scholarly journals Nonsmooth and multivalued analysis with applications in optimization

1983 ◽  
Author(s):  
Νικόλαος Παπαγεωργίου
Keyword(s):  
Optimization Problems ◽  
Recent Theory ◽  
Subdifferential Calculus ◽  
Infinite Dimensional ◽  
Third Stage ◽  
Existence And Stability ◽  
Nonconvex Analysis ◽  
Vector Valued ◽  
Stability Results ◽  
Convex Operators

The object of this thesis is two - fold . In the first part, we develop analogs of convex and nonconvex analysis for vector - valuedoperators, while in the second part we study the theory of Banach valued multifunctions. In the first part, we start with a study of convex operators. We introduce the notion of algebraic and topological subdifferentials and then derive conditions for those two to be equal. Also we develop a complete subdifferential and e-subdifferential calculus. In the sequence, we deal with the duality theory of convex operators. For that purpose, we introduc a notion of lower semicontinuity of operators and we show that this class is identical with the class of operators that are the upper envelope of continuous affine operators . This allows us to study analogs of the major duality schemes for vector optimization problems. Finally, we conclude our study of convex operators with some probabilistic results on Caratheodory convex integrands. Then we pass to nonconvex operators and introduce the class of locally o-Lipschitz operators. For those operators, we define a generalized subdifferential and develop a corresponding calculus that extends Clarke's theory to a vectorial context. Furthermore, we show that this extension is consistent with the convex theory. Applications to vectorial optimization are given. In the third stage of the process, we consider general operators and using geometric notions we introduce a new subdifferential calculus and provide applications in optimization. We close the first part of the thesis with a detailed study of infinite dimensional Pareto optimization problems and obtain existence and stability results for such problems. In the second part of the thesis, we pass to multivalued analysis. We introduce the vector valued Aumann integral and study its properties.Multifunctions depending on parameters are studied and results are obtained determining which properties of the integrand multifunction are preserved by integration. Then, using a notion of set valued conditional expectation, we introduce set valued martingales and obtain several convergence theorems. Also we study properties of the profile of a multifunction and weakly convergent sequences of multifunctions. Then we consider set valued measures, study their properties and define an integral with respect to a set valued measure and determine its properties. Finally, in the last chapter of our thesis, motivated by Ioffe's recent theory of fans, we introduce the notion of a "normal fan " and develop an integral theory for such fans.

Stability results for convex vector-valued optimization problems

Positivity ◽  
2010 ◽  
Vol 15 (3) ◽  
pp. 441-453 ◽  
Author(s):  
J. Zeng ◽  
S. J. Li ◽  
W. Y. Zhang ◽  
X. W. Xue
Keyword(s):  
Optimization Problems ◽  
Vector Valued ◽  
Stability Results

scholarly journals On quantitative stability in infinite-dimensional optimization under uncertainty

2021 ◽  
Author(s):  
M. Hoffhues ◽  
W. Römisch ◽  
T. M. Surowiec
Keyword(s):  
Stochastic Optimization ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Optimization Under Uncertainty ◽  
Optimal Solutions ◽  
Probability Metrics ◽  
Pde Constrained Optimization ◽  
Quantitative Stability ◽  
Infinite Dimensional ◽  
Optimal Value

AbstractThe vast majority of stochastic optimization problems require the approximation of the underlying probability measure, e.g., by sampling or using observations. It is therefore crucial to understand the dependence of the optimal value and optimal solutions on these approximations as the sample size increases or more data becomes available. Due to the weak convergence properties of sequences of probability measures, there is no guarantee that these quantities will exhibit favorable asymptotic properties. We consider a class of infinite-dimensional stochastic optimization problems inspired by recent work on PDE-constrained optimization as well as functional data analysis. For this class of problems, we provide both qualitative and quantitative stability results on the optimal value and optimal solutions. In both cases, we make use of the method of probability metrics. The optimal values are shown to be Lipschitz continuous with respect to a minimal information metric and consequently, under further regularity assumptions, with respect to certain Fortet-Mourier and Wasserstein metrics. We prove that even in the most favorable setting, the solutions are at best Hölder continuous with respect to changes in the underlying measure. The theoretical results are tested in the context of Monte Carlo approximation for a numerical example involving PDE-constrained optimization under uncertainty.

scholarly journals EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

2019 ◽  
pp. 341
Author(s):  
Mohamed Houas ◽  
Mohamed Bezziou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Nonlocal Boundary ◽  
Stability Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Stability Results

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

scholarly journals Quasi-regularity in Optimization

1986 ◽  
Vol 41 (2) ◽  
pp. 188-192 ◽  
Author(s):  
Kung-Fu Ng ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Lagrange Multiplier ◽  
Optimization Problems ◽  
General Situation ◽  
Lagrange Multiplier Rule ◽  
Infinite Dimensional ◽  
Space Setting ◽  
Multiplier Rule ◽  
Dimensional Version ◽  
Definition Of

AbstractThe notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

scholarly journals Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
P. Rueda ◽  
E. A. Sánchez Pérez
Keyword(s):  
Type Theorem ◽  
Classical Case ◽  
Function Spaces ◽  
Dual Systems ◽  
Banach Function Spaces ◽  
Infinite Dimensional ◽  
Local Compactness ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Vector Valued

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

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