The computation of Lagrange-multiplier estimates for constrained minimization

Abstract In this paper, constrained minimization for the point of closest approach of two conic sections is developed. For this development, we considered the nine cases of possible conics, namely, (elliptic–elliptic), (elliptic–parabolic), (elliptic–hyperbolic), (parabolic–elliptic), (parabolic–parabolic), (parabolic–hyperbolic), (hyperbolic–elliptic), (hyperbolic–parabolic), and (hyperbolic–hyperbolic). The developments are considered from two points of view, namely, analytical and computational. For the analytical developments, the literal expression of the minimum distance equation (S) and the constraint equation (G), including the first and second derivatives for each case, are established. For the computational developments, we construct an efficient algorithm for calculating the minimum distance by using the Lagrange multiplier method under the constraint on time. Finally, we compute the closest distance S between two conics for some orbits. The accuracy of the solutions was checked under the conditions that L| solution ≤ ɛ1; G| solution ≤ ɛ2, where ɛ1,2 < 10−10. For the cases of (parabolic–parabolic), (parabolic–hyperbolic), and (hyperbolic–hyperbolic), we studied thousands of comets, but the condition of the closest approach was not met.

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Mass-constrained minimization of a one-homogeneous functional arising in strain-gradient plasticity

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2012.07.054 ◽

2013 ◽

Vol 397 (1) ◽

pp. 381-401 ◽

Cited By ~ 5

Author(s):

Micol Amar ◽

Maria Chiricotto ◽

Lorenzo Giacomelli ◽

Giuseppe Riey

Keyword(s):

Strain Gradient ◽

Strain Gradient Plasticity ◽

Constrained Minimization ◽

Gradient Plasticity

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Computation of fold and cusp bifurcation points in a system of ordinary differential equations using the Lagrange multiplier method

International Journal of Dynamics and Control ◽

10.1007/s40435-021-00821-4 ◽

2021 ◽

Author(s):

Livia Owen ◽

Johan Matheus Tuwankotta

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Lagrange Multiplier ◽

Lagrange Multiplier Method ◽

Multiplier Method ◽

Bifurcation Points

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Lagrange Multiplier Characterizations of Constrained Best Approximation with Infinite Constraints

Journal of Optimization Theory and Applications ◽

10.1007/s10957-021-01856-5 ◽

2021 ◽

Author(s):

Hassan Bakhtiari ◽

Hossein Mohebi

Keyword(s):

Lagrange Multiplier ◽

Best Approximation ◽

Constrained Best Approximation

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A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽

10.3390/math9080890 ◽

2021 ◽

Vol 9 (8) ◽

pp. 890

Author(s):

Suthep Suantai ◽

Kunrada Kankam ◽

Prasit Cholamjiak

Keyword(s):

Image Processing ◽

Minimization Problem ◽

Convex Functions ◽

Lower Semicontinuous ◽

Image Inpainting ◽

Constrained Minimization ◽

Convex Minimization Problem ◽

Mild Conditions ◽

Constrained Minimization Problem ◽

Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

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Use of the Lagrange Multiplier Test for Assessing Measurement Invariance Under Model Misspecification

Educational and Psychological Measurement ◽

10.1177/00131644211020355 ◽

2021 ◽

pp. 001316442110203

Author(s):

Lucia Guastadisegni ◽

Silvia Cagnone ◽

Irini Moustaki ◽

Vassilis Vasdekis

Keyword(s):

Lagrange Multiplier ◽

False Positive ◽

Latent Variable ◽

Model Misspecification ◽

Cross Product ◽

Small Sample ◽

Type I ◽

Local Dependence ◽

Lagrange Multiplier Test ◽

Product Approach

This article studies the Type I error, false positive rates, and power of four versions of the Lagrange multiplier test to detect measurement noninvariance in item response theory (IRT) models for binary data under model misspecification. The tests considered are the Lagrange multiplier test computed with the Hessian and cross-product approach, the generalized Lagrange multiplier test and the generalized jackknife score test. The two model misspecifications are those of local dependence among items and nonnormal distribution of the latent variable. The power of the tests is computed in two ways, empirically through Monte Carlo simulation methods and asymptotically, using the asymptotic distribution of each test under the alternative hypothesis. The performance of these tests is evaluated by means of a simulation study. The results highlight that, under mild model misspecification, all tests have good performance while, under strong model misspecification, the tests performance deteriorates, especially for false positive rates under local dependence and power for small sample size under misspecification of the latent variable distribution. In general, the Lagrange multiplier test computed with the Hessian approach and the generalized Lagrange multiplier test have better performance in terms of false positive rates while the Lagrange multiplier test computed with the cross-product approach has the highest power for small sample sizes. The asymptotic power turns out to be a good alternative to the classic empirical power because it is less time consuming. The Lagrange tests studied here have been also applied to a real data set.

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