Lagrange multipliers Subject

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. 

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... 

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 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... 

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

The method of Lagrange Multipliers finds an extremum subject to some constraint on the variables. (Franklin, Methods of Advanced Calculus, 67, 1944 Wylie Barrett, Advanced Engineering Mathematics, 841, 1982). 

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... 

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schrodinger equation it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by... 

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... 

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 Schrodinger variational principle requires (T //1T) to be stationary subject to constant normalization ( ). Introducing a Lagrange multiplier, the variational condition is... 

The reference state T is a Slater determinant constructed as a normalized antisymmetrized product of N orthonormal spin-indexed orbital functions orbital energy functional E = To + Ec is to be made stationary, subject to the orbital orthonormality constraint (i j) = StJ, imposed by introducing a matrix of Lagrange multipliers. The variational condition is... 

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

Optimal Control. Optimal control is extension of the principles of parameter optimization to dynamic systems. In this case one wishes to optimize a scalar objective function, which may be a definite integral of some function of the state and control variables, subject to a constraint, namely a dynamic equation, such as Equation (1). The solution to this problem requires the use of time-varying Lagrange multipliers for a general objective function and state equation, an analytical solution is rarely forthcoming. However, a specific case of the optimal control problem does lend itself to analytical solution, namely a state equation described by Equation (1) and a quadratic objective function given by... 

The problem of maximizing the objective J subject to the above constraints can be transformed into an unconsbained problem by using Lagrange multipliers. According to this standard procedure, we multiply Eq. (4.11) by an unknown number X and Eq. (4.12) by an unknown two-component state vector ... 

In Eq. (16), yci stands for the number of segments of a chain in conformation c located in layer i. The two equations express the obvious conditions that each lattice layer must be occupied and that the total number of chains is constant. The Lagrange multiplier method is used to calculate the minimum free energy subject to the above constraints. By introducing the multipliers at, for each of the constraints given by Eq. (16), and (i for the constraint expressed by Eq. (17), one can write... 

With the method of Lagrange multipliers, it can be shown that is minimized subject to equation 140 provided for all k... 

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... 

Treatment of Raw Data. The method of Lagrange multipliers was used to normalize the experimental weight, sulfur, nickel, and vanadium data. A best fit of the experimental data subject to the constraints of 100% recoveries was obtained, using weighting factors determined by the analytical precision of each measurement. 

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... 

Lagrange multipliers Terms in a method used to find the maximum or minimum of a function that is subject to constraints. The function is written so that... 

The mathematical formulation leads to a constrained Gibbs energy minimization problem, subject to conservation of the total amounts of the individual chemical elements that make up the chemical species This constraint is incorporated into the problem via the method of Lagrange multipliers Details of the procedure are given elsewhere W ... 

The non-stoichiometric formulation, in which the stoichiometric equations are not used, instead the material balance constraints are treated by means of Lagrange multipliers. In these direct free energy minimization methods the problem is usually expressed as minimizing G, for fixed T and p, subject to the material balance constraint. 

We now impose not only a given overall strength but also a given envelope. The average local intensity is thus also specified. This requires the introduction of the set of N constraints (3.13), each of which is assigned the Lagrange multiplier yf. The distribution of maximal entropy subject to the three constraints (1)—(3)is... 

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