Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation

1979 ◽  
Vol 3 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Optimization ◽  
Lagrange Multiplier ◽  
Best Approximation ◽  
Duality Theorem ◽  
Multiplier Theorems ◽  
New Applications

scholarly journals Duality theorem for approximation problem with fuzzy statement

2015 ◽  
Vol 23 ◽  
pp. 88
Author(s):  
V.I. Ruban
Keyword(s):  
Best Approximation ◽  
Duality Theorem ◽  
Convex Set ◽  
Approximation Problem ◽  
The Best Approximation

We obtained fuzzy analogue of duality theorem for the best approximation by a convex set.

scholarly journals Sensitivity functionals in convex optimization problem

Filomat ◽  
2016 ◽  
Vol 30 (14) ◽  
pp. 3681-3687
Author(s):  
Robert Namm ◽  
Gyungsoo Woo
Keyword(s):  
Variational Inequality ◽  
Convex Optimization ◽  
Lagrange Multiplier ◽  
Optimization Problem ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
Convex Optimization Problem ◽  
Infinite Dimensional ◽  
Finite Dimensional

We consider sensitivity functionals and Lagrange multiplier method for solving finite dimensional convex optimization problem.An analysis based on this property is also applied for semicoercive infinite dimensional variational inequality in mechanics.

scholarly journals ε-Properly Efficiency of Multiobjective Semidefinite Programming with Set-Valued Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Chun Hong Yuan ◽  
Wei Dong Rong
Keyword(s):  
Saddle Point ◽  
Semidefinite Programming ◽  
Lagrange Multiplier ◽  
Alternative Theorem ◽  
Generalized Cone Subconvexlikeness ◽  
Multiplier Theorems

In this paper, we study the ε-properly efficiency of multiobjective semidefinite programming with set-valued functions. Firstly, we obtain the scalarization theorems under the condition of the generalized cone-subconvexlikeness. Then, we establish the alternative theorem which contains matrixes and vectors, the ε-Lagrange multiplier theorems, and the ε-proper saddle point theorems of the primal programming under some suitable conditions.

scholarly journals Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4669-4684
Author(s):  
H. Mohebi
Keyword(s):  
Lagrange Multiplier ◽  
Best Approximation ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Dual Cone ◽  
The Best Approximation ◽  
Constraint Set ◽  
Nonconvex Constraint ◽  
Constrained Best Approximation

In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j = 1,2,...,m} of inequalities with nonsmooth nonconvex constraint functions gj : Rn ? R (j = 1,2,...,m).We show that under Abadie?s constraint qualification the ?perturbation property? of the best approximation to any x in Rn from a convex set ?K := C ? K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn: By using the idea of tangential subdifferential and a non-smooth version of Abadie?s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set ?K is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie?s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results.

1250-kilovolt scanning transmission electron microscope

1984 ◽  
Vol 42 ◽  
pp. 624-625
Author(s):  
T. Imura ◽  
S. Maruse ◽  
K. Mihama ◽  
M. Iseki ◽  
M. Hibino ◽  
...  
Keyword(s):  
High Voltage ◽  
Electron Probe ◽  
Scanning Transmission Electron Microscope ◽  
Pressure Vessels ◽  
Practical Applications ◽  
Voltage Generator ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
New Applications ◽  
High Voltage Generator

Ultra high voltage STEM has many inherent technical advantages over CTEM. These advantages include better signal detectability and signal processing capability. It is hoped that it will explore some new applications which were previously not possible. Conventional STEM (including CTEM with STEM attachment), however, has been unable to provide these inherent advantages due to insufficient performance and engineering problems. Recently we have developed a new 1250 kV STEM and completed installation at Nagoya University in Japan. It has been designed to break through conventional engineering limitations and bring about theoretical advantage in practical applications.In the design of this instrument, we exercised maximum care in providing a stable electron probe. A high voltage generator and an accelerator are housed in two separate pressure vessels and they are connected with a high voltage resistor cable.(Fig. 1) This design minimized induction generated from the high voltage generator, which is a high frequency Cockcroft-Walton type, being transmitted to the electron probe.

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