Slater's Condition and Strong Duality

Many optimization problems in portfolio construction — minimizing risk subject to constraints on turnover, sector exposure, or leverage — can be attacked two ways: solve the original ("primal") problem directly, or solve a related ("dual") problem built from the constraints, which is often easier. In general, the dual problem's best value is only a lower bound on the primal's ("weak duality"); it can genuinely fall short. Strong duality is the stronger property that the two values are exactly equal, which makes the dual a fully reliable stand-in for the primal.

Slater's condition is a simple sufficient check for strong duality in convex problems: if there exists at least one point that satisfies every inequality constraint strictly — with some room to spare, not sitting exactly on a boundary — then strong duality is guaranteed to hold. It's a mild requirement in practice: most realistic portfolio constraints (a leverage cap, a sector limit) leave room for at least some strictly feasible portfolio, so Slater's condition is usually satisfied without special effort.

Worked example

Minimizing portfolio variance subject to a leverage constraint $\sum |w_i| \le 2$ is a convex problem. The all-cash portfolio ($w = 0$) satisfies the constraint strictly ($0 < 2$), so Slater's condition holds and strong duality is guaranteed — solving the dual problem (often faster or with a cleaner structure) gives exactly the same optimal variance as solving the primal problem directly, with zero gap to worry about.