1573-2916, Bd. 82 (2022), 3, S. 615-626

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.
Eichfelder, Gabriele; Warnow, Leo
An approximation algorithm for multi-objective optimization problems using a box-coverage. - In: Journal of global optimization, ISSN 1573-2916, Bd. 83 (2022), 2, S. 329-357

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.
Baier, Robert; Eichfelder, Gabriele; Gerlach, Tobias
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. - In: Optimization, ISSN 1029-4945, Bd. 71 (2022), 8, S. 2273-2314

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.
Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane
On Clarke's subdifferential of marginal functions. - In: Applied set-valued analysis and optimization, ISSN 2562-7783, Bd. 3 (2021), 3, S. 281-292

In this paper, we derive an upper estimate of the Clarke subdifferential of marginal functions in Banach spaces. The structure of the upper estimate is very similar to other results already obtained in the literature. The novelty lies on the fact that we derive our assertions in general Banach spaces, and avoid the use of the Asplund assumption.
Eichfelder, Gabriele;
Twenty years of continuous multiobjective optimization in the twenty-first century. - In: EURO journal on computational optimization, ISSN 2192-4414, Bd. 9 (2021), 100014, insges. 15 S.

The survey highlights some of the research topics which have attracted attention in the last two decades within the area of mathematical optimization of multiple objective functions. We give insights into topics where a huge progress can be seen within the last years. We give short introductions to the specific sub-fields as well as some selected references for further reading. Primarily, the survey covers the progress in the development of algorithms. In particular, we discuss publicly available solvers and approaches for new problem classes such as non-convex and mixed integer problems. Moreover, bilevel optimization problems and the handling of uncertainties by robust approaches and their relation to set optimization are presented. In addition, we discuss why numerical approaches which do not use scalarization techniques are of interest.