# 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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### What is the maximum of the function in this feasibility?

g(x,y)=xy2+x2−4xy. The region is only feasible if y≥−1, x≤3 and y≤x. I found the minimum in (2,2) but the other points were saddle points. Is there any maximum extrema in this region?
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In general, we know that for strongly convex functions for which we can compute the Hessian and find the Lipschitz constant $L$ of the gradient, gradient descent will converge provided that the step ...
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### What do you call the block-diagonal entries of a hessian?

Consider some twice differentiable function $$f: \mathbb{R}^k \to \mathbb{R}$$ and some partition $k = m+n$, such that we may see this as a function $f(x_1,...x_m,y_1,....y_n)$. Is there some common ...
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### Multivariable Function Optimization with Semidefinite Hessian Matrix

I'm writing to ask for support in carrying out an exercise about the optimization of a multivariable function. The function is: $f(x,y,z)=x^2+y^2z+z^2-2x$ Clearly the domain is $\mathbb{R}^3$, and $f$...
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### Complexity of numerical derivation for general nonlinear functions [closed]

In classical optimization literature numerical derivation of functions is often mentioned to be a computationally expensive step. For example Quasi-Newton methods are presented as a method to avoid ...
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### Guaranteed invertibility of the approximated Hessian in Levenberg-Marquardt

I need to show that the approximated Hessian in the Levenberg-Marquardt algorithm is guaranteed to be invertible, whereas in the Gauss-Newton algorithm, this is not always required to be true. ...
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### Convexity analysis

I know that summation of two convex functions is also convex but I would like to know that does the same holds true for non-convex functions also such that the summation of two non-convex functions ...
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### Showing that Lagrangian is concave using negative definiteness of the Hessian

I'm trying to solve a homework exercise where I am required to use the fact that, if a Lagrangian is concave, and a point $x^*$ satisfies the condition $\nabla \mathcal L (\mathbf x,\lambda)=0$, ...
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### Can we interpret the determinant of a $2×2$ Hessian matrix in terms of the area of a parallelogram?

As we known, the determinant of a $2×2$ matrix is the area of a parallelogram? In this case, can we interpret the determinant of a $2×2$ Hessian matrix in terms of the area of a parallelogram? If so, ...
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### Unconstrained function minimization when only given samples?

Let's say we want to minimize a function $f$ of the form: $$f(x) = \sum_{i=1}^n f(x_i)$$ where $x_i$ are samples. One way to solve these types of problem is via identifying a search direction via ...