saddlepoint

A saddlepoint in mathematics is a point in the domain of a function of two or more variables where the gradient is zero but the point is not an extremum. At a saddlepoint, the function increases in some directions and decreases in others, so the point is neither a local minimum nor a local maximum.

In several variables, a common diagnostic uses the Hessian matrix, which collects second partial derivatives. For a function f(x, y), let D = f_xx f_yy − (f_xy)^2. If D > 0 and f_xx > 0, the point is a local minimum; if D > 0 and f_xx < 0, it is a local maximum. If D < 0, the point is a saddlepoint. If D = 0, the test is inconclusive. A classic example is f(x, y) = x^2 − y^2, which has a saddlepoint at (0, 0): f increases with x and decreases with y.

Saddlepoints appear in various contexts beyond elementary calculus. In optimization, they are critical points that are not extrema and can complicate convergence for gradient-based methods. In game theory, a saddle point of a payoff function corresponds to a Nash equilibrium in a two-player zero-sum game; it satisfies f(x*, y) ≥ f(x*, y*) ≥ f(x, y*) for all x and y. In such settings, the point (x*, y*) represents stability against unilateral deviations by either player.

In statistics, the term also appears in saddlepoint approximations, a family of techniques for approximating probability distributions and tail probabilities using the saddlepoint of a cumulant generating function, often providing accurate results with small samples.

Overall, saddlepoints describe critical points with mixed curvature and arise in optimization, game theory, and statistical approximation.