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Show The weight X has to be larger than the largest absolute value of the optimal Lagrange multipliers. The problem EPP is then solved by replacing f( x) and g( x) by their first order Taylor series approximation about a current solution x \ This leads to an approximation to the function P( x, X) termed PL( s; x, X), given as : PL ( s ; x , X ) « f( x') + 5f( x') * s + X ] T viol ( g( x') + 8g( x') * s) ( 3.17) em This approximation is valid for small step bounds s on each variable x and so an additional restriction is imposed: s < s < su ( 3.18) In addition the new point x + s is required to satisfy feasibility x1 < x + s < x u ( 3.19) These two restrictions can be combined to form Max ( s , s' - x') < s < Min ( s", xu - x ) ( 3.20) The region defined by this constraint is termed a trust region and it is assumed that in this region the linear function PL is a good approximation to P, the penalty function. The SLP algorithm then solves a succession of linear programs ( LP) with objective PL. The LP problem is then defined as: Minimize 6f( x') * s + X^ ( s + sn ) ( 3.21) p em Subject to: g1 - sn < g( x') + 6g( x') * s < g u + sp ( 3.22) Max ( s, x1 - x') < s < Min ( s", x" - x ' ) ( 3.23) sfi k 0 ( 3.24) sp £ 0 ( 3.25) where Sp, s n and s are decision variables and Sp, sn are deviation variables representing constraint violations. At most one of these variables can be positive in each constraint in an optimal LP solution. Sp represents the amount by which the upper bound of a constraint is violated and sn the amount by which the lower bound is violated. For Sp, sn equal to zero a feasible solution exists. The algorithm compares the changes in the exact penalty function P to its linear approximation PL to evaluate algorithm performance. A succession of LP's is then solved until one of four stopping criteria are satisfied. Figure 3.7 presents a flow chart delineating the major steps and stopping rules used in this algorithm. The SLP computer program of Lasdon and Kim ( 1983) was obtained and used for model applications. Computational experience with this algorithm is described in the section on model application. 52 |