Lagrange multipliers constrained

Manaa MR, Yarkony DR (1993) On the intersection of two potential energy surfaces of the same symmetry. Systematic characterization using a lagrange multiplier constrained procedure. J Chem Phys 99 5251... 

Yarkony, D. R. (1993). Systematic determination of intersections of potential energy surfaces using a Lagrange multiplier constrained procedure. Journal of Physical Chemistry, 97(17), 4407-4412. 

Potential Energy Surfaces of the Same Symmetry. Systematic Characterization Using a Lagrange Multiplier Constrained Procedure. 

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 constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

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... 

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. 

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. 

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... 

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... 

For the sake of completeness we mention here an alternative definition of eigenvalue decomposition in terms of a constrained maximization problem which can be solved by the method of Lagrange multipliers ... 

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 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 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,... 

In constrained simulations, the Hamiltonian M is supplemented by a Lagrange multiplier... 

The best quality to be found may be a temperature, a temperature program or profile, a concentration, a conversion, a yield of preferred product, kind of reactor, size of reactor, daily production, profit or cost — a maximum or minimum of some of these factors. Examples of some of these cases are in this group of problems. When mathematical equations can be formulated, peaks or valleys are found by elementary mathematics or graphically. With several independent variables quite sophisticated mathematical procedures are available to find optima. Here a case of two variables occurs in problem P4.12.ll that is solved graphically. The application of Lagrange Multipliers for finding constrained optima is made in problem P4.ll.19. 

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

Figure 3.13 Constrained minimization the minimum of a function f x) submitted to the constraint g(x)=0 occurs at M on the constraint subspace, here on the curve (x)=0 where Vf(x)+XVg(x)=0. P is the unconstrained minimum of/( ). This principle is the base for the method of Lagrange multipliers.

PVVaik) is therefore the direction of constrained minimization. As in the case of Lagrange multipliers, no progress can be made and search will stop when the (k + l)th minimization direction PV (k+1) is orthogonal to the fcth minimization direction PV inner product of these vectors becomes less than an arbitrarily small value. 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

The augmented Lagrange multiplier algorithm finds the energy minimum of the constrained problem with an iterative, three-step procedure ... 

A similar constrained optimization problem has been solved in Section 2.5.4 by the method of Lagrange multipliers. Using the same method we look for the stationary point of the Lagrange function... 

Here n is the number of moles of the fcth component. The equilibrium composition is found as the set of nt values (k = 1,..., K) that minimizes the function G, with the constraints of the mass balance of the system. The problem of a constrained minimization can be solved by a number of methods [369] frequently the method of undetermined Lagrange multipliers is used [368]. 

Let us now return to the original problem of maximizing the entropy function (5.7) subject to the constraints (5.6b-d). With Lagrange multipliers Av, Ay, and AN, the constrained function S is... 

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