# Questions tagged [hessian-matrix]

The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.

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### How to find the Jacobian and Hessian of a function involving multiple Kronecker products?

I am having trouble finding the Jacobian and Hessian of this function involving the Kronecker product. I have a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ and a vector $\mathbf{x}\in\mathbb{R}^{n^4}$...
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### Non-negative second derivative of two-variable function implies convexity

I am trying to show that a function of two variables in $C^2$ is convex if its second derivative is non-negative. A twice differentiable function of several variables is convex iff its Hessian matrix ...
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### How to implement Newton's method for optimization in a reliable way.

I'm trying to implement Newton's method for optimization. I want to find the local minima of a non-convex function (specifically a negative log likelihood) of $m$ variables where $m\sim 10$. Here is ...
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### Hessenberg matrix and eigenvalue problem

Let $A_{1} \in \mathbb{R}^{n \times n}$ be an unreduced Hessenberg matrix. Given $\mu$ (for simplicity, we assume that it is real), we compute the QR factorization $A_{1}-\mu I=Q_{1} R_{1}$ by Givens ...
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### Negative curvature towards a certain direction

If $u^T\nabla^2 f(z)u$ is strictly negative, can we say that the Hessian $\nabla^2 f(z)$ has at least one negative eigenvalue corresponding to $u$?
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### How to compute the Hessian of a vector valued function?

The best learning rate of a cost function is strictly less than 2λ, where λ is the largest eigenvalue of the Hessian. I want to get the best learning rate of my gradient descent algorithm, which is: <...
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### Is the following function convex or nonconvex?

Suppose $\mathbf x=\left(\mathbf x_1,\mathbf x_2\right)^T$. Is the function $f(\mathbf x)=\|\mathbf x_1\mathbf x_1^H+\mathbf A\mathbf x_2\mathbf x_2^H\mathbf A^H\|^2$ convex with respect to $\mathbf x$...
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### Matrix “divided” by a matrix. Can this be done?

So I am working on a problem and which basically looks like this $- \frac{x^t A^{-1}x}{M x^TA^{-2}x}$ where $M$ is a scalar, $x$ is a vector and $A$ is a matrix (which in the actual problem is the ...
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### Is $g(x)=f(Ax)$ strongly convex if $f(z)$ is strongly convex?

Suppose that f(z) has gradient $\nabla f(z)$ and hessian h(f(z)). Let g(x)=f(Ax). Suppose that f(z) is strongly convex. Is g(x) necessarily strongly convex? If so, prove, if not find a counterexample....
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### How to expand this tensorial Taylor expansion to the $n$th term?

Wikipedia describes the use of the Hessian matrix in a Taylor series expension. I've noted that the first term is written in terms of the original function, the second term uses the gradient of the ...
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### Geometric intuition for Hessian matrices

I have been learning about multivariate calculus on my own. I have learned, notation-wise the properties of concave and quasi-concave functions. But I am finding it very difficult to find geometric ...
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### Gradient, Hessian, and minimum of a vector function?

Given the function $f(\vec{x})=(\frac{1}{2})\vec{x}^TP^TP\vec{x}+q^T\vec{x}+r$, where $\vec{x}$ and $q \in \mathbb{R}^n$, and $P \in \mathbb{R}^{n x n}$ is full rank and $r \in \mathbb{R}$, I am ...
I am trying to show the convexity of functions which include exponential variables with different powers. My question is, for quadratic functions($x^TQx$) it is easier to show as showing Q is ...