Affine hull vs. convex hull

The affine hull and convex hull are closely related concepts. Let be a set in . The affine hull of is the set of all affine combinations of elements of : The convex hull of is the set of all convex combinations of the elements of : Putting the definitions side by side, we see […]

Affine hull vs. convex hull

Exact line search and backtracking line search

Assume we are trying to minimize some convex function which is differentiable. One way to do this is to use a descent method. A general descent method can be described as follows: Given a starting point in the domain of , iterate over the following 3 steps until some stopping criterion is satisfied: Choose a […]

Exact line search and backtracking line search

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?