Equality Lagrange multipliers

Iris type of constrained minimisation problem can be tackled using the method of Lagrange nultipliers. In this approach (see Section 1.10.5 for a brief introduction to Lagrange nultipliers) the derivative of the function to be minimised is added to the derivatives of he constraint(s) multiplied by a constant called a Lagrange multiplier. The sum is then et equal to zero. If the Lagrange multiplier for each of the orthonormality conditions is... 

The constraint force can be introduced into Newton s equations as a Lagrange multipli (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we wri F y as —A and so F Ar equals Am. Thus ... 

Ajt is the Lagrange multiplier and x represents one of the Cartesian coordinates two atoms. Applying Equation (7.58) to the above example, we would write dajdx = Xm and T y = Xdajdy = —X. If an atom is involved in a number of lints (because it is involved in more than one constrained bond) then the total lint force equals the sum of all such terms. The nature of the constraint for a bond in atoms i and j is ... 

Equality Constrained Problems—Lagrange Multipliers Form a scalar function, called the Lagrange func tion, by adding each of the equality constraints multiplied by an arbitrary iTuiltipher to the objective func tion. 

Conditions in Eq. (3-86), called complementaiy slackness conditions, state that either the constraint gj(z) = 0 and/or its corresponding multipher is zero. If constraint gj(z) is zero, it is behaving hke an equality constraint, and its multiplier pi is exactly the same as a Lagrange multiplier for an equality constraint. If the constraint is... 

The first term in this equation describes the suppression of the probability of the fluctuation with the correlator Eq. (3.22) (the weight />[//(a)] of the disorder configuration is exp (— J da/2 (x))), while the second term stems from the condition that the energy c+[t/(x)] of the lowest positive-energy single-electron state for the disorder realization t/(x) equals c. The factor /< is a Lagrange multiplier. It can be shown that the disorder fluctuation //(a) that minimizes A [//(a)] has the form of the soliton-anlisolilon pair configuration described by [48] ... 

Further Comments on General Programming.—This section will utilize ideas developed in linear programming. The use of Lagrange multipliers provides one method for solving constrained optimization problems in which the constraints are given as equalities. 

Alternatively p can be seen as a Lagrange multiplier introduced to solve the constrained problem minimize 0prior(x) subject to (/>ml(x) be equal to some... 

The above constrained parameter estimation problem becomes much more challenging if the location where the constraint must be satisfied, (xo,yo), is not known a priori. This situation arises naturally in the estimation of binary interaction parameters in cubic equations of state (see Chapter 14). Furthermore, the above development can be readily extended to several constraints by introducing an equal number of Lagrange multipliers. 

The foregoing inequality constraints must be converted to equality constraints before the operation begins, and this is done by introducing a slack variable q, for each. The several equations are then combined into a Lagrange function F, and this necessitates the introduction of a Lagrange multiplier, X, for each constraint. 

Thanks to the particular choice made for the NUP, taken equal to a superposition of spherical atoms, it is for the first time possible within the present approach to compute MaxEnt deformation maps in a straightforward manner. Once the Lagrange multipliers X have been obtained, the deformation density is simply... 

The total number of degrees of freedom (NDoF = Ncm + 2NAaDtli,) was 4439 this is also equal to the number of Lagrange multipliers. The constrained maximisation of the Bayesian score converged in less than 40 iterations sufficient memory and disk... 

In this example a and are the Lagrange multipliers, which can be determined from the expressions for the constraints, although it is not always necessary to do so. Clearly, this approach can be generalized and the number of multipliers is equal to the number of constraints on the system. 

In addition to providing optimal x values, both simplex and barrier solvers provide values of dual variables or Lagrange multipliers for each constraint. We discuss Lagrange multipliers at some length in Chapter 8, and the conclusions reached there, valid for nonlinear problems, must hold for linear programs as well. In Chapter 8 we show that the dual variable for a constraint is equal to the derivative of the optimal objective value with respect to the constraint limit or right-hand side. We illustrate this with examples in Section 7.8. 

The KTC are closely related to the classical Lagrange multiplier results for equality constrained problems. Form the Lagrangian... 

Define Lagrange multipliers A, associated with the equalities and Uj for the inequalities, and form the Lagrangian function... 

The Lagrange multipliers have vanished because there are equals numbers of A s and s in the numerator and denominator. The equation is identical to what we have obtained in Sections 3.1.3 for the quasi-chemical approximation. There are however two differences. First, we have another way to determine the 2-site probabilities. Instead of using eqn. (48) or (15) we can use the equation we get by substituting (47) in the restriction (25). Second, this way of determining the 2-site probabilities insures automatically that the solutions fulfill the sum rules. 

To cany out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled least squares estimator. For the combined sample, we obtain... 

Construct the Lagrange multiplier statistic for testing the hypothesis that all of the slopes (but not the constant term) equal zero in the binomial logit model. Prove that the Lagrange multiplier statistic is iiR2 in the regression of (y - P) on the xs, where P is the sample proportion of ones. 

The equilibrium order parameters X g and rjeq minimize AF subject to any system constraints. Supposing that the system s composition is fixed, the method of Lagrange multipliers leads to a common-tangent construction for AF with respect to XB—or equivalently, equality of chemical potentials of both A and B. Two compositions, Xjj and X +, will coexist at equilibrium for average compositions XB in the composition range Xe < XB < -X Bq+ if they satisfy... 

Here V(m ) is the probability distribution for the generalized mean size in the first phase, taken over partitions with fixed and N with equal a priori probabilities. Note that given m, irP is fixed in the second phase by the moment equivalent of particle conservation iV W1) + N mPl = Nm(° The integral in (17) can be replaced by the maximum of the integrand in the thermodynamic limit, because In V(m ) is an extensive quantity. Introducing a Lagrange multiplier pm for the above moment constraint then shows that the quantity pm has the same status as the density p = p0 itself Both are thermodynamic density variables. This reinforces the discussion in the introduction, where we showed that moment densities can be regarded as densities of quasi-species of particles. 

Recall that in the moment approach, each phase a is parameterized by Lagrange multipliers kf for the original moments (the ones appearing in the excess free energy of the system) and the fraction of system volume that it occupies. If extra moments are used, there is one additional Lagrange multiplier A, for each of them these are common to all phases. These parameters have to be chosen such that the pressure (44) and the moment chemical potentials nt given by Eq. (42) are equal in all phases. Furthermore, the (fractional) phase volumes v have to sum to one, and the lever rule has to be satisfied for all moments (both original and extra) ... 

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