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.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250054
Author(s):  
Nader Kanzi
Keyword(s):  
Constraint Qualification ◽  
Necessary Conditions ◽  
Natural Extension ◽  
Clarke Subdifferential ◽  
Optimal Solutions ◽  
First Order ◽  
Locally Lipschitz ◽  
Necessary Conditions For Optimality ◽  
Infinite Programming ◽  
The Clarke Subdifferential

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

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).

Lagrangean conditions and quasiduality

1977 ◽  
Vol 16 (3) ◽  
pp. 325-339 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Local Minimum ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Weak Duality ◽  
Partially Ordered Vector Space ◽  
Counter Example ◽  
Cone Constraints ◽  
Constrained Minimization Problem

For a constrained minimization problem with cone constraints, lagrangean necessary conditions for a minimum are well known, but are subject to certain hypotheses concerning cones. These hypotheses are now substantially weakened, but a counter example shows that they cannot be omitted altogether. The theorem extends to minimization in a partially ordered vector space, and to a weaker kind of critical point (a quasimin) than a local minimum. Such critical points are related to Kuhn-Tucker conditions, assuming a constraint qualification; in certain circumstances, relevant to optimal control, such a critical point must be a minimum. Using these generalized critical points, a theorem analogous to duality is proved, but neither assuming convexity, nor implying weak duality.

Finite Strain Elastostatics With Stiffening Materials: A Constrained Minimization Model

1997 ◽  
Vol 64 (2) ◽  
pp. 440-442 ◽  
Author(s):  
S. J. Hollister ◽  
J. E. Taylor ◽  
P. D. Washabaugh
Keyword(s):  
Boundary Value Problem ◽  
Minimization Problem ◽  
Finite Strain ◽  
Necessary Conditions ◽  
Piecewise Linear ◽  
Process Parameter ◽  
Constrained Minimization ◽  
Problem Statement ◽  
Stress Strain ◽  
Constrained Minimization Problem

Finite strain elastostatics is expressed for general anisotropic, piecewise linear stiffening materials, in the form of a constrained minimization problem. The corresponding boundary value problem statement is identified with the associated necessary conditions. Total strain is represented as a superposition of variationally independent constituent fields. Net stress-strain properties in the model are implicit in terms of the parameters that define the constituents. The model accommodates specification of load fields as functions of a process parameter.

AN EIGENVALUE PROBLEM FOR A CLASS OF NONLINEAR ELLIPTIC OPERATORS

2005 ◽  
Vol 03 (01) ◽  
pp. 27-44
Author(s):  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Minimization Problem ◽  
Constraint Qualification ◽  
Lagrange Equation ◽  
Generalized Derivatives ◽  
Real Numbers ◽  
Nonlinear Elliptic ◽  
Generalized Multipliers ◽  
Derivatives Of

Let a: [0, +∞) → [0, +∞) be an increasing continuous function with a(t) = 0 if and only if t = 0 and limt→+∞a(t) = +∞, Ω ⊂ ℝN be a bounded domain having the segment property and T[u,u] a nonnegative quadratic form involving the only generalized derivatives of order m of the function u: Ω → ℝ. Let p ≥ 1, μi ≠ 0 be real numbers, [Formula: see text], 1 ≤ i ≤ p and [Formula: see text] Put [Formula: see text] and [Formula: see text] Under certain hypotheses on Gi, we show that the minimization problem [Formula: see text] has a solution. Moreover, due to the well-known theorem on generalized multipliers involving the Robinson constraint qualification condition, the solution of the preceeding minimization problem is a weak solution of the corresponding Euler–Lagrange equation (1.1)–(1.2) below. We emphasize that no Δ2-condition on A or [Formula: see text] is imposed. One application to mechanics is given.

scholarly journals Invex functions and constrained local minima

1981 ◽  
Vol 24 (3) ◽  
pp. 357-366 ◽  
Author(s):  
B.D. Craven
Keyword(s):  
Vector Function ◽  
Minimization Problem ◽  
Convex Cone ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Constrained Minimization ◽  
Sufficient Condition ◽  
Local Minima ◽  
Invex Functions ◽  
Constrained Minimization Problem

If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K-invex, of a vector function in relation to a convex cone K. Necessary conditions and sufficient conditions are obtained for a function f to be K-invex. This leads to a new second order sufficient condition for a constrained minimum.

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.

Sign in / Sign up

Export Citation Format

Share Document