I am studying Andrew Ng's note on Support Vector Machines. On page 10 it showed the Karush-Kuhn-Tucker (KKT) conditions that $w^∗, α^∗\text{ and } β^∗$ should satisfy when $d^{*} = p^{*}$.
Later on page 12, It is written we should be able to verify the KKT conditions are holding true as it is shown in the red box image shown below.
I would appreciate if anyone explain more how should I verify 6$^{th}$ KKT condition in this case.
Thanks in advance.
See the paragraph at the bottom of page 9, which lays out what is known as the Slater condition for strong duality of a convex optimization problem. You need to have a convex objective to be minimized, affine functions in any equality constraints, convex functions in any (<=) inequality constraints, and there needs to be at least one vector x that is strictly feasible (that is g_i(x) < b_i for each constraint) with respect to the inequality constraints and feasible with respect to the equality constraints. If those conditions are satisfied then strong duality will hold.
The author is asking you to verify that the optimization problem stated at the start of section 6 satisfies these conditions. Is the objective a convex function? Are the g functions in the constraints (after rewriting them as <= inequalities) convex? Is there a strictly feasible solution?