{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 ,
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}