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A Goldstein stationary point is defined via the norm of the gradient mapping: if the supremum of \|\nabla f(x)\| = \|x - prox_{\eta f}(x)\|/\eta does not exceed \delta, then x is called a \delta-Goldstein stationary point.
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10/15/2026
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UnverifiedThe Goldstein stationary point condition requires that within a neighborhood of some radius δ, the lower bound on the gradient norm and the amount of function value decrease at all points satisfy a certain inequality.74% similarUnverifiedThe derivative of the squaring operation |x|² is 2x, which is linear, so gradients will not explode or vanish sharply at extreme values as they do with exponential functions, giving it a natural advantage in numerical stability.65% similarUnverifiedThe Toeplitz–Hausdorff theorem guarantees that the numerical range of a matrix is always a convex compact set in the complex plane.63% similarUnverifiedThe proximal operator prox_{ηh}(v) = argmin_x { h(x) + (1/(2η))||x-v||^2 } is equivalent to performing one step of implicit gradient descent on h at the point v, and does not require h to be differentiable.61% similarUnverifiedThe basic form of a variational inequality is to find a vector x* such that ⟨F(x*), x-x*⟩≥0 holds for all feasible points x, where F is an operator.61% similar
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