Nonzero lagrange multipliers

Solutions of constrained minimization problems give rise to Lagrange multiplier rules. In this paper, we show that a simple condition on a specific constraint implies that the associated coefficient in the Lagrange multiplier rule is not zero. We conclude with an example which shows that such knowledge increases the information available about the solution of a problem of minimal curvature.This work supported in part by NSF Grant No. MCS-75-05581-A01.     

Vector maximisation and lagrange multipliers

The Lagrangean function for scalar constrained optimisation problems is extended in a directly analogous manner to constrained vector optimisation problems. Some simple saddle point results are presented for vector maxima sets. Conditions are given for the characterisation of the vector maximum set of the original vector problem in terms of the vector maximum sets with respect to the vector Lagrangeans. Finally some attention is given to Lagrangean relaxation for vector optimisation problems as an extension of a result of Everett.     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.     

Implicit function theorems for mathematical programming and for systems of inequalities

Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn—Tucker theorem are motivated and rigorously demonstrated. The special case of a convex objective function with linear constraints is also treated with emphasis on computational details. An example, an application to chemical equililibrium problems, is given.Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.     

Existence theorems for lagrange control problems with unbounded time domain

Existence theorems are proved for usual Lagrange control systems, in which the time domain is unbounded. As usual in Lagrange problems, the cost functional is an improper integral, the state equation is a system of ordinary differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables. It is shown that the boundary conditions at infinity require a particular analysis. Problems of this form can be found in econometrics (e.g., infinite-horizon economic models) and operations research (e.g., search problems).The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.     

Quasi interiors,lagrange multipliers,andL p spectral estimation with lattice bounds

Lagrange multipliers useful in characterizations of solutions to spectral estimation problems are proved to exist in the absence of Slater's condition provided a new constraint involving the quasi-relative interior holds. We also discuss the quasi interior and its relation to other generalizations of the interior of a convex set and relationships between various constraint qualifications. Finally, we characterize solutions to theL ^p spectral estimation problem with the added constraint that the feasible vectors lie in a measurable strip [, ].The authors wish to thank Jonathan M. Borwein and Adrian S. Lewis for many enlightening discussions and useful suggestions. The duality approach to the general problem inL ^p was suggested by J. M. Borwein.     

Passage to lagrange multipliers for determination of the high-order constant for Pontryagin minimum

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, Sbornik Trudov VNIISI, No. 14, pp. 42–52, 1990.     

An alternative approach to understanding second-order conditions for constrained optimization—lagrange multipliers

This paper presents an alternative approach to solving a standard problem, frequently encountered in advanced microeconomics, using the technique of Lagrange multipliers. The objective is to enhance the understanding of students as to the derivation of the second-order conditions.     

Computable numerical bounds for lagrange multipliers of stationary points of non-convex differentiable non-linear programs

It is shown that the satisfaction of a standard constraint qualification of mathematical programming [5] at a stationary point of a non-convex differentiable non-linear program provides explicit numerical bounds for the set of all Lagrange multipliers associated with the stationary point. Solution of a single linear program gives a sharper bound together with an achievable bound on the 1-norm of the multipliers associated with the inequality constraints. The simplicity of obtaining these bounds contrasts sharply with the intractable NP-complete problem of computing an achievable upper bound on the p-norm of the multipliers associated with the equality constraints for integer $\text{p ≧ 1}$.     

Implicit multifunction theorems with positively homogeneous maps

We extend Robinson’s and Ledyaev and Zhu’s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps, allowing us to obtain results that more closely resemble the classical (single-valued) implicit function theorem. We highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results. As part of our analysis, we study perturbations of generalized linear openness and metric regularity. Finally, we discuss how our methods may also be adapted to study generalized calmness.     

Implicit multifunction theorems in complete metric spaces

In this paper, we establish some new characterizations of metric regularity of implicit multifunctions in complete metric spaces by using lower semicontinuous envelopes of the distance functions to set-valued mappings. Through these new characterizations it is possible to investigate implicit multifunction theorems based on coderivatives and on contingent derivatives as well as the perturbation stability of implicit multifunctions.     

On the influence of the critical lagrange multipliers on the convergence rate of the multiplier method

The paper is devoted to the analysis of the influence of the critical Lagrange multipliers on the convergence rate of the multiplier method and the efficiency of various techniques for accelerating the final stage of this method.     

Two-parameter Vilenkin multipliers and a square function

Two-parameter Vilenkin systems will be investigated. First we give a general sufficient condition for multipliers to be bounded between two-dimensional Hardy spaces H ^q(0<q1). By means of interpolation and duality argument, this theorem can be extended to other spaces. As a consequence, we can prove the (H ^q , L ^q)-boundedness of the Sunouchi operator U with respect to two-parameter Vilenkin systems for all 0 <q 1. Moreover, the equivalence f{H_q} ~ Uf_q (f H^q)follows for 1/2<q 1.     

Implicit multifunction theorems for the sensitivity analysis of variational conditions

We study implicit multifunctions (set-valued mappings) obtained from inclusions of the form 0∈M(p,x), whereM is a multifunction. Our basic implicit multifunction theorem provides an approximation for a generalized derivative of the implicit multifunction in terms of the derivative of the multifunctionM. Our primary focus is on three special cases of inclusions 0∈M(p,x) which represent different kinds of generalized variational inequalities, called “variational conditions”. Appropriate versions of our basic implicit multifunction theorem yield approximations for generalized derivatives of the solutions to each kind of variational condition. We characterize a well-known generalized Lipschitz property in terms of generalized derivatives, and use our implicit multifunction theorems to state sufficient conditions (and necessary in one case) for solutions of variational conditions to possess this Lipschitz, property. We apply our results to a general parameterized nonlinear programming problem, and derive a new second-order condition which guarantees that the stationary points associated with the Karush-Kuhn-Tucker conditions exhibit generalized Lipschitz continuity with respect to the parameter.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).