On uniqueness of Kuhn-Tucker multipliers in nonlinear programming

Recently Fujiwara, Han and Mangasarian introduced a new constraint qualification which is a slight tightening of the well-known Mangasarian—Fromovitz constraint qualification. We show that this new qualification is a necessary and sufficient condition for the uniqueness of Kuhn—Tucker multipliers. We also show that it implies the satisfaction of second order necessary optimality conditions at a local minimum.     

Convex optimization and lagrange multipliers

The duality theorem of linear programming is used to prove several results on convex optimization. This is done without using separating hyerplane theorems.This work was supported in part by a grant from Investors in Business Education.     

Stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization

This paper introduces an algorithm for minimizing a single-variable locally Lipschitz function subject to a like function being nonpositive. The method combines polyhedral and quadratic approximation, a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point. The convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases. The algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms. Research sponsored, in part, by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-83-0210. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon.     

A finite procedure for determining if a quadratic form is bounded below on a closed polyhedral convex set

Mathematical Programming -     

Nonlinear programming methods in the presence of noise

The problem of minimizing a nonlinear function with nonlinear constraints when the values of the objective, the constraints and their gradients have errors, is studied. This noise may be due to the stochastic nature of the problem or to numerical error.Various previously proposed methods are reviewed. Generally, the minimization algorithms involve methods of subgradient optimization, with the constraints introduced through penalty, Lagrange, or extended Lagrange functions. Probabilistic convergence theorems are obtained. Finally, an algorithm to solve the general convex (nondifferentiable) programming problem with noise is proposed.Originally written for presentation at the 1976 Budapest Symposium on Mathematical Programming.     

Exact penalty functions and stability in locally Lipschitz programming

In this paper we extend the theory of exact penalty functions for nonlinear programs whose objective functions and equality and inequality constraints are locally Lipschitz; arbitrary simple constraints are also allowed. Assuming a weak stability condition, we show that for all sufficiently large penalty parameter values an isolated local minimum of the nonlinear program is also an isolated local minimum of the exact penalty function. A tight lower bound on the parameter value is provided when certain first order sufficiency conditions are satisfied. We apply these results to unify and extend some results for convex programming. Since several effective algorithms for solving nonlinear programs with differentiable functions rely on exact penalty functions, our results provide a framework for extending these algorithms to problems with locally Lipschitz functions.     

A smoothing-out technique for min—max optimization

In this paper, we suggest approximations for smoothing out the kinks caused by the presence of max or min operators in many non-smooth optimization problems. We concentrate on the continuous-discrete min—max optimization problem. The new approximations replace the original problem in some neighborhoods of the kink points. These neighborhoods can be made arbitrarily small, thus leaving the original objective function unchanged at almost every point ofR ⁿ . Furthermore, the maximal possible difference between the optimal values of the approximate problem and the original one, is determined a priori by fixing the value of a single parameter. The approximations introduced preserve properties such as convexity and continuous differentiability provided that each function composing the original problem has the same properties. This enables the use of efficient gradient techniques in the solution process. Some numerical examples are presented.     

Exact penalty functions in nonlinear programming

It is shown that the existence of a strict local minimum satisfying the constraint qualification of [16] or McCormick's [12] second order sufficient optimality condition implies the existence of a class of exact local penalty functions (that is ones with a finite value of the penalty parameter) for a nonlinear programming problem. A lower bound to the penalty parameter is given by a norm of the optimal Lagrange multipliers which is dual to the norm used in the penalty function.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS74-20584 A02.     

An elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions in nonlinear programming

In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets.     

Necessary and sufficient conditions for regularity of constraints in convex programming

This paper derives some necessary and sufficient conditions for (Lagrangian) regularity of the nondifferentiable convex programming problem. Furthermore, some weakest constraint qualifications are presented using the supporting functions and their derivatives, the outer normal cones, the single constraint function and its directional derivatives and epigraph and the projections of the outer normal cones     

Optimality condition: Constraint regularization

Necessity and sufficiency conditions are given for the existence of a finite set of redundant constraints which together with a given set of constraints in an optimization problem satisfy the Kuhn—Tucker, and the Guignard Constraint qualifications when the original set of constraints fails to satisfy either or both of them.This work was carried out at the Department of Statistics, University College of Wales, Aberystwyth while the author was sponsored by the Commonwealth Scholarship Commission in the United Kingdom, through the Department of Statistics, University of Nigeria, Nsukka.     

Solutions to the Navier–Stokes equations with mixed boundary conditions in two‐dimensional bounded domains

In this paper we consider the system of the non‐steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Fréchet derivative of at . We prove that is one‐to‐one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W^{2, 2}‐regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.