notes2 - FIRST-ORDER OPTIMALITY CONDITIONS In this section,...

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In this section, we state frst-order necessary conditions For x to be a local minimizer and show how these conditions are satisfed on a small example. The prooF oF the result is presented in subsequent sections. As a preliminary to stating the necessary conditions, we defne the Lagrangian Function For the general problem (12.1). L ( x ) ± f ( x ) ± i E I λ i c i ( x ) . (12.33) (We had previously defned special cases oF this Function For the examples oF Section 1.) FIRST-ORDER OPTIMALITY CONDITIONS
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The necessary conditions defned in the Following theorem are called frst-order con- ditions because they are concerned with properties oF the gradients (frst-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 satisfed at ( x ) x L ( x ) ± 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 , For all i I , (12.34d)
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notes2 - FIRST-ORDER OPTIMALITY CONDITIONS In this section,...

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