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).

scholarly journals On quasidifferentiable optimization

1986 ◽  
Vol 41 (1) ◽  
pp. 64-78 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Minimization Problem ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Directional Derivatives ◽  
Generalized Derivatives ◽  
Differentiable Functions ◽  
Locally Lipschitz ◽  
Necessary Conditions For Optimality ◽  
Constrained Minimization Problem ◽  
Convexity Properties

AbstractLagrangian necessary conditions for optimality, of both Fritz John and Kuhn Tucker types, are obtained for a constrained minimization problem, where the functions are locally Lipschitz and have directional derivatives, but need not have linear Gâteaux derivatives; the variable may be constrained to lie in a nonconvex set. The directional derivatives are assumed to have some convexity properties as functions of direction; this generalizes the concept of quasidifferentiable function. The convexity is not required when directional derivatives are replaced by Clarke generalized derivatives. Sufficient Kuhn Tucker conditions, and a criterion for the locally solvable constraint qualification, are obtained for directionally differentiable functions.

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

Mathematics ◽  
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.

scholarly journals Identification of mechanical properties of arteries with certification of global optimality

2021 ◽  
Author(s):  
Jan-Lucas Gade ◽  
Carl-Johan Thore ◽  
Jonas Stålhand
Keyword(s):  
Global Solution ◽  
Branch And Bound ◽  
Clinical Data ◽  
Minimization Problem ◽  
Branch And Bound Algorithm ◽  
Stationary Points ◽  
Model Parameters ◽  
Global Optimality ◽  
Clinical Value ◽  
Non Linear

AbstractIn this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

An eigenvalue problem for generalized Laplacian in Orlicz—Sobolev spaces

1999 ◽  
Vol 129 (1) ◽  
pp. 153-163 ◽  
Author(s):  
Vesa Mustonen ◽  
Matti Tienari
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Minimization Problem ◽  
Lagrange Equation ◽  
Image Position ◽  
Generalized Laplacian

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problemwhere Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equationfor some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

2018 ◽  
Vol 35 (01) ◽  
pp. 1850008
Author(s):  
Na Xu ◽  
Xide Zhu ◽  
Li-Ping Pang ◽  
Jian Lv
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Accumulation Point ◽  
Stationary Points ◽  
Nondegeneracy Condition ◽  
Equilibrium Constraints ◽  
Convergence Properties ◽  
Constant Rank Constraint Qualification ◽  
Relaxation Schemes

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

A Geometric Theory for Formulation of Form, Profile and Orientation Tolerances: Problem Formulation

1998 ◽  
Author(s):  
J. B. Gou ◽  
Y. X. Chu ◽  
H. Wu ◽  
Z. X. Li
Keyword(s):  
Configuration Space ◽  
Minimization Problem ◽  
Geometric Theory ◽  
Problem Formulation ◽  
Euclidean Group ◽  
Geometric Properties ◽  
Symmetry Subgroup ◽  
Minimum Zone ◽  
Constrained Minimization Problem ◽  
Linear Programming Problems

Abstract This paper develops a geometric theory which unifies the formulation and evaluation of form (straightness, flatness, cylindricity and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. In the paper, based on an an important observation that a toleranced feature exhibits a symmetry subgroup G0 under the action of the Euclidean group, SE(3), we identify the configuration space of a toleranced (or a symmetric) feature with the homogeneous space SE(3)/G0 of the Euclidean group. Geometric properties of SE(3)/G0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/G0, with G0 being the symmetry subgroup of the underlying feature. We transform the non-differentiable minimization problem into a differentiable minimization problem over an extended configuration space. Using geometric properties of SE(3)/G0, we derive a sequence of linear programming problems whose solutions can be used to approximate the minimum zone solutions.

scholarly journals On general curves lying on a quadric

1927 ◽  
Vol 1 (1) ◽  
pp. 19-30 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Rational Curves ◽  
Stationary Points ◽  
Image Position

Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric isTwo proofs of this result are given, in §§ 4 and 5.

Smart Polygeneration Grid: Control and Optimization System

2012 ◽  
Author(s):  
Maximiliano Bozzo ◽  
Francesco Caratozzolo ◽  
Alberto Traverso
Keyword(s):  
Control System ◽  
Working Conditions ◽  
Minimization Problem ◽  
Power Supply ◽  
Supply And Demand ◽  
Software Tool ◽  
Test Cases ◽  
Urban District ◽  
Constrained Minimization Problem ◽  
Different Levels

This study aims at the development of a software tool for supply and demand matching of electrical and thermal energy in an urban district. In particular, the tool has been developed for E-NERDD, the experimental district that TPG-DIMSET is going to build in Savona, Italy. E-NERDD is an acronym for Energy and Efficiency Research Demonstration District. It is one of the districts that will be used within the project to demonstrate how different software tools and algorithms perform in thermodynamic, economic and environmental terms. The software tool originally developed for and implemented in this work, called E-NERDD Control System, is targeted on enabling the operation of the hardware, when connected in a district mode. Supply and demand are matched to reach a thermoeconomic optimum. An optimization algorithm is organized into two different levels of optimization: a first level that resolves a constrained minimization problem in planning power supply for each generator on the basis of day-before forecasting; and a second level that distributes among the different machines the gap between planned and real-time demand. The algorithm developed is demonstrated in four test cases in order to test it in different working conditions.

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