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In this section, we state first-order necessary conditions for x to be a local minimizer and show how these conditions are satisfied on a small example. The proof of the result is presented in subsequent sections. Asa preliminarytostatingthenecessaryconditions, wedefinetheLagrangianfunction for the general problem (12.1). L ( x , λ ) f ( x ) i E I λ i c i ( x ) . (12.33) (We had previously defined special cases of this function for the examples of Section 1.) FIRST-ORDER OPTIMALITY CONDITIONS
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The necessary conditions defined in the following theorem are called first-order con- ditions because they are concerned with properties of the gradients (first-derivative vectors) of the objective and constraint functions. These conditions are the foundation for many of the algorithms described in the remaining chapters of the book. Theorem 12.1 (First-Order Necessary Conditions). Suppose that x is a local solution of (12.1), that the functions f and c i in (12.1) are continuously differentiable, and that the LICQ holds at x . Then there is a Lagrange multiplier vector λ , with components λ i , i E I , such that the following conditions are satisfied at ( x , λ ) x L ( x , λ ) 0 , (12.34a) c i ( x ) 0 , for all i E , (12.34b) c i ( x ) 0 , for all i I , (12.34c) λ i 0 ,
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