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- Generalized differentiation (4)
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- Generalized differentiation. (1)
- Image reconstruction (1)
- Mathematical programs with equilibrium constraints (1)
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- Mixed boundary controls (1)
- Multi-channel model (1)
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Articles 1 - 10 of 10
Full-Text Articles in Mathematics
Relative Pareto Minimizers To Multiobjective Problems: Existence And Optimality Conditions, Truong Q. Bao, Boris S. Mordukhovich
Relative Pareto Minimizers To Multiobjective Problems: Existence And Optimality Conditions, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers to general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient …
Image Reconstruction In Multi-Channel Model Under Gaussian Noise, Veera Holdai, Alexander Korostelev
Image Reconstruction In Multi-Channel Model Under Gaussian Noise, Veera Holdai, Alexander Korostelev
Mathematics Research Reports
The image reconstruction from noisy data is studied. A nonparametric boundary function is estimated from observations in N independent channels in Gaussian white noise. In each channel the image and the background intensities are unknown. They define a non-identifiable nuisance "parameter" that slows down the typical minimax rate of convergence. The large sample asymptotics of the minimax risk is found and an asymptotically optimal estimator for boundary function is suggested.
Necessary Conditions For Super Minimizers In Constrained Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Necessary Conditions For Super Minimizers In Constrained Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
This paper concerns the study of the so-called super minimizers related to the concept of super efficiency in constrained problems of multiobjective optimization, where cost mappings are generally set-valued. We derive necessary conditions for super minimizers on the base of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.
Optimization And Feedback Design Of State-Constrained Parabolic Systems, Boris S. Mordukhovich
Optimization And Feedback Design Of State-Constrained Parabolic Systems, Boris S. Mordukhovich
Mathematics Research Reports
The paper is devoted to optimal control and feedback design of stateconstrained parabolic systems in uncertainty conditions. Problems of this type are among the most challenging and difficult in dynamic optimization for any kind of dynamical systems. We pay the main attention to considering linear multidimensional parabolic'systems with Dirichlet boundary controls and pointwise state constraints, while the methods developed in this study are applicable to other kinds of boundary controls and dynamical systems of the parabolic type. The feedback design problem is formulated in the minimax sense to ensure stabilization of transients within the prescribed diapason and robust stability of …
Suboptimality Conditions For Mathematical Programs With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich
Suboptimality Conditions For Mathematical Programs With Equilibrium Constraints, Truong Q. Bao, Panjak Gupta, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal …
Existence Of Minimizers And Necessary Conditions In Set-Valued Optimization With Equilibrium Constraints, Truong Q. Bao, Boris S. Mordukhovich
Existence Of Minimizers And Necessary Conditions In Set-Valued Optimization With Equilibrium Constraints, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
In this paper we study set-valued optimization problems with equilibrium constraints (SOPEOs) described by parametric generalized equations in the form 0 is an element of the set G(x) + Q(x) where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the PalaisSmale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation.
Suboptimal Minimax Design Of Constrained Parabolic Systems With Mixed Boundary Control, Boris S. Mordukhovich
Suboptimal Minimax Design Of Constrained Parabolic Systems With Mixed Boundary Control, Boris S. Mordukhovich
Mathematics Research Reports
The paper concerns minimax control problems for linear multidimensional parabolic systems with distributed uncertain perturbations and control functions acting in mixed (Robin) boundary conditions. The main goal is to design a feedback control regulator that ensures the required state performance and robust stability under any feasible perturbations and minimize an energy-type functional under the worst perturbations from the given area. We design and justify an easily implemented suboptimal structure of the feedback boundary regulator and compute its optimal parameters ensuring the required state performance and robust stability of the nonlinear closed-loop control system on the infinite horizon.
Variational Analysis In Bilevel Programming, S Dempe, J Dutta, Boris S. Mordukhovich
Variational Analysis In Bilevel Programming, S Dempe, J Dutta, Boris S. Mordukhovich
Mathematics Research Reports
The paper is devoted to applications of advanced tools of modern variational analysis and generalized differentiation to problems of optimistic bilevel programming. In this way, new necessary optimality conditions are derived for two major classes of bilevel programs: those with partially convex and with fully convex lower-level problems. We provide detailed discussions of the results obtained and their relationships with known results in this area.
Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Variational Principles For Set-Valued Mappings With Applications To Multiobjective Optimization, Truong Q. Bao, Boris S. Mordukhovich
Mathematics Research Reports
This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for …
Discrete Approximations, Relaxation, And Optimization Of One-Sided Lipschitzian Differential Inclusions In Hilbert Spaces, Tzanko Donchev, Elza Farkhi, Boris S. Mordukhovich
Discrete Approximations, Relaxation, And Optimization Of One-Sided Lipschitzian Differential Inclusions In Hilbert Spaces, Tzanko Donchev, Elza Farkhi, Boris S. Mordukhovich
Mathematics Research Reports
We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modi- fied one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W^1,p-norm as p greater than or equal to 1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under. consideration by …