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Show 3.2.6 Solution Technique The formulation results in a moderately large constrained, nonlinear optimization problem. The model has N* W bounded decision variables ( e. g., 10 years with 50 wells leads to 500 decision variables). The number of linear constraints varies from ( K* N+ B) to ( K* N+ K) depending on which type of constraint ( equation 3.11, or 3.12) is used to describe the water rights constraint.- For the example above, if there are 10 water supply agencies, 50 agency boundary segments, the model has 110 to 150 linear constraints. In addition there are W+ C nonlinear constraints ( 70 for the example above, if there are 20 contaminated area boundary segments), and a nonlinear objective function. The penalty successive linear programming ( SLP) algorithm of Palacios- Gomez et al. ( 1982) was adopted for model solutions, since it was well suited for the model's largely linear characteristics. This algorithm solves nonlinear programming ( NLP) problems of the general form: Minimize f( x) Subject to j g £ g( x) < g Ax< b x '< x < xu ( 3.14) where x is an n x 1 vector of decision variables, defined on the real space [ x', xu], f( x) is a nonlinear objective function, g( x) is a set of m nonlinear constraint functions bounded by limits gl and gu, A is a k x n matrix of coefficients for a linear subset of constraints, and b is a k x 1 vector of right hand sides for the linear constraint subset. The successive linear programming ( SLP) algorithm of Palacios- Gomez et al. ( 1982) is described in the context of the formulation presented. The nonlinear programming ( NLP) model stated is solved by optimizing an exact penalty function formed as the sum of objective function plus a weighted sum of constraint violations. The penalty function is optimized subject to the bounds on the variables, and the k linear constraints. The exact penalty problem ( EPP) is stated as: Minimize P( x, X) « f( x) + XjjT viol( g( x)) m Subject to Ax < b x < x £ xu ( 3.15) where X is a sufficiently large weight, and 0 ifg< g( x)< gu viol( g( x))= - ^ ( g- g( x)) ifg( x)< g ( 3.16) ( g( x)- gU) ifg( x)< g° 51 |