However, all curvature superfields and

Lagrangian multipliers exhibit the same covariant BRST transformation: s(x) = -[[C.sup.ab], (x)], where [C.sub.ab] is a zero-form superfield ghost associated with [[OMEGA].sup.ab].

Let v denote the equilibrium link flow pattern of the addressed problem if and only if there exists a vector of

Lagrangian multipliers corresponding to environmental constraints, denoted by [u.sup.*] = [([u.sup.*.sub.e], e [member of] E).sup.T].

Each iteration includes four steps: first, convert the optimal problem into a dual problem and find the dual solution of the dual problem; secondly, update the

Lagrangian multipliers by the subgradient algorithm; thirdly, based on the updated

Lagrangian multipliers , find the feasible solution for the primal optimal problem to obtain the upper bound; fourthly, check whether the duality gap between the feasible solution and the dual solutions achieve some value or the maximum iteration time arrives.

Denoting {[[lambda].sub.t](i), [[pi].sub.t](i), [[mu].sub.t](i), [[phi].sub.t](i)} as the

Lagrangian multipliers of constraints (A.2) through (A.5), respectively, the firm's first-order conditions for {[i.sub.t](i), [k.sub.t + 1](i), [b.sub.t + 1](i)} are given, respectively, by

In the equation, p, [gamma], [lambda] = {[[lambda].sub.1], [[lambda].sub.2]} stands for

Lagrangian multipliers and constant [beta] is the penalty factor that depends on mesh size h.

Given an initial [[lambda].sup.0], the kth iteration of the method of the

Lagrangian multipliers for solving (2.1) has the form

(2), (3) and (4) are the three possible constraints, which can be relaxed using the respective

Lagrangian multipliers. The approach to relax the capacity constraint (4) has been tried by many researchers in the past, viz.

Elimination of

Lagrangian Multipliers. Mechanics Research Communications, Vol.

with [[lambda].sub.i] and [[micro].sub.j] being the appropriate

LaGrangian multipliers.

The

Lagrangian multipliers [Mu] and [Alpha] can now be found from condition (6) and (7) respectively:

That is, shadow input prices are theoretically equal to (1) Mathematical Expression Omitted] where [[Mu].sub.i] and endogenous

Lagrangian multipliers.(1) Changing all shadow input prices proportionately while holding observed input prices constant is impossible.

where [lambda.sub.31] and [lambda.sub.32] are

Lagrangian multipliers. Partial differentiation with respect to [R.sub.11] , [R.sub.12] , [R.sub.22] , [R.sub.23] , [L.sub.12] , and [L.sub.22] yields the first-order conditions.