Hessian matrix concavity

Sarah Johnson

May 19, 2025

1 min read

The function (x1,x2) ↦ x1 ( x 1, x 2) ↦ x 1 is an affine function, and hence is concave (and convex). The Hessian matrix of a multivariable function f ( x, y, z, . To be more fancy, say X is a section of TRn. Then the second partial derivative test goes as follows: These tests are irreducible one with the other. I Example: Check whether the f(x;y) = xy + x2y2 + x3y3 de ned on <2 + is quasiconcave.Hessian matrix, Convexity and Concavity. I need to check the convexity of f(x) for these two questions, using the Hessian matrix. In summary, the .The Hessian matrix, or simply Hessian, is an n×n square matrix composed of the second-order partial derivatives of a function of n variables.A smooth vector field in TRn is an object of the form X = n ∑ j = 1aj ∂ ∂xj, where aj are smooth functions.

optimization

Let D(x, y) = fxxfyy − f 2 xy .For example, the Hessian matrix can be used to detect corners and blob-like structures in images, which are important features for object recognition and tracking.Stack Exchange Network.

Hessian to show convexity

The Cobb-Douglas function is defined on $\mathbb R^2_+=[0,\infty)^2$ , (continuous also on that domain), and differentiable on $(0,\infty)^2$ .Balises :The Hessian MatrixConcave Hessian MatrixHessian Matrix Calculator helps you to calculate Hessian matrix of a function.Geometrically, the linear transformation performed by the Hessian matrix is a transformation that makes the basic bowl-shaped function more convex or .Overview

How to calculate the Hessian Matrix (formula and examples)

First, we find closed-form expressions for the Hessian matrix of the mutual information, differential entropy and entropy power with respect to arbitrary parameters of the system and, second, we study the concavity properties of these quantities. 6 Answers to the exercises 8.Balises :Hessian and ConvexityCritical PointSee here for proofs (concavity proofs are the same as convexity proofs, with the inequalities reversed). Enter function.Balises :The Hessian MatrixMatrix Computations The methods presented in this chapter are general, such that many results can be derived using the introduced framework.

I know what is a Hessian matrix, and mathematical calculations required to calculate if a matrix is positive-semidefinite. I am struggling to find some proof about concavity and hessian matrix, How can I proof: If the Hessian Matrix of f f is Negative Definite, then f f is concave. Asked 9 years, 10 months ago. In the case of twice differentiable functions, the most usual tests of concavity and quasi- concavity are those concerning the monotonicity (with respect to l) property of the signs of the l th principal minors or the l th principal bordered minors of the hessian matrix.This follows from the following theorem: f1 is non-decreasing, non-negative real valued function defined on Rn+ and f1 is quasi-concave and homogeneous of degree 1. For initial data in the form of log-Lipschitz . Background: Second partial derivatives. Summing two concave functions produces a concave function, and every concave function is quasiconcave. Since the leading principal minors are D1 = 2 and D2 = 5, the Hessian is neither positive semide nite or negative semide nite. Calculate the Hessian matrix at the point (1,1) of the following .The Hessian is a matrix that organizes all the second partial derivatives of a function.

The Hessian Matrix: Eigenvalues, Concavity, and Curvature

Examples of the Hessian Matrix

A function f, twice differentiable, is concave if and only if for all x the subdeterminants |Hi| of the Hessian matrix have the property |H1| ≤ 0, |H2| ≥ 0, |H3| ≤ 0, and so on.minor of this matrix is the det of the top-left (k +1) ×(k +1) submatrix. First question: f(x) = {x1x2 x ∈ Rn + + ∞ otherwise then the hessian: H = (0 1 1 0) Note that f(x) = g(u(x;y)) where u(x;y) = xy and g(z) = z + z2 + z3 . Viewed 2k times. Notice that since f is a quadratic form, we could also have used the symmetric matrix of the quadratic form to conclude this. In simpler terms, using the Hessian matrix allows us to determine whether a specific point of a function is concave upward, concave downward, or a saddle point.Computes the Hessian Matrix of a three variable function. The Hessian matrix.Balises :MatricesHessian and ConvexityLinear AlgebraConvexity and Concavity3 The eigenvalues of the Hessian matrix 3. Viewed 12k times.Table 1: Eigenvalues of the Hessian matrix of f (x, y) = sin x sin y at selected critical points, with concavity.Hessian test for local max/min. Thus, the formula for the Hessian . All rights reserved. It is an online calculator which provides you a step-by-step easy solution.The Hessian matrix: Eigenvalues, concavity, and curvature.To check the strict concavity, one can form find the related Hessian matrix and check to see whether the Hessian matrix is negative definite or not. Both goals are intimately related since concavity can be assessed through the The analysis of quadratic functions from Chapter 1 becomes a fundamental tool for describing behavior that is beyond the linear approximation, such as bending (convexity/concavity).The Concavity test by Hessian is a mathematical method used to determine the concavity of a function at a given point. Brown, BYU{Idaho Dept. Berechnet die Hesse-Matrix einer Funktion mit drei Variablen.First, we find closed-form expressions for the Hessian matrix of the mutual information, differential entropy and entropy power with respect to arbitrary parameters of the system . Understanding the Gradient and Hessian Matrix in Multivariate Optimization. That is, X is a smooth function X: V ⊂ Rn → TRn such that X(p) ∈ TpRn. It should be fairly easy to apply the definition to handle this case.

Lecture 5 Principal Minors and the Hessian

The Hessian matrix was named after Ludwig Otto Hesse, a 19th-century German mathematician who made very important contributions to the field of linear algebra. Critical point (x0, y0) Hf (x0, y0) Eigenvalues of Hf (x0, y0) Concavity at .checking for concavity.

All About the Hessian Matrix, Convexity, and Optimization

I will appreciate it if you can share with me the link to the proof thanks. I am aware the function can be said to be convex if over the domain of f the hessian is defined and is positive semidefinite.Balises :MatricesHessian and ConvexityConvexity and ConcavityHessian Matrix Su cient conditions. • Understanding the problem of the vanishing gradient • Defn of pseudo-concavity: f is pseudo-concave if . I have a function, u(x1,x2) = α . hessian-matrix. 5 Conclusion 7. Multivariate optimization problems require a solution to . This also means that if a monotonic transformation of f is concave, then f is concave.

This is because defining differentiability on boundary points requires extra care that's usually unnecessary for the issue at hand (such as defining concavity of a function via the Hessian matrix).Balises :The Hessian MatrixHessian Matrix ConvexConcave Hessian Matrix

Hessian Matrix and Convex Functions

Asked 5 years, 6 months ago.03, of April 21, 2014.The complex Hessian might also be used to accelerate the convergence of iterative optimization algorithms, to study the stability of iterative algorithms, and to study convexity and concavity of an objective function. Usually we write Xp rather than X(p).

Proving a function is quasi-concave

Can someone explain geometrically, why do we check if the Hessian matrix of a multivariable function is positive semidefinite, to determine if the function is convex? real-analysis ; positive-semidefinite; hessian-matrix; Share.

Modified 9 years, 10 months ago. Different from normal . Using the Hessian matrix to find convex functions. Added Oct 7, 2018 by vik_31415 in Mathematics.Balises :The Hessian MatrixHessian Matrix ConvexHessian Matrix and Convex Functions. The Hessian matrix, denoted by H, related to h(t, w) is given by H = [− 1 t 1 w 1 w − t w2], To check the negative definiteness of H, I chose an arbitrary two-dimensional column vector x = [x1 .$\begingroup$ @LinAlg I can draw it, but how can I make a conclusion about the concavity of the function from the Hessian matrix analysis? $\endgroup$ – sequence Nov 3, 2018 at 15:34From Figures 5 and 6, it can be easily understood that the Hessian matrix can be used to determine the concavity or convexity of a second-order derivative.Analyzing the Hessian.

multivariable calculus

(i) If d2f(x 0) >0, x 0 is a strict local min. Ask Question Asked 6 months ago.Convexity and concavity with Hessian matrices.Hessian matrix/Hesse-Matrix. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Computes the Hessian Matrix of a three variable function. Sufficient conditions For any function that is twice differentiable, a sufficient condition for concavity unimodality is that its Hessian matrix \(H(\x)\) is negative definite for all \(\x\):

multivariable calculus

At (0, 0) you just need to finess things a bit, but if you think about what concavity means physically, you can see that it is impossible to fail at a single point.

Hessian matrix

Specifically, you start by computing this quantity: H = f x x ( x 0, y 0) f y y ( x 0, y 0) − f x y ( x 0, y 0) 2.of this work is twofold.Balises :The Hessian MatrixHessian Matrix ConvexMatricesLinear AlgebraIt is the usual practice to check the concavity or quasi concavity of utility function in consumer theory, which is the basic property of utility function. Similarly, the tangent bundle of M is TM = ⋃ p ∈ MTpM. Students of courses in multivariable calculus are often taught the so-called “D-test” for optimizing functions of two variables: Theorem 1.

Analyzing the Hessian

Both goals are intimately related since concavity can be assessed through the Suppose z = f (x, y) has continuous second partial derivatives.It is given by. Viewed 34 times 0 $\begingroup$ I have a function and would like to get its convexity or concavity property.<0, x 0 is a strict local max. Find more Mathematics widgets in Wolfram|Alpha. Cadence CFD Solutions. The second partial derivative test tells us how to verify whether this stable point is a local maximum, local minimum, or a saddle point. It involves calculating the Hessian matrix, which is a matrix of second-order partial derivatives of the function, and using its eigenvalues to determine the concavity.Balises :The Hessian MatrixCritical PointHessian Matrix Eigenvalues

THE HESSIAN AND CONVEXITY

This means that f is neither convex nor concave.The complex Hessian might also be used to accelerate the convergence of iterative optimization algorithms, to study the stability of iterative algorithms, and to study .

Indefinite Hessian and concavity

Neither of the above functions are strictly (quasi)concave, so we need a separate .) , which different . Send feedback | Visit Wolfram|Alpha.

Then f1 is a concave function.

4 Concavity and curvature 6.

The Hessian matrix: Eigenvalues, concavity, and curvature

1 Introduction.In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation.Using a Hessian Matrix for Convexity Determination of a Function. The matrix of second derivatives, known as the Hessian, is symmetric when the derivatives are all continuous.Balises :The Hessian MatrixConcave Hessian MatrixLinear Algebra

optimization