Search results
Results from the WOW.Com Content Network
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is ...
Theorem — (sufficiency) If there exists a solution to the primal problem, a solution (,) to the dual problem, such that together they satisfy the KKT conditions, then the problem pair has strong duality, and , (,) is a solution pair to the primal and dual problems.
The example function (,) = (+) illustrates that the equality does not hold for every function. A theorem giving conditions on f , W , and Z which guarantee the saddle point property is called a minimax theorem .
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
If D(a, b) < 0 then (a, b) is a saddle point of f. If D(a, b) = 0 then the point (a, b) could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive). Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at (x, y) implies that f xx and f yy ...
Walter Hayman's 1956 paper "A Generalisation of Stirling's Formula" is considered one of the earliest examples of the saddle-point method. [ 7 ] [ 8 ] [ 9 ] In 1990, Philippe Flajolet and Andrew Odlyzko developed the theory of singularity analysis.
For saddle point problems, however, many discretizations are unstable, giving rise to artifacts such as spurious oscillations. The LBB condition gives criteria for when a discretization of a saddle point problem is stable. The condition is variously referred to as the LBB condition, the Babuška–Brezzi condition, or the "inf-sup" condition.