1,282 research outputs found
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
Robust optimality and duality for composite uncertain multiobjective optimization in Asplund spaces with its applications
This article is devoted to investigate a nonsmooth/nonconvex uncertain
multiobjective optimization problem with composition fields
((\hyperlink{CUP}{\mathrm{CUP}}) for brevity) over arbitrary Asplund spaces.
Employing some advanced techniques of variational analysis and generalized
differentiation, we establish necessary optimality conditions for weakly robust
efficient solutions of (\hyperlink{CUP}{\mathrm{CUP}}) in terms of the
limiting subdifferential. Sufficient conditions for the existence of (weakly)
robust efficient solutions to such a problem are also driven under the new
concept of pseudo-quasi convexity for composite functions. We formulate a
Mond-Weir-type robust dual problem to the primal problem
(\hyperlink{CUP}{\mathrm{CUP}}), and explore weak, strong, and converse
duality properties. In addition, the obtained results are applied to an
approximate uncertain multiobjective problem and a composite uncertain
multiobjective problem with linear operators.Comment: arXiv admin note: substantial text overlap with arXiv:2105.14366,
arXiv:2205.0114
Set-based Robust Optimization of Uncertain Multiobjective Problems via Epigraphical Reformulations
In this paper, we study a method for finding robust solutions to
multiobjective optimization problems under uncertainty. We follow the set-based
minmax approach for handling the uncertainties which leads to a certain set
optimization problem with the strict upper type set relation. We introduce,
under some assumptions, a reformulation using instead the strict lower type set
relation without sacrificing the compactness property of the image sets. This
allows to apply vectorization results to characterize the optimal solutions of
these set optimization problems as optimal solutions of a multiobjective
optimization problem. We end up with multiobjective semi-infinite problems
which can then be studied with classical techniques from the literature
Qualitative Characteristics and Quantitative Measures of Solution's Reliability in Discrete Optimization: Traditional Analytical Approaches, Innovative Computational Methods and Applicability
The purpose of this thesis is twofold. The first and major part is devoted to
sensitivity analysis of various discrete optimization problems while the second
part addresses methods applied for calculating measures of solution stability
and solving multicriteria discrete optimization problems.
Despite numerous approaches to stability analysis of discrete optimization
problems two major directions can be single out: quantitative and qualitative.
Qualitative sensitivity analysis is conducted for multicriteria discrete optimization
problems with minisum, minimax and minimin partial criteria. The main
results obtained here are necessary and sufficient conditions for different stability
types of optimal solutions (or a set of optimal solutions) of the considered
problems.
Within the framework of quantitative direction various measures of solution
stability are investigated. A formula for a quantitative characteristic called
stability radius is obtained for the generalized equilibrium situation invariant
to changes of game parameters in the case of the H¨older metric. Quality of the
problem solution can also be described in terms of robustness analysis. In this
work the concepts of accuracy and robustness tolerances are presented for a
strategic game with a finite number of players where initial coefficients (costs)
of linear payoff functions are subject to perturbations.
Investigation of stability radius also aims to devise methods for its calculation.
A new metaheuristic approach is derived for calculation of stability
radius of an optimal solution to the shortest path problem. The main advantage
of the developed method is that it can be potentially applicable for
calculating stability radii of NP-hard problems.
The last chapter of the thesis focuses on deriving innovative methods based
on interactive optimization approach for solving multicriteria combinatorial
optimization problems. The key idea of the proposed approach is to utilize
a parameterized achievement scalarizing function for solution calculation and
to direct interactive procedure by changing weighting coefficients of this function.
In order to illustrate the introduced ideas a decision making process is
simulated for three objective median location problem.
The concepts, models, and ideas collected and analyzed in this thesis create
a good and relevant grounds for developing more complicated and integrated
models of postoptimal analysis and solving the most computationally challenging
problems related to it.Siirretty Doriast
A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions
In a general Hilbert framework, we consider continuous gradient-like
dynamical systems for constrained multiobjective optimization involving
non-smooth convex objective functions. Our approach is in the line of a
previous work where was considered the case of convex di erentiable objective
functions. Based on the Yosida regularization of the subdi erential operators
involved in the system, we obtain the existence of strong global trajectories.
We prove a descent property for each objective function, and the convergence of
trajectories to weak Pareto minima. This approach provides a dynamical
endogenous weighting of the objective functions. Applications are given to
cooperative games, inverse problems, and numerical multiobjective optimization
An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces
This paper explores new notions of approximate minimality in set optimization using a set approach. We propose characterizations of several approximate minimal elements of families of sets in real linear spaces by means of general functionals, which can be unified in an inequality approach. As particular cases, we investigate the use of the prominent Tammer–Weidner nonlinear scalarizing functionals, without assuming any topology, in our context. We also derive numerical methods to obtain approximate minimal elements of families of finitely many sets by means of our obtained results
Optimization and Equilibrium Problems with Equilibrium Constraints in Infinite-Dimensional Spaces
The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding sequential normal compactness (SNC) properties
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