Optimality Conditions and Sensitivity
Complementary slacknessWe consider a primal convex optimization problem (without equality constraints for simplicity):
We assume that the duality gap is zero: , and that both primal and dual values are attained, by a primal-dual pair . (This is guaranteed for example if both the primal and dual are strictly feasible, in the weak sense of Slater’s.) We have
This implies that minimizes the function :
KKT optimality conditionsAssume that the functions are differentiable. Consider the so-called KKT (The acronym comes from the names Karush, Kuhn and Tucker, researchers in optimization around 1940-1960) conditions on a primal-dual pair .
If, in addition the problem is convex, then the conditions are also sufficient. To see this, note that the first two conditions imply that are feasible for the primal and dual problems, respectively. Since is convex, the fourth condition (which we called Lagrangian stationarity) states that is a minimizer of , hence
Primal solutions from dual variablesAssume that the problem has a zero duality gap, with dual values attained. Now assume that is optimal for the dual problem, and assume that the minimization problem
This allows to compute the primal solution when a dual solution is known, by solving the above problem. ExamplesXXX |