Inmathematics, asaddle point orminimax point[1] is apoint on thesurface of thegraph of a function where theslopes (derivatives) inorthogonal directions are all zero (acritical point), but which is not alocal extremum of the function.[2] An example of a saddle point is when there is a critical point with a relativeminimum along one axial direction (between peaks) and a relativemaximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the-direction.
The name derives from the fact that the prototypical example in two dimensions is asurface thatcurves up in one direction, andcurves down in a different direction, resembling a ridingsaddle. In terms ofcontour lines, a saddle point in two dimensions gives rise to a contour map with, in principle, a pair of lines intersecting at the point. Such intersections are rare in contour maps drawn with discrete contour lines, such as ordnance survey maps, as the height of the saddle point is unlikely to coincide with the integer multiples used in such maps. Instead, the saddle point appears as a blank space in the middle of four sets of contour lines that approach and veer away from it. For a basic saddle point, these sets occur in pairs, with an opposing high pair and an opposing low pair positioned in orthogonal directions. The critical contour lines generally do not have to intersect orthogonally.
A simple criterion for checking if a givenstationary point of a real-valued functionF(x,y) of two real variables is a saddle point is to compute the function'sHessian matrix at that point: if the Hessian isindefinite, then that point is a saddle point. For example, the Hessian matrix of the function at the stationary point is the matrix
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point is a saddle point for the function but the Hessian matrix of this function at the origin is thenull matrix, which is not indefinite.
In the most general terms, asaddle point for asmooth function (whosegraph is acurve,surface orhypersurface) is a stationary point such that the curve/surface/etc. in theneighborhood of that point is not entirely on any side of thetangent space at that point.
In a domain of one dimension, a saddle point is apoint which is both astationary point and apoint of inflection. Since it is a point of inflection, it is not alocal extremum.
Asaddle surface is asmooth surface containing one or more saddle points.
Classical examples of two-dimensional saddle surfaces in theEuclidean space are second order surfaces, thehyperbolic paraboloid (which is often referred to as "the saddle surface" or "the standard saddle surface") and thehyperboloid of one sheet. ThePringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape.
Saddle surfaces have negativeGaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature. A classical third-order saddle surface is themonkey saddle.[3]
In a two-playerzero sum game defined on a continuous space, theequilibrium point is a saddle point.
For a second-order linear autonomous system, acritical point is a saddle point if thecharacteristic equation has one positive and one negative realeigenvalue.[4]
In optimization subject to equality constraints, the first-order conditions describe a saddle point of theLagrangian.
Indynamical systems, if the dynamic is given by adifferentiable mapf then a point is hyperbolic if and only if the differential ofƒn (wheren is the period of the point) has no eigenvalue on the (complex)unit circle when computed at the point. Thenasaddle point is a hyperbolicperiodic point whosestable andunstable manifolds have adimension that is not zero.
A saddle point of a matrix is an element which is both the largest element in its column and the smallest element in its row.
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