KKT condition

Consider the general optimization problem

Suppose that the object function and the constraint functions and are continuously differentiable at a point If is a local optimum and optimization problem satisfies some conditions, then there exist constant and , called KKT multipliers, such that

KKT condition

• Stationarity:
  
• Primal feasibility:
  
• Dual feasibility:
  
  Complementary slackness:
  

Note:

• KKT condition is a necessary condition.
• KKT condition is necessary and sufficient condition under Slater condition.
• Slater condition: For a convex problem(i.e. are convex and is affine), there exists a point such that and

KKT.png

Lagrange function