The Hessian matrix and critical points of a scalar field

Depends on

• $C^k$ maps and multi-index derivative notation in Euclidean space
• The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
• The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations

Used by

• Damped Newton method Algorithm
• Finite-budget inexact Newton–Krylov with a descent safeguard Algorithm
• Ideal exact gradient line-search step Algorithm
• Sample-based constrained policy optimization update Algorithm
• Second-order Taylor expansion f(a+h)=f(a)+∇ f(a)· h+1/2h^TH_f(a)h+o(‖h‖²) Corollary
• The Hessian determinant of each C² holomorphic component is nonpositive; a nondegenerate critical point is a saddle Corollary
• The Hessian of a C² scalar field is symmetric Corollary
• A strictly convex function can have a singular Hessian Counterexample
• A zero Hessian occurs at a strict minimum, a strict maximum, and a saddle Counterexample
• Two equal modes contribute two Gaussian terms Counterexample
• x²+y²(1-x)³ has a unique critical point, a strict local but nonglobal minimum Counterexample
• Forward and central finite-difference Jacobian-vector and Hessian-vector products Definition
• Local and strict local extrema for scalar fields on Euclidean open sets Definition
• Mirror maps, Bregman divergence, and local Hessian norms Definition
• Positive definite, negative definite, semidefinite, and indefinite quadratic forms Definition
• Quadratic trust-region models and the actual-to-predicted reduction ratio Definition
• The cubic-regularized quadratic model Definition
• Validated ordered eigenvalue enclosures for an index-one Hessian Definition
• A curved double well has an exact circular minimum-energy path Example
• A damped Newton step precedes the local quadratic phase Example
• A scalar inexact Newton solve trades inner work for outer error Example
• An indefinite Rosenbrock Hessian makes the raw Newton direction ascend Example
• Dimer finite-difference bias competes with force-evaluation error Example
• The monkey saddle x³-3xy² has an indefinite higher-order critical point Example
• The standard Rosenbrock start accepts a full Newton step Example
• A Hessian lower bound gives strong convexity and gradient-gap bounds Lemma
• Cubic regularization bounds consecutive unsuccessful trials Lemma
• Exponential Lyapunov drift for the formal Langevin generator Lemma
• Locally Lipschitz Hessians give the BFGS Dennis–Moré condition Lemma
• Paired force evaluations give a second-order Hessian-vector estimate Lemma
• The Hessian is negative semidefinite at an interior local maximum Lemma
• Uniform Hessian bounds control gradient changes and Taylor remainders Lemma
• An exact index-one eigenvector-following step is the Newton correction at a saddle Proposition
• A C² function is convex exactly when its Hessian is positive semidefinite Theorem
• A gap-controlled dimer mode gives a local saddle-translation error recurrence Theorem
• Adaptive cubic regularization reaches first-order accuracy in O(ε⁻³ᐟ²) evaluations Theorem
• An inexact gradient residual certifies distance under strong convexity Theorem
• Armijo gradient descent has an explicit global linear rate Theorem
• Damped Newton reaches its local quadratic phase in finitely many continued updates Theorem
• Exact gradient line search has a global objective-gap contraction Theorem

…and 8 more results.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Analysis, Convexity, and Optimization (standard reference, not scraped)