Lagrange multipliers existence

Clearly as ARHS approaches zero the limit of this ratio does not exist the ratio approaches infinity because A OV remains —0.32. Hence the function OV (RHS) is not differentiable at RHS = 5.4, so no Lagrange multiplier exists at this point. 

Brezzi, F., 1974. On the existence, uniqueness and approximation of saddle point problems arising with Lagrange multipliers. RAIRO, Serie Rouge 8R-2, 129-151. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

The preceding results may be stated in algebraic terms. For V/ to lie within the cone described earlier, it must be a nonnegative linear combination of the negative gradients of the binding constraints that is, there must exist Lagrange multipliers u such that... 

Show that Lagrange multipliers do not exist for the following problem ... 

The existence of Lagrange multipliers depends on the form of the constraints and is not always guaranteed. To illustrate instances in which Lagrange multipliers may not have finite values (i.e., no existence), we will study problem (3.6), and we will assume that we have identified a candidate optimum point, x, which satisfies the equality constraints that is,... 

Thus, the Lagrange multiplier A cannot take a finite value (i.e., it does not exist). 

Let it be a local optimum of problem (3.3), the functions f(x),h(x),g(x) be twice continuously differentiable, and the second-order constraint qualification holds at x. If there exist Lagrange multipliers A, fi satisfying the KKT first-order necessary conditions ... 

Remark 3 If the primal problem (P) has an optimal solution and it is stable, then using the theorem of existence of optimal multipliers (see section 4.1.4), we have an alternative interpretation of the optimal solution (A, p) of the dual problem (D) that (A, p) are the optimal Lagrange multipliers of the primal problem (P). 

Constraints in optimization problems often exist in such a fashion that they cannot be eliminated explicitly—for example, nonlinear algebraic constraints involving transcendental functions such as exp(x). The Lagrange multiplier method can be used to eliminate constraints explicitly in multivariable optimization problems. Lagrange multipliers are also useful for studying the parametric sensitivity of the solution subject to the constraints. 

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

We seek extrema (maxima, minima, or saddle points) of/, subject to these two conditions. We shall show that there exist two constants, defined as a and ft (these two are known as the Lagrange multipliers), such that the system of n + 2 equations... 

When equality constraints or restrictions on certain variables exist in an optimization situation, a powerful analytical technique is the use of Lagrange multipliers. In many cases, the normal optimization procedure of setting the partial of the objective function with respect to each variable equal to zero and solving the resulting equations simultaneously becomes difficult or impossible mathematically. It may be much simpler to optimize by developing a Lagrange expression, which is then optimized in place of the real objective function. 

Thus we seek the conditional minimum. We will find it using the Lagrange multipliers method (see Appendix N available at booksite.elsevier.com/978-0-444-59436-5, p. el21). In this method, the equations of the constraints multiphed by the Lagrange multipliers are added (or subtracted, does not matter) to the original function that is to be minimized. Then we minimize the function as if the constraints did not exist. 

Last decades the non-extensive thermodynamics has being developed to describe the properties of systems where the thermodynamic limit conditions are violated. It is not a purpose of this paper to give one more review of non-extensive thermodynamics, its methods and formalism. The reader can found it in numerous papers, reviews and books (see, for example, (Abe Okamoto, 2001 Abe et al., 2007 Gell-Mann Tsallis, 2004 Tsallis, 2009)). Here we would like to underline only that the definition of temperature is very close related to the existence of thermal equilibrium and is very sensitive to the thermodynamic limit conditions, so a researcher should be veiy careful in the prescribing the meaning of temperature to a Lagrange multiplier when entropy maximum is looked for. 

From (3.40), 8 is a Hermitian matrix. It is always possible, therefore, to find a unitary matrix U such that the transformation (3.67) diagonalizes 8. We are not concerned with how to obtain such a matrix, only that such a matrix exists and is unique. There must exist, then, a set of spin orbitals for which the matrix of Lagrange multipliers is diagonal. 

If more than one restraint exists and actually limits the state of the system, there will be a corresponding number of additional Lagrange multipliers, each of which must satisfy the condition expressed in Eq. (52). The condition is unchanged when the existence of control constraints is taken into account. 

Finite systems, described by differential equations, require some sort of boundary conditions at the edges of the system for full specification. From our point of view, it is sufficient to recognize that they exist and to say that nonzero boundary conditions can be included in the form of Eq. (87) by means of a suitable delta function. We must also introduce a distributed (generalized) Lagrange multiplier or adjoint function, N (x, t). 

The siHailest value of for a given pressure difference iPi — Po) obviopsly exists when P has a tnajaihihit valhe. The method of Lagrange multipliers may be uwd for determining such a constrained maximum (34). This method indicates that the maximum value of F exists when... 

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