Nó chỉ ra rằng điều kiện cần thiết của u ≥ 0 đảm bảo các gradient của chi phí và điểm hạn chế chức năng theo hướng ngược nhau. F cách này không thể giảm hơn nữa bằng cách bước theo hướng dốc tiêu cực mà không vi phạm các hạn chế. | It turns out that the necessary condition u 0 ensures that the gradients of the cost and the constraint functions point in opposite directions. This way f cannot be reduced any further by stepping in the negative gradient direction without violating the constraint. That is any further reduction in the cost function leads to leaving the feasible region at the candidate minimum point. This can be observed in Fig. 4-19. The necessary conditions for the equality and inequality constraints can be summed up in what are commonly known as the Karush-Kuhn-Tucker KKT first-order necessary conditions displayed in Theorem Theorem Karush-Kuhn-Tucker KKT Optimality Conditions Let x be a regular point of the feasible set that is a local minimum for f x subject to h x 0 i 1 to p gj x 0 j 1 to m. Then there exist Lagrange multipliers v a p-vector and u an m-vector such that the Lagrangian function is stationary with respect to xj Vị uj and sj at the point x . 1. Lagrangian Function p m L x v u s f x  Vihi x  uj gj x sj f x vTh x uT g x s2 i 1 j 1 2. Gradient Conditions dL dXk g- v h- u g- 0 k 1to n dXk i 1 dxk j 1 dxk 77 0 fi hi x 0 i 1 to p dVi 7 0 fi gj x sj 0 j 1 to m Wj 3. Feasibility Check for Inequalities sj 0 or equivalently gj 0 j 1 to m 4. Switching Conditions V 0 fi 2u sj 0 j 1to m dsj 5. Nonnegativity of Lagrange Multipliers for Inequalities u 0 j 1 to m 6. Regularity Check Gradients of active constraints should be linearly independent. In such a case the Lagrange multipliers for the constraints are unique. 130 INTRODUCTION TO OPTIMUM DESIGN It is important to understand the use KKT conditions to i check possible optimality of a given point and ii determine the candidate local minimum points. Note first from Eqs. to that the candidate minimum point must be feasible so we must check all the constraints to ensure their satisfaction. The gradient conditions of Eq. must also be satisfied simultaneously. These .
