Lagrange problem solving

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. 

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. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

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... 

There is a standard mathematical tool for solving the problem of maximizing a quantity that has to satisfy constraints, namely the method of Lagrange undetermined... 

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 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. 

Use the method of Lagrange multipliers to solve the following problem. Find the values of jcj, x2, and a> that... 

Solve the following problem via the Lagrange multiplier method ... 

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]. 

The basic idea in OA/ER is to relax the nonlinear equality constraints into inequalities and subsequently apply the OA algorithm. The relaxation of the nonlinear equalities is based upon the sign of the Lagrange multipliers associated with them when the primal (problem (6.21) with fixed y) is solved. If a multiplier A is positive then the corresponding nonlinear equality hi(x) = 0 is relaxed as hi x) <0. If a multiplier A, is negative, then the nonlinear equality is relaxed as -h (jc) < 0. If, however, A = 0, then the associated nonlinear equality constraint is written as 0 ht(x) = 0, which implies that we can eliminate from consideration this constraint. Having transformed the nonlinear equalities into inequalities, in the sequel we formulate the master problem based on the principles of the OA approach discussed in section 6.4. 

Step 3 Solve the primal problem P(y1) [i.e., problem (6.35)] to find the upper bound UBD — P(yJ) as well as the Lagrange multipliers. 

Alternative (ii) Add to the relaxed master problem the linearizations around the infeasible continuous point. Note though that to treat the relaxed master problem we need to have information on the Lagrange multipliers. To obtain such information, a feasibility problem needs to be solved and Viswanathan and Grossmann (1990) suggested one formulation of the feasibility problem that is,... 

Step 4 Solve a relaxed master problem. We distinguish two cases, the relaxed primal master and the relaxed Lagrange relaxation master problem ... 

Execution times for the overall ammonia plant model, of which the C02 capture system is a small part, are on the order of 30 s for the parameter estimation case, and less than a minute for an Optimize case. The model consists of over 65,000 variables, 60,000 equations, and over 300,000 nonzero Jacobian elements (partial derivates of the equation residuals with respect to variables). This problem size is moderate for RTO applications since problems over four times as large have been deployed on many occasions. Residuals are solved to quite tight tolerances, with the tolerance for the worst scaled residual set at approximately 1.0 x 10 9 or less. A scaled residual is the residual equation imbalance times its Lagrange multiplier, a measure of its importance. Tight tolerances are required to assure that all equations (residuals) are solved well, even when they involve, for instance, very small but important numbers such as electrolyte molar balances. 

Trade-Off Between Two Objectives. By increasing e a family of e-constraint problems is solved successively using the max-sensitive method ( ). Using the usual e-constraint method the problems are solved separately for the various values of e. However with this method the optimal solution for the new value of e is pursued into the neighbourhood of the optimal solution for the last value by examining the sensitivities for all parameters. Incidentally, the Lagrange multiplier for the e-constraint is obtained from among the sensitivities for the parameters. ... 

PROBLEM 2.6.1. Solve the simple harmonic motion problem by using (i) Lagrange s and (ii) Hamilton s equations of motion. 

The constant p, the chemical potential, is a Lagrange parameter that is introduced to ensure proper normalization, as in Hartree-Fock theory. At this stage, Kohn and Sham noted that Eq. (3.36) is the Euler equation for noninteracting electrons in the external potential V ff. Thus, finding the total energy and the density of the system of electrons subject to the external potential V is equivalent to finding these quantities for a noninteracting system in the potential Veir- Such a problem can in principle be solved exactly, but we have to know E and the potential V c-The one-particle problem can be solved as ... 

The Lagrangian formalism is widely applied to solve mechanical problems. But besides Lagrange s formalism, there is a formalism first developed by Hamilton. Sometimes the Hamiltonian formalism presents certain advantages in solving mechanical problems. But the real power... 

Stationary points of the functional [c] should be calculated through variation of the coefficients c. Kohn s variational principle requires the wave function on dS to remain fixed during the variation, 6fa = 0. In view of Eq. (27), this means that variation of the Ck is subject to the additional condition Y.k Tak Sck — 0. The standard way to solve a variational problem with constraints is to use undetermined Lagrange multipliers [234]. A technical realization of this method, which we do not describe here, is given in Ref. 60. Using it, one obtains a compact expression for a set of coefficients c which render [c] stationary, namely... 

Two other problems are solved by the preceding analysis. The first is the Lagrange multiplier formulation of the problem in which the maximum of c — c/ — would be sought. Here the process should cease as soon as the value of the reaction rate at the exit drops to X that is, Ce is given by... 

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