A meshfree method for a PDE-constrained optimization problem. - In: SIAM journal on numerical analysis, ISSN 1095-7170, Bd. 59 (2021), 4, S. 1896-1917
We describe a new approximation method for solving a PDE-constrained optimization problem numerically. Our method is based on the adjoint formulation of the optimization problem, leading to a system of weakly coupled, elliptic PDEs. These equations are then solved using kernel-based collocation. We derive an error analysis and give numerical examples.
https://doi.org/10.1137/20M1363510
A steepest descent method for set optimization problems with set-valued mappings of finite cardinality. - In: Journal of optimization theory and applications, ISSN 1573-2878, Bd. 190 (2021), 3, S. 711-743
In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).
https://doi.org/10.1007/s10957-021-01887-y
A decision space algorithm for multiobjective convex quadratic integer optimization. - In: Computers & operations research, ISSN 0305-0548, Bd. 134 (2021), 105396, S. 1-13
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.
https://doi.org/10.1016/j.cor.2021.105396
A general branch-and-bound framework for continuous global multiobjective optimization. - In: Journal of global optimization, 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.
https://doi.org/10.1007/s10898-020-00984-y
Optimality conditions in discrete-continuous nonlinear optimization. - In: Minimax theory and its applications, 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, 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.
https://doi.org/10.1007/s10898-020-00971-3
Nonconvex constrained optimization by a filtering branch and bound. - In: Journal of global optimization, 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.
https://doi.org/10.1007/s10898-020-00956-2
Expensive multi-objective optimization of electromagnetic mixing in a liquid metal. - In: Optimization and engineering, ISSN 1573-2924, Bd. 22 (2021), 2, S. 1065-1089
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.
https://doi.org/10.1007/s11081-020-09561-4
Coordinate transformation and its uncertainty under consideration of a non-orthogonal coordinate base. - In: Measurement science and technology, ISSN 1361-6501, Bd. 32 (2021), 4, 045001, insges. 6 S.
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.
https://doi.org/10.1088/1361-6501/aba3f5
Solving multiobjective mixed integer convex optimization problems. - In: SIAM journal on optimization, 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.
https://doi.org/10.1137/19M1264709