Karush Lagrange function

Each of the inequality constraints gj(z) multiphed by what is called a Kuhn-Tucker multiplier is added to form the Lagrange function. The necessaiy conditions for optimality, called the Karush-Kuhn-Tucker conditions for inequality-constrained optimization problems, are... 

This section presents first the formulation and basic definitions of constrained nonlinear optimization problems and introduces the Lagrange function and the Lagrange multipliers along with their interpretation. Subsequently, the Fritz John first-order necessary optimality conditions are discussed as well as the need for first-order constraint qualifications. Finally, the necessary, sufficient Karush-Kuhn-Dicker conditions are introduced along with the saddle point necessary and sufficient optimality conditions. 

Definition 3.2.8 (Karush-Kuhn-Dicker saddle point) Let the Lagrange function of problem... 

Remark 1 Note that the saddle point sufficiency conditions do not require either additional convexity assumptions or a constraint qualification like condition. Note also that the saddle point sufficiency conditions do not require any differentiability on the Lagrange function. If in addition, the functions /(jc), h(x),g(x) are differentiable, and hence the Lagrange function is differentiable, and (3c, A,p) is a Karush-Kuhn-Tucker Saddle point, then it is a Karush-Kuhn-Tucker point [i.e., it is a solution of (3.3) and it satisfies the constraint qualification]. 

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