Lagrange objective function

At any point where the functions h(z) are zero, the Lagrange func tion equals the objective function. 

In Appendix A, we follow the derivation of Shi and Rabitz and carry out the functional variation of the objective functional [Eq. (1)] so as to obtain the equations that must be obeyed by the wave function (vl/(t)), the undetermined Lagrange multiplier (x(0)> the electric field (e(t)). Since the results discussed in Section IV.B focus on controlled excitation of H2, where molecular polarizability must be considered, the penalty term given by Eq. (3) is used and the equations that must be obeyed by these functions are (see Appendix A for a detailed derivation) ... 

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 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... 

Find the minimum value of the objective function using Lagrange multipliers for the case in which K = cr — 2. 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

Therefore, the Lagrange multipliers A provide information on the sensitivity of the objective function with respect to the perturbation vector b at the optimum point 3c. 

In the definition of the Lagrange function L(x, A, m) (see section 3.2.2) we associated Lagrange multipliers with the equality and inequality constraints only. If, however, a Lagrange multiplier Mo is associated with the objective function as well, the definition of the weak Lagrange function L (x, A, fi) results that is,... 

Remark 2 In the Fritz John first-order necessary optimality conditions, the multiplier /x0 associated with the objective function can become zero at the considered point x without violating the optimality conditions. In such a case, the Lagrange function becomes independent of f(x) and the conditions are satisfied for any differentiable objective function f(x) whether it exhibits a local optimum at x or not. This weakness of the Fritz John conditions is illustrated in the following example. 

Remark 4 Note that the objective function in D(pk) corresponds to a Lagrange relaxation. 

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... 

A quadratic Taylor Series is used to approximate the Gibbs free energy as a function of composition (Equation (1)). Then, an unconstrained objective function, using Lagrange multipliers and mass balance constraints, is minimized to give improved compositions. ... 

The equations of constraint are divided into two groups. One set of constraints, referred to as substitution constraints, are used to eliminate selective dependent variables from the objective function, j. The other set, called Lagrange constraints are used directly in the... 

With the selection of the objective function and Lagrange constraints, the Lagrangian, L, is defined by... 

The problem must be set up such that the objective function and the Lagrange constraint equations are functions of the state and decision variables (Equations 4 and 5). A major deviation from the procedure outlined by Tribus and El-Sayed (5) is in the selection of the Lagrange constraint equations and state variables. The added complexity of having steam as the working fluid (compared to an ideal gas in the gas turbine optimization performed by Tribus and El-Sayed) makes it impractical to select state variables that correspond to available-energy flows. Consequently, this requirement was relaxed entirely. This gives the designer the opportunity to use any variable as a state variable,... 

The trade-off curve in the objective function space is non-convex as shown in Figure 3. Figure U shows the profiles of the sensitivities, Sj s (j=Ts,Ti,Ti), along the non-inferior solution curve. The sensitivity profile drawn in bold strokes shows the changes in the Lagrange multiplier. It is equal to the trade-off ratio along the non-inferior solution curve based on Equation (13). It is not continuous at point 3. 

When equality constraints or restrictions on certain variables exist in an optimization situation, a powerful analytical technique is the use of Lagrange multipliers. In many cases, the normal optimization procedure of setting the partial of the objective function with respect to each variable equal to zero and solving the resulting equations simultaneously becomes difficult or impossible mathematically. It may be much simpler to optimize by developing a Lagrange expression, which is then optimized in place of the real objective function. 

In applying this technique, the Lagrange expression is defined as the real function to be optimized (i.e., the objective function) plus the product of the Lagrangian multiplier (A) and the constraint. The number of Lagrangian multipliers must equal the number of constraints, and the constraint is in the form of an equation set equal to zero. To illustrate the application, consider the situation in which the aim is to find the positive value of variables X and y which make the product xy a maximum under the constraint that x2 + y2 = 10. For this simple case, the objective function is xy and the constraining equation, set equal to zero, is x1 + y2 - 10 = 0. Thus, the Lagrange expression is... 

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

Sufficiency conditions to assure that a Kuhn-Tucker point is a local minimum point require one to prove that the objective function will increase for any feasible move away from such a point. To cany out such a test, one has to generate the matrix of second derivatives of the Lagrange function with respect to all the variables z evaluated at z. The test is seldom done, as it requires too much work. 

For this simple ease, the objective function is xy and the constraining equation, set equal to zero, is + y 10 = 0. Thus, the Lagrange expression is... 

The problem may be seen as that of minimizing an objective function, subject to a set of nonlinear constraints. It may be solved by Lagrange s method... 

In the OCT formulation, the TDSE written as a 2 x 2 matrix in a BO basis set, equation tAl. 6.72). is introduced into the objective functional with a Lagrange multiplier, j x, t) [54]. The modified objective functional may now be written as... 

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