Satulapiste

Satulapiste, known in English as a saddle point, is a concept in calculus and optimization describing a stationary point that is not a local extremum. At a saddle point, the function has directions of increase and decrease: it rises along some paths and falls along others.

In the standard setting of a twice-differentiable function f: R^n -> R, a point x* where the gradient is zero (the critical point) can be classified using the Hessian matrix H of second derivatives at x*. If H is positive definite, x* is a local minimum; if H is negative definite, x* is a local maximum. If H is indefinite (it has both positive and negative eigenvalues), x* is a non-degenerate saddle point. If H is semidefinite or the test is inconclusive, higher-order derivatives or alternative methods are needed to decide the point’s nature. Degenerate saddles, where the Hessian is zero or semidefinite, require more detailed analysis.

Example: f(x, y) = x^2 − y^2 has a saddle at (0, 0). The gradient vanishes there, and the Hessian diag(2, −2) is indefinite, indicating increase in x-direction and decrease in y-direction.

Saddle points also appear in constrained optimization and in game theory, where a saddle point of a payoff function f(x, y) satisfies f(x*, y) ≤ f(x*, y*) ≤ f(x, y*) for all x, y. In numerical optimization, saddle points can cause stagnation for gradient-based methods, motivating techniques that exploit curvature information.