Lagrange multipliers, constraints

The set of equations in the previous section 4 is solved numerically using the NIM (natural iteration method). When we use the constraint Lagrange multipliers, the NIM consists of the "Major Iteration" and the "Minor Iteration". The major iteration solves the set of basic of equations in (11), the reduction relations in (2) and the normalization relation. It was proved that this iteration always converges from whatever initial guess values it starts with. 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

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. 

Lagrange multipliers are often referred to as shadow prices, adjoint variables, or dual variables, depending on the context. Suppose the variables are at an optimum point for the problem. Perturb the variables such that only constraint hj changes. We can write... 

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... 

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

Here a denotes the set of constraints that directly involve r and the /( are the Lagrange multipliers introduced into the problem. 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

This is an example of a constrained optimization, the energy should be minimized under the constraint that the total Cl wave function is normalized. Introducing a Lagrange multiplier (Section 14.6), this can be written as... 

Using a Lagrange multiplier to take care of the constraint... 

We next find a Pf(y) that maximizes Eq. (4-189) subject to the constraints of Eqs. (4-186) and (4-187). Using the method of Lagrange multipliers, we find a stationary point with respect to Pf y) of he function... 

The constants Aj and A2 are known as Lagrange multipliers. As we have already seen two of the variables can be expressed as functions of the third variable hence, for example, dxx and dx2 can be expressed in terms of dx3, which is arbitrary. Thus Ax and A2 may be chosen so as to cause the vanishing of the coefficients of dxx and dx2 (their values are obtained by solving the two simultaneous equations). Then since dx3 is arbitrary, its coefficient must vanish in order that the entire expression shall vanish. This gives three equations that, together with the two constraint equations gt = 0 ( = 1,2), can be used to determine the five unknowns xx, x2, Xg, Xx, and A2. 

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. 

Now, consider independently small changes in e(r), / and x- These variations are now allowed to be independent because of the use of the Lagrange multiplier, x(0 to impose the constraint equation. 

The point where the constraint is satisfied, (x0,yo), may or may not belong to the data set (xj,yj) i=l,...,N. The above constrained minimization problem can be transformed into an unconstrained one by introducing the Lagrange multiplier, to and augmenting the least squares objective function to form the La-grangian,... 

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 problem of minimizing Equation 14.24 subject to the constraint given by Equation 14.26 or 14.28 is transformed into an unconstrained one by introducing the Lagrange multiplier, to, and augmenting the LS objective function, SLS(k), to yield... 

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. 

With this choice of constraint functions and Lagrange multipliers, we can rewrite formula (6) and express the MaxEnt distribution of electrons as... 

Minimising this expression subject to the constraint tp+ + tp + 4+ + 4- = T using the Lagrange multipliers, one obtains the optimal counting-time proportions ... 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

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