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On the lagrange functions of quadratic models that are defined by interpolation

OPTIMIZATION METHODS & SOFTWARE 2001年第1-4期16卷 289-309页

作者： Powell, MJD Univ Cambridge Dept Appl Math & Theoret Phys Cambridge CB3 9EW England

Quadratic models are of fundamental importance to the efficiency of many optimization algorithms when second derivatives of the objective function influence the required values of the variables. They may be constructed by interpolation to function values for suitable choices of the interpolation points. We consider the lagrange functions of this technique, because they have some highly useful properties. In particular, they show whether a change to an interpolation point preserves nonsingularity of the interpolation equations, and they provide a bound on the error of the quadratic model. Further, they can be updated efficiently when an interpolation point is moved. These features are explained. Then it is shown that the error bound can control the adjustment of a trust region radius in a way that gives excellent convergence properties in an algorithm for unconstrained minimization calculations. Finally, a convenient procedure for generating the initial interpolation points is described.

关键词： lagrange functions quadratic interpolation unconstrained minimization

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Structure-Preserving Kernel-Based Meshless Methods for Solving Dissipative PDEs on Surfaces

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JOURNAL OF SCIENTIFIC COMPUTING 2025年第3期102卷 1-25页

作者： Sun, Zhengjie Ling, Leevan Chen, Meng Nanjing Univ Sci & Technol Sch Math & Stat Nanjing Peoples R China Hong Kong Baptist Univ Dept Math Kowloon Hong Kong Peoples R China Nanchang Univ Inst Math & Interdisciplinary Sci Sch Math & Comp Sci Nanchang Peoples R China

In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties. Furthermore, we provide numerical evidence demonstrating that the lagrange function and the coefficients generated by a restricted kernel decay exponentially, even though a comprehensive theory has not yet been developed.

关键词： Kernel-based method Dissipative PDEs Structure-preserving scheme lagrange functions Surface PDEs

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Locally supported, quasi-interpolatory bases for the approximation of functions on graphs

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LINEAR ALGEBRA AND ITS APPLICATIONS 2024年 703卷 369-394页

作者： Fuselier, E. Ward, J. P. High Point Univ High Point NC 27268 USA North Carolina A&T State Univ Greensboro NC 27411 USA

Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of lagrange bases on graphs, where the lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present B infinity error estimates between the local basis and the corresponding interpolatory lagrange basis functions in cases where the underlying graph satisfies an assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate ||Al||(infinity) <= root n ||A||(2). (c) 2024 Published by Elsevier Inc.

关键词： Graph Laplacian lagrange functions Approximation on graphs Variational splines on graphs Graph basis functions

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APPLICATION OF SOME SPECIFICATION OF THE DUBOVITSKII-MILYUTIN METHOD TO PROBLEMS OF OPTIMAL-CONTROL

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 1988年第2期12卷 101-108页

作者： LEDZEWICZKOWALEWSKA, U Institute of Mathematics Lodz University ul. Stefana Banacha 22 90-238 Lodz Poland

The Dubovitskii-Milyutin method is useful to obtain a necessary condition for the extremal problems with only one equality constraint. A generalization of the Dubovitskii-Milyutin method is the case of n equality constraints in any form under some assumptions about the cones has been previoulsy applied to problems of an optimal control with equality constraints on the phase coordinates in and to a problem with no-operator equality constraint. Specification of the Dubovitskii-Milyutin method has been given without any additional assumption about the cones but for the case of n equality constraints given in the operator form. This specification is applied here to obtain a necessary condition for the problem of an optimal control with equality constraints onthe phase coordinates and the control.

关键词： Dubovitskii-Milyutin method operator equality constraint optimal control local extremum principle lagrange functions cone of tangent directions

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OPTIMALITY, DUALITY AND SADDLE POINT CRITERIA FOR A ROBUST FRACTIONAL INTERVAL-VALUED OPTIMIZATION PROBLEM WITH UNCERTAIN INEQUALITY CONSTRAINTS VIA CONVEXIFICATORS

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RAIRO-OPERATIONS RESEARCH 2023年第3期57卷 1397-1416页

作者： Kummari, Krishna Jaichander, Rekha R. Ahmad, Izhar Sch Sci GITAM Hyderabad Campus Dept Math Hyderabad 502329 India St Francis Coll Women Begumpet Dept Math Hyderabad 500006 India King Fahd Univ Petr & Minerals Dept Math Dhahran 31261 Saudi Arabia King Fahd Univ Petr & Minerals Ctr Intelligent Secure Syst Dhahran 31261 Saudi Arabia

This article focuses on optimality conditions for a robust fractional interval-valued optimization problem with uncertain inequality constraints (RNFIVP) based on convexificators. Using the tools of convexity, an example of sufficient optimality conditions is demonstrated. Robust parametric duality for (RNFIVP) is formulated and utilizing the concept of convexity, usual duality results between the primal and dual problems are investigated. Further, the equivalence between the saddle point criteria of a Lagrangian type function and a robust LU-optimal solution for (RNFIVP) with convexity is also examined.

关键词： Robust optimization Convexificator Fractional interval-valued problem Robust parametric duality convexity LU-optimal solution lagrange functions Saddle point

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Isogeometric analysis enhanced by the scaled boundary finite element method

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2015年 283卷 733-762页

作者： Natarajan, Sundararajan Wang, JunChao Song, Chongmin Birk, Carolin Indian Inst Technol Dept Mech Engn Madras 600036 Tamil Nadu India Univ New S Wales Sch Civil & Environm Engn Sydney NSW 2052 Australia

By leveraging the information of a typical CAD model in the analysis, the intensive process of discretization can be circumvented. This unification has led to the 'Isogeometric Analysis' (IGA) (Hughes et al., 2005). However, as the CAD model provides information only of the boundary, a 2D/3D stress analysis is still one major step away. In this work, the concepts of isogeometric analysis and the scaled boundary finite element method (SBFEM) are combined. The SBFEM requires only the boundary information and hence provides a seamless integration with the CAD modeling. Within the proposed framework, the NURBS basis functions are used to discretize the unknown fields in the circumferential direction, whilst analytical solution is sought in the radial direction. We further extend the framework to problems with singularities and to dynamic analysis. The accuracy and the convergence properties of the proposed method are demonstrated with benchmark problems in the context of linear elasticity and linear elastic fracture mechanics. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Scaled boundary finite element method Isogeometric analysis Dynamic analysis Stress intensity factor NURBS lagrange functions

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A radial basis function Galerkin method for inhomogeneous nonlocal diffusion

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2016年第Feb.1期299卷 366-380页

作者： Lehoucq, R. B. Rowe, S. T. Sandia Natl Labs Computat Math Albuquerque NM 87185 USA Texas A&M Univ Dept Math College Stn TX 77843 USA

We introduce a meshfree discretization for a nonlocal diffusion problem using a localized basis of radial basis functions. Our method consists of a conforming radial basis of local lagrange functions for a variational formulation of a volume constrained nonlocal diffusion equation. We also establish an L-2 error estimate on the local lagrange interpolant. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, sparse, symmetric positive definite stiffness matrix. We demonstrate that both the continuum and discrete problems are well-posed and present numerical results for the convergence behavior of the radial basis function method. We explore approximating the solution to inhomogeneous differential equations by solving inhomogeneous nonlocal integral equations using the proposed radial basis function method. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Radial basis functions Nonlocal diffusion lagrange functions Volume constraint

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Interpolating splines on graphs for data science applications

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2020年第2期49卷 540-557页

作者： Ward, John Paul Narcowich, Francis J. Ward, Joseph D. North Carolina A&T State Univ Greensboro NC 27411 USA Texas A&M Univ College Stn TX USA

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the lagrange function, showing that they scalewith the density of the data. These results indicate that lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Local basis functions Kernel-based machine learning Interpolation lagrange functions

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PARAMETRIC SADDLE POINT CRITERIA IN SEMI- INFINITE MINIMAX FRACTIONAL PROGRAMMING PROBLEMS UNDER ( p, r)- INVEXITY

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2015年第1期36卷 1-28页

作者： Antczak, Tadeusz Univ Lodz Fac Math & Comp Sci PL-90238 Lodz Poland

A semi-infinite minimax fractional programming problem with both inequality and equality constraints is considered. Characterizations of an optimal solution by a saddle point of the scalar lagrange function and the vector-valued lagrange function defined for such an optimization problem are given. Hence, the equivalence between an optimal solution and a saddle point of the scalar lagrange function and the vector-valued lagrange function in the considered semi-infinite minmax fractional programming problem is established under various (p, r)-invexity assumptions.

关键词： (p r)-Invex function lagrange functions Saddle point Semi-infinite minimax fractional programming problem 90C34 90C29 90C26 90C30

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NEW DECOMPOSITION AND CONVEXIFICATION ALGORITHM FOR NONCONVEX LARGE-SCALE PRIMAL DUAL OPTIMIZATION

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1990年第2期67卷 279-296页

作者： FENG, X MUKAI, H BROWN, RH WASHINGTON UNIV DEPT SYST SCI & MATHST LOUISMO 63130

A new algorithm for solving nonconvex, equality-constrained optimization problems with separable structures is proposed in the present paper. A new augmented Lagrangian function is derived, and an iterative method is presented. The new proposed Lagrangian function preserves separability when the original problem is separable, and the property of linear convergence of the new algorithm is also presented. Unlike earlier algorithms for nonconvex decomposition, the convergence ratio for this method can be made arbitrarily small. Furthermore, it is feasible to extend this method to algorithms suited for inequality-constrained optimization problems. An example is included to illustrate the method.

关键词： Primal-dual methods lagrange functions decomposition nonlinear programming convergence analysis

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