Lagrange Multiplier Theorem Research Articles

Stress as a Constraint Reaction in Rigid Bodies

The unique construction of stress as a constraint reaction in a rigid body loaded on its boundary in a state of equilibrium is described through the use of the elementary variational problem of minimum potential energy. We suppose that the body is naturally constrained to be strain free and this leads to a non-trivial constrained minimization problem. Our construction is based on an application of a Riesz-like representation theorem within the context of the classical Lagrange multiplier theorem of variational calculus.

Super efficiency in vector optimization with nearly convexlike set-valued maps

In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency.

The Lagrange multipliers and hyperstress constraint reactions in incompressible multipolar elasticity theory

Multipolar elasticity is a pseudo spatial-dependent theory which allows for higher deformation gradients to affect the value of the stored energy function and, in particular, it introduces the concept of multipolar traction as a source of contact interaction between adjacent surfaces in material bodies and at their boundaries. As a result of the presence of higher deformation gradients in the constitutive structure, the mathematical set-up for such a theory of material behavior generally requires deformation fields to lie in a Sobolev space W n,p( B, R 3) , where n is the order of the highest deformation gradient that is present. The standard such norm is nonuniformly scale dependent because of the presence of higher gradients and so we introduce an equivalent weighted norm which, for n=2, requires the introduction of a single intrinsic length scale l. For this case, we then study the necessary first variation condition for a sufficiently smooth minimizer y ∗∈W n,p( B, R 3) in a sufficiently smooth domain and prove a related Lagrange multiplier theorem in Theorem 4.1. This theorem depends upon the validity of a “Riesz-like” representation theorem for a continuous linear functional on W 1,p( B, R) , which we consider in Lemma 4.1. The strange nature of the space dual to W 1,p( B, R) requires certain special technical considerations and these lead us to propose a natural scheme for constructing the unique Lagrange multiplier fields. Finally, in the last section of this work, we solve an elementary example problem and apply our earlier conclusions on existence and uniqueness to uniquely determine the constraint reaction stress and hyperstress fields. We show how these fields depend upon the length scale l.

Comments on "Controller design with multiple objectives"

We point out that the proof of one of the theorems presented in the above paper by Elia et al. (see ibid. vol.42 (1997)) is invalid and provide a correct proof. The flawed proof is long and complex, and incorrectly tries to make use of duality results via a version of the Lagrange multiplier theorem. The proof we presented, based on an elementary convexity argument, is very simple.

Lagrange Duality in Multiobjective Fractional Programming Problems with n-Set Functions

In this paper, Lagrange multiplier theorems are developed for the multiobjective fractional programming problem involving n-set functions. A Lagrange dual is introduced and duality results in terms of efficient solutions are established.

Optimality Conditions for State-Constrained Dirichlet Boundary Control Problems

In this paper, we prove that the combination of representation theorems for additive measures introduced in Ref. 1, together with a Lagrange multiplier theorem, leads to a short and direct proof of the optimality conditions for Dirichlet control problems with pointwise state constraints.

Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained

The continuation inverse problem revisited

SUMMARY The non-uniqueness of the continuation of a finite collection of harmonic potential field data to a level surface in the source-free region forces its treatment as an inverse problem. A formalism is proposed for the construction of continuation functions which are extremal by various measures. The problem is cast in such a form that the inverse problem solution is the potential function on the lowest horizontal surface above all sources, serving as the boundary function for the Dirichlet problem in the upper halfplane. The desired continuation, at the higher level of interest, must then be in the range of the upward continuation operator acting on this boundary function, rather than being allowed the full freedom of itself being part of a Dirichlet problem boundary function. Extremal solutions minimize non-linear functionals of the continuation function, which are re-expressed as diVerent functionals of the boundary function. A crux of the method is that there is no essential distinction between the upward and downward continuation inverse problems to levels above or below data locations. Casting the optimization as a Lagrange multiplier problem leads to an integral equation for the boundary function, which is readily solved in the Fourier domain for a certain class of functionals. The desired extremal continuation is then given by upward continuation. It is found that for some functionals, application of the Lagrange multiplier theorem requires a further restriction on the set of allowable boundary functions: bandlimitedness is a natural choice for the continuation problem. With this imposition, the theory is developed in detail for semi-norm functionals penalizing departure from a constant potential, in the 2-norm and Sobelev norm senses, and illustrated by application for a small synthetic Deep Tow magnetic field data set.

A lagrange multiplier theorem for control problems with state constraints

We prove an existence theorem of Lagrange multipliers for an abstract control problem in Banach spaces. This theorem may be applied to obtain optimality conditions for control problems governed by partial differential equations in the presence of pointwise state constraints.

Regularity Properties of the Phase for Multivariable Systems

For multivariable, input-output systems that are represented as rational, transfer-function matrices, the most frequently used measures of relative stability are gain based. However, there are a number of important physical applications where the phase of a perturbation can also have a significant effect on relative stability. Such applications led J. R. Bar-on and E. A. Jonckheere [J. R. Bar-on, {Phase and Gain Margins for Multivariable Control Systems, Ph.D. thesis, University of Southern California, 1990; J. R. Bar-on and E. A. Jonckheere, Internat. J. Control, 52 (1990), pp. 485--498] to define precisely the notions of phase, minimum-phase mapping, and phase margin for multivariable systems. The objective of this paper is to establish conditions under which the phase and minimum-phase mappings have certain desired regularity properties (e.g., continuity or differentiability). After a review of the definitions of the phase concepts under consideration, we collect a few well-known results about set-valued maps that have direct applications to parametrized families of constrained optimization problems. Using these results we show that, under very mild conditions, the minimum-phase mapping is lower semicontinuous as a function of frequency; as a consequence, the phase margin (initially defined as the infimum of the phase of all destabilizing unitary perturbations in the range of frequencies where destabilizing perturbations can occur) is achieved as the phase of a specific destabilizing unitary perturbation. We then establish sufficient conditions of gradually increasing strength for the minimum-phase mapping to be continuous and real analytic as a function of frequency. The proof of the real analyticity of the minimum-phase mapping relies on the implicit function theorem and the Lagrange multiplier theorem.

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.

Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints

We consider optimal control problems governed by semilinear par- abolic equations with nonlinear boundary conditions and pointwise constraints on the state variable. In Robin boundary conditions considered here, the nonlinear term is neither necessarily monotone nor Lipschitz with respect to the state variable. We derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces. The adjoint state must satisfy a parabolic equation with Radon measures in Robin boundary conditions, in the terminal condition and in the distributed term. We give a precise meaning to the adjoint equation with measures as data and we prove the existence of a unique weak solution for this equation in an appropriate space.

Lagrange multipliers saddle points and scalarizations in composite multiobjective nonsmooth programming

A Lagrange multiplier theorem is established for a nonsmooth constrained multiobjective optimizatioi problems where the objective function and the constraints are compositions of F-invex functions, anc locally Lipschitz and Gateaux differentiable functions. Furthermore, a vector valued Lagrangian is intro duced and vector valued saddle point results are presented. A scalarization result and a characterizatior of the set of all conditionally properly efficient solutions for V-invex composite problems are alsc discussed under appropriate conditions.

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.

Material tailoring of structures to achieve a minimum radiation efficiency

A strategy is developed for designing structures that radiate sound inefficiently in light fluids. The problem is broken into two steps. First, given a frequency and overall geometry of the structure, a surface velocity distribution is found that produces a minimum radiation efficiency. This particular velocity distribution is referred to as the ‘‘weak radiator’’ velocity profile. A finite element adaptation of the integral wave equation is combined with the Lagrange multiplier theorem to obtain this surface velocity distribution. Second, a distribution of Young’s modulus and density distribution is found for the structure such that it exhibits the weak radiator velocity profile as one of its mode shapes. Extensive use of structural finite element modeling as well as linear programming techniques is made to find this distribution. The result is a weak radiator structure. When compared to a structure with uniform material properties, the weak radiator structural response is found to exhibit lower wave‐numb...

Smallest Nonnegative Solutions to Linear Inverse Problems

Physical considerations often dictate nonnegativity for solutions to linear geophysical inverse problems. Applying the Lagrange multiplier theorem, the nature of that nonnegative solution is deduced, which is smallest in the 2-norm sense. The two-data inverse problem of inference of a planetary density profile, from measurements of mass and moment-of-inertia, serves to show the existence of nonextremal, nonnegative solutions that also satisfy the Lagrange multiplier condition. Numerical exploration indicates which of this family is the true extremal solution.

Material tailoring of structures to achieve a minimum radiation condition

A strategy is developed for designing structures that radiate sound inefficiently in light fluids. The problem is broken into two steps. First, given a frequency and overall geometry of the structure, a surface velocity distribution is found that produces a minimum radiation condition. This particular velocity distribution is referred to as the ‘‘weak radiator’’ velocity profile. Second, a distribution of Young’s modulus and density distribution is found for the structure such that it exhibits the weak radiator velocity profile as one of its mode shapes. In the first step, a finite element adaptation of the integral wave equation is combined with the Lagrange multiplier theorem to obtain a surface velocity distribution that minimizes the radiated sound power. In the second step, extensive use of structural finite element modeling as well as linear programming techniques is made. The result is a weak radiator structure. When compared to a structure with uniform material properties, the weak radiator structural response is found to exhibit three important characteristics. First, the weak radiator structure shows lower vibration amplitude near its boundaries. Second, the weak radiator structure exhibits lower wave number content in the supersonic region. Third, the distribution of surface acoustic intensity on the weak radiator structure is very small at all the points along its surface. The effect of modal overlap on the performance of the weak radiator structures is found to be negligible. While the method employed here is general, the example of a simple beam radiating in a rigid baffle is used for the purpose of illustration.

A hahn-banach theorem in ordered linear spaces and its applications

In this paper, a weak version of Hahn-Banach theorem is derived for convex relations in ordered linear spaces in which a separation theorem of convex sets in [11] is used. Then this result is applied to derive existence of weak Lagrange multiplier theorem for the following vector optimization problem: where F,H are convex relations from a linear space X to an ordered linear space Y.We also prove existence of weak subgradient for convex relations by using Hahn-Banach theorem. A partial extension of the subgradient formula in [2] to ordered space is obtained

A design method for achieving weak radiator structures using active vibration control.

A general strategy is devised for achieving minimum radiation of sound from structures subjected to a harmonic excitation force. A quadratic expression is written for the total sound power radiated from a structure in terms of the primary excitation and actuator (control) forces. Using the Lagrange multiplier theorem and the penalty approach, a single control force vector is found which results in minimum radiated sound power (weak radiator). A general formation is represented for the single frequency excitation case. This formulation is then extended to the case of a structure subjected to a broadband excitation force. The numerical implementation of these formulations is discussed for a baffled beam. Application of this formulation is demonstrated for shaker actuators as well as PZT surface actuators. Results for a single frequency weak radiator show that when the response of the structure with control is compared to that of the structure without control, four important observations can be made. First, the controlled structure exhibits lower response amplitudes at frequencies near structural resonances. Second, the controlled structural response amplitude decreases as near to the boundaries. Third, the wave-number content of the controlled structural response shifts from the supersonic to subsonic region. Fourth, the surface acoustic pressure and velocity of the controlled structure are nearly in quadrature. Broadband weak radiators are also achieved for a 1000-Hz bandwidth. Improvements of up to 40 dB are achieved.

Lagrange multiplier theorem of multiobjective programming problems with set functions

In this paper, Lagrange multiplier theorems are developed for the cases of single-objective and multiobjective programming problems with set functions. Properly efficient solutions are also characterized by subdifferentials and zero-like functions.