What are the KKT conditions?

Consider an optimization problem in standard form: with the variable . Assume that the ‘s and ‘s are differentiable. (At this point, we are not assuming anything about their convexity.) As before, define the Lagrangian as the function Let and be the primal and dual optimal points respectively (i.e. points where the primal and dual […]

What are the KKT conditions?

Lagrange dual, weak duality and strong duality

Consider an optimization problem in standard form: with the variable . Let be the domain for , i.e. the intersection of the domains of the ‘s and the ‘s. Let denote the optimal value of the problem. The Lagrange dual function is the function defined as the minimum value of the Lagrangian over : The […]

Lagrange dual, weak duality and strong duality

The Lagrange dual function is always concave

Consider an optimization problem in standard form: with the variable . Let be the domain for , i.e. the intersection of the domains of the ‘s and the ‘s. The Lagrangian associated with this problem is the function defined as with domain . The Lagrange dual function is the function defined as the minimum value […]

The Lagrange dual function is always concave