Topological Stability of Feasible Sets in Semi-infinite Optimization: A Tutorial

2020 ◽  
pp. 339-361
Author(s):  
Jan-J. Rückmann
Keyword(s):  
Topological Stability ◽  
Feasible Sets

scholarly journals Surface Stabilized Topological Solitons in Nematic Liquid Crystals

Crystals ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 840
Author(s):  
Inge Nys ◽  
Brecht Berteloot ◽  
Guilhem Poy
Keyword(s):  
Liquid Crystal ◽  
Topological Stability ◽  
Light Modulator ◽  
Topological Solitons ◽  
Bottom Substrate ◽  
Closed Ring ◽  
Versatile Tool ◽  
Photo Alignment ◽  
Strong Confinement ◽  
Careful Design

Photo-alignment is a versatile tool to pattern the alignment at the confining substrates in a liquid crystal (LC) cell. Arbitrary alignment patterns can be created by using projection with a spatial light modulator (SLM) for the illumination. We demonstrate that a careful design of the alignment patterns allows the stabilization of topological solitons in nematic liquid crystal (NLC) cells, without the need for chirality or strong confinement. The created LC configurations are stabilized by the anchoring conditions imposed at the substrates. The photo-aligned background at both substrates is uniformly planar aligned, and ring-shaped regions with a 180° azimuthal rotation are patterned with an opposite sense of rotation at the top and bottom substrate. A disclination-free structure containing a closed ring of vertically oriented directors is formed when the patterned rings at the top and bottom substrate overlap. Thanks to the topological stability, a vertical director orientation in the bulk is observed even when the centra of both patterned rings are shifted over relatively large distances. The combination of numerical simulations with experimental measurements allows identification of the 3D director configuration in the bulk. A finite element (FE) Q-tensor simulation model is applied to find the equilibrium director configuration and optical simulations are used to confirm the correspondence with experimental microscopy measurements. The created LC configurations offer opportunities in the field of optical devices, light guiding and switching, particle trapping and studies of topological LC structures.

scholarly journals Predicting relative topological stability of mobile users in a P2P mobile cloud

2020 ◽  
Vol 2 (11) ◽  
Author(s):  
Pijush Kanti Dutta Pramanik ◽  
Gautam Bandyopadhyay ◽  
Prasenjit Choudhury
Keyword(s):  
Mobile Users ◽  
Mobile Cloud ◽  
Topological Stability

An alternative theorem for set-valued maps via set relations and its application to robustness of feasible sets

Optimization ◽  
2018 ◽  
Vol 67 (7) ◽  
pp. 1067-1075 ◽  
Author(s):  
Yuto Ogata ◽  
Tamaki Tanaka ◽  
Yutaka Saito ◽  
Gue Myung Lee ◽  
Jae Hyoung Lee
Keyword(s):  
Alternative Theorem ◽  
Feasible Sets ◽  
Set Relations

On Some Problems With Modeling of Coulomb Friction in Self-Locking Speed Reducers

2014 ◽  
Author(s):  
Marek Wojtyra
Keyword(s):  
Rigid Body ◽  
Coulomb Friction ◽  
Equations Of Motion ◽  
Velocity Ratio ◽  
Friction Forces ◽  
Friction Law ◽  
Rigid Body Model ◽  
Feasible Sets ◽  
Unstable Solutions ◽  
Coulomb Friction Law

A simple mathematical model of friction in speed reducers is presented and discussed. A rigid body approach, typical for multibody simulations, is adopted. The model is based on the Coulomb friction law and exploits the analogy between reducers and wedge mechanisms. The first version of the model is purely rigid, i.e. no deflections of the mechanism bodies are allowed. Constraints are introduced to maintain the ratio between input and output velocity. It is shown that when friction is above the self-locking limit, paradoxical situations may be observed when kinetic friction is investigated. For some sets of parameters of the mechanism (gearing ratio, coefficient of friction and inertial parameters) two distinct solutions of normal and friction forces can be found. Moreover, for some combinations of external loads, a solution that satisfies equations of motion, constraints and Coulomb friction law does not exist. Furthermore, for appropriately chosen loads and parameters of the mechanism, infinitely many feasible sets of normal and friction forces can be found. Examples of all indicated paradoxical situations are provided and discussed. The second version of the model allows deflection of the frictional contact surface, and forces proportional to this deflection are applied to contacting bodies (no constraints to maintain the input-output velocity ratio are introduced). In non-paradoxical situations the obtained results are closely similar to those predicted by the rigid body model. In previously paradoxical situations no multiple solutions of friction force are found, however, the amended model does not solve all problems. It is shown that in regions for which the paradoxes were observed only unstable solutions are available. Numerical examples showing behavior of the model are provided and analyzed.

scholarly journals A Comparison of Variational Bounds for the Information Bottleneck Functional

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1229 ◽  
Author(s):  
Bernhard C. Geiger ◽  
Ian S. Fischer
Keyword(s):  
Cost Function ◽  
Short Note ◽  
General Setting ◽  
Conditional Entropy ◽  
Variational Bounds ◽  
Information Bottleneck ◽  
Feasible Sets ◽  
Shed Light ◽  
Relevant Cost ◽  
General Setup

In this short note, we relate the variational bounds proposed in Alemi et al. (2017) and Fischer (2020) for the information bottleneck (IB) and the conditional entropy bottleneck (CEB) functional, respectively. Although the two functionals were shown to be equivalent, it was empirically observed that optimizing bounds on the CEB functional achieves better generalization performance and adversarial robustness than optimizing those on the IB functional. This work tries to shed light on this issue by showing that, in the most general setting, no ordering can be established between these variational bounds, while such an ordering can be enforced by restricting the feasible sets over which the optimizations take place. The absence of such an ordering in the general setup suggests that the variational bound on the CEB functional is either more amenable to optimization or a relevant cost function for optimization in its own regard, i.e., without justification from the IB or CEB functionals.

scholarly journals Definite determinantal representations of multivariate polynomials

2019 ◽  
Vol 19 (07) ◽  
pp. 2050129 ◽  
Author(s):  
Papri Dey
Keyword(s):  
Semidefinite Programming ◽  
Matrix Polynomial ◽  
Heuristic Method ◽  
Linear Matrix ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Determinantal Representation ◽  
Representation Problem ◽  
Feasible Sets

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

Dynamics of G-processes

2020 ◽  
Vol 20 (01) ◽  
pp. 2050037
Author(s):  
W. Jung ◽  
K. Lee ◽  
C.A. Morales
Keyword(s):  
Dynamical Systems ◽  
Group Structure ◽  
Global Solutions ◽  
Topological Stability ◽  
Natural Group ◽  
Shadowing Property ◽  
Infinite Dimensional ◽  
Mathematical Sciences ◽  
The Relationship

A G-process is briefly a process ([A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, Vol. 182 (Springer, 2013)], [C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971) 239–252], [P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176 (Amer. Math. Soc., 2011)]) for which the role of evolution parameter is played by a general topological group [Formula: see text]. These processes are broad enough to include the [Formula: see text]-actions (characterized as autonomous [Formula: see text]-processes) and the two-parameter flows (where [Formula: see text]). We endow the space of [Formula: see text]-processes with a natural group structure. We introduce the notions of orbit, pseudo-orbit and shadowing property for [Formula: see text]-processes and analyze the relationship with the [Formula: see text]-processes group structure. We study the equicontinuous [Formula: see text]-processes and use it to construct nonautonomous [Formula: see text]-processes with the shadowing property. We study the global solutions of the [Formula: see text]-processes and the corresponding global shadowing property. We study the expansivity (global and pullback) of the [Formula: see text]-processes. We prove that there are nonautonomous expansive [Formula: see text]-processes and characterize the existence of expansive equicontinuous [Formula: see text]-processes. We define the topological stability for [Formula: see text]-processes and prove that every expansive [Formula: see text]-process with the shadowing property is topologically stable. Examples of nonautonomous topologically stable [Formula: see text]-processes are given.

scholarly journals On stability and stationary points in nonlinear optimization

1986 ◽  
Vol 28 (1) ◽  
pp. 36-56 ◽  
Author(s):  
J. Guddat ◽  
H. Th. Jongen ◽  
J. Rueckmann
Keyword(s):  
Stability Condition ◽  
Minimization Problem ◽  
Level Sets ◽  
Constraint Qualification ◽  
Stationary Points ◽  
Feasible Set ◽  
Manifold With Boundary ◽  
Feasible Sets ◽  
Constrained Minimization Problem ◽  
Image Position

This paper presents three theorems concerning stability and stationary points of the constrained minimization problem:In summary, we provethat, given the Mangasarian-Fromovitz constraint qualification (MFCQ), the feasible setM[H, G] is a topological manifold with boundary, with specified dimension; (ℬ) a compact feasible setM[H, G] is stable (perturbations ofHandGproduce homeomorphic feasible sets) if and only if MFCQ holds;under a stability condition, two lower level sets offwith a Kuhn-Tucker point between them are homotopically related by attachment of ak-cell (kbeing the stationary index in the sense of Kojima).

On feasible sets defined through Chebyshev approximation

1998 ◽  
Vol 47 (2) ◽  
pp. 255-264 ◽  
Author(s):  
Francisco Guerra ◽  
Miguel A. Jim�nez
Keyword(s):  
Chebyshev Approximation ◽  
Feasible Sets
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