Computation of the Solution of the Approximate Problem

1979 ◽  
pp. 52-68
Author(s):  
Bertrand Mercier
Keyword(s):  
Approximate Problem

scholarly journals AUV-Method for a Class of Constrained Minimized Problems of Maximum Eigenvalue Functions

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Wei Wang ◽  
Ming Jin ◽  
Shanghua Li ◽  
Xinyu Cao
Keyword(s):  
Exact Penalty ◽  
Maximum Eigenvalue ◽  
Convex Approximation ◽  
Equivalent Problem ◽  
Composite Function ◽  
Primal Problem ◽  
Approximate Problem ◽  
Constrained Problem ◽  
Constrained Minimization Problem ◽  
Unconstrained Problem

In this paper, we apply theUV-algorithm to solve the constrained minimization problem of a maximum eigenvalue function which is the composite function of an affine matrix-valued mapping and its maximum eigenvalue. Here, we convert the constrained problem into its equivalent unconstrained problem by the exact penalty function. However, the equivalent problem involves the sum of two nonsmooth functions, which makes it difficult to applyUV-algorithm to get the solution of the problem. Hence, our strategy first applies the smooth convex approximation of maximum eigenvalue function to get the approximate problem of the equivalent problem. Then the approximate problem, the space decomposition, and theU-Lagrangian of the object function at a given point will be addressed particularly. Finally, theUV-algorithm will be presented to get the approximate solution of the primal problem by solving the approximate problem.

scholarly journals A duality-type method for the obstacle problem

2013 ◽  
Vol 21 (3) ◽  
pp. 181-196 ◽  
Author(s):  
Diana Rodica Merlusca
Keyword(s):  
Sobolev Spaces ◽  
Obstacle Problem ◽  
Optimization Problem ◽  
Quadratic Optimization ◽  
Higher Order ◽  
Type Method ◽  
Quadratic Optimization Problem ◽  
Approximate Problem ◽  
Duality Property ◽  
New Type

Abstract Based on a duality property, we solve the obstacle problem on Sobolev spaces of higher order. We have considered a new type of approximate problem and with the help of the duality we reduce it to a quadratic optimization problem, which can be solved much easier.

A LOCALLY SMOOTHING METHOD FOR MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

2015 ◽  
Vol 56 (3) ◽  
pp. 299-315 ◽  
Author(s):  
YU CHEN ◽  
ZHONG WAN
Keyword(s):  
Stationary Point ◽  
Sufficient Conditions ◽  
Accumulation Point ◽  
Smoothing Method ◽  
Stationary Points ◽  
Local Perturbation ◽  
Complementarity Constraints ◽  
Approximate Problem ◽  
Mathematical Programs ◽  
Strongly Stationary

We propose a locally smoothing method for some mathematical programs with complementarity constraints, which only incurs a local perturbation on these constraints. For the approximate problem obtained from the smoothing method, we show that the Mangasarian–Fromovitz constraints qualification holds under certain conditions. We also analyse the convergence behaviour of the smoothing method, and present some sufficient conditions such that an accumulation point of a sequence of stationary points for the approximate problems is a C-stationary point, an M-stationary point or a strongly stationary point. Numerical experiments are employed to test the performance of the algorithm developed. The results obtained demonstrate that our algorithm is much more promising than the similar ones in the literature.

scholarly journals Numerical computations of the PCD method

2017 ◽  
Vol 37 (1) ◽  
pp. 39-54
Author(s):  
Ahmed Tahiri
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Stiffness Matrix ◽  
Boundary Value ◽  
Elasticity Problem ◽  
Numerical Computations ◽  
Piecewise Constant ◽  
Approximate Problem ◽  
Discretization Technique ◽  
Triangular Elements

The PCD (piecewise constant distributions) method is a discretization technique of the boundary value problems in which the unknown distribution and its derivatives are represented by piecewise constant distributions but on distinct meshes. It has the advantage of producing the most sparse stiffness matrix resulting from the approximate problem. In this contribution, we propose a general PCD triangulation by combining rectangular elements and triangular elements. We also apply this discretization technique for the elasticity problem. We end with presentation of numerical results of the proposed method for the 2D diffusion equation.

A Logical Function Method for Dynamic and Design Sensitivity Analysis of Mechanical Systems With Intermittent Motion

1982 ◽  
Vol 104 (1) ◽  
pp. 90-100 ◽  
Author(s):  
P. E. Ehle ◽  
E. J. Haug
Keyword(s):  
Sensitivity Analysis ◽  
Design Sensitivity ◽  
Step Function ◽  
Design Sensitivity Analysis ◽  
Logical Function ◽  
Smooth Functions ◽  
Heaviside Step Function ◽  
Approximate Problem ◽  
Logical Functions ◽  
Intermittent Motion

Dynamic and design sensitivity analysis of mechanisms and machines with intermittent motion are accomplished through introduction of “logical functions” to approximate discontinuities and special features of system motion. The Heaviside step function and the delta and unit doublet distributions are introduced to represent discontinuities and to determine the values of certain functions of isolated times. These functions and distributions are approximated by smooth functions, and validity of the approximation is argued both mathematically and physically. A numerical method is then presented for analysis of the approximate problem. An elementary and a complex, realistic example are presented to illustrate applications of the method.

scholarly journals ANALYSIS OF AN INVERSE PROBLEM ARISING IN PHOTOLITHOGRAPHY

2012 ◽  
Vol 22 (05) ◽  
pp. 1150026 ◽  
Author(s):  
LUCA RONDI ◽  
FADIL SANTOSA
Keyword(s):  
Inverse Problem ◽  
Variational Formulation ◽  
Diffractive Optics ◽  
Shape Design ◽  
Two Dimensional ◽  
Convergence Properties ◽  
Approximate Problem ◽  
The Relationship ◽  
Well Posed ◽  
Dimensional Domain

We consider the inverse problem of determining an optical mask that produces a desired circuit pattern in photolithography. We set the problem as a shape design problem in which the unknown is a two-dimensional domain. The relationship between the target shape and the unknown is modeled through diffractive optics. We develop a variational formulation that is well-posed and propose an approximation that can be shown to have convergence properties. The approximate problem can serve as a foundation to numerical methods, much like the Ambrosio–Tortorelli's approximation of the Mumford–Shah functional in image processing.

scholarly journals On the approximate problem for the incremental thermoelasticity

2021 ◽  
Vol 44 (5) ◽  
pp. 619-633
Author(s):  
N. Bazarra ◽  
J. R. Fernández ◽  
R. Quintanilla
Keyword(s):  
Approximate Problem
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