# Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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### Minimizing the trace of a SPD-matrix subject to two trace equality constraints

I've encountered a problem where I need to check many (approximately $10^4$-$10^5$) minimum trace solutions subject to two trace inequality constraints. I've written this problem as an SDP and my gut ...
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### Maximum Likelihood Estimation via Numerical Optimization

I am reviewing the maximum likelihood estimation where we derive the likelihood of the sample data parametrized by the unknown underlying distribution parameters and we maximize the likelihood with ...
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### Optimizing $\arg\min_{U}||R-UMU^\dagger||$

Consider the optimization problem $$\arg\min_{U}||R-UMU^\dagger||$$ with R and M hermitian matrix and U unitary and ||.|| is the Frobenius norm. Are there known numerical method to solve it ? I ...
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### gas network optimal design (using pyomo): Which optimization textbook?

I am currently working on the optimal design of a gas network (actually hydrogen) and particularly trying to optimally size pipelines' diameters (using pyomo python package) for a given graph. The ...
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### Time complexity of conjugate gradients applied to normal equations

Consider a least squares problem $\min_x \|Ax-b\|^2$ with a rectangular matrix $A$. One way to solve it is to use the conjugate gradients method applied to the normal equations $A^\top A x = A^\top b$,...
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### Sparse PCA vs Orthogonal Matching Pursuit

Can't wrap my head around the difference between Sparse PCA and OMP. Both try to find a sparse linear combination. Of course, the optimization criteria is different. In Sparse PCA we have: \begin{...
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### What criteria guarantee that gradient descent tends to a local minimum?

Is there a set of criteria that a function $f$ can have so as to ensure that the gradient descent algorithm leads us to a local minimum? If so, why do such criteria ensure reaching a minimum?
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### $k^{-k}$ converges superlinearly but not quadratically.

A sequence $(x_k)$ converges to 0 superlinearly if $||x_{k+1}/x_k||\to0$ as $k\to\infty$. A sequence $(x_k)$ converges to 0 quadratically if there exist $N,M>0$ such that $||x_{k+1}/x_k^2||<M$ ...
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### How to minimize this function of two variables

I'm trying to minimize this function of $s$ over some bounds of $x$ and $y$. The function is $$\frac{s x y + 1}{a b s^{3} x y + a s^{2} \left(b + x\right) + s x y + 1}$$ $a$ and $b$ are constants, and ...
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### $f(x) = \frac{1}{2}x^TQx+q^Tx$ has a unique minimum if and only if Q is positive definite.

I am trying to prove the above result. I can show one direction but am having trouble with the direction "unique minimum implies positive definite". Edit: I have seen related questions such ...
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### Derivation of Gradient Descent Update Rule

I am trying to understand the interpretation of Gradient Descent provided in Ryan Tibshirani's 2019 Convex Optimization course scribed notes: http://www.stat.cmu.edu/~ryantibs/convexopt-S15/scribes/05-...
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### Numerically dense but dynamically sparse optimization problems

From the Ceres Solver's documentation on nonlinear least-squares: ellipse_approximation.cc fits points randomly distributed on an ellipse with an approximate line ...
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### Obtaining $h_i$ by evaluating $f(s)=\sum_{i=1}^n \exp(-h_i s)h_i$

Suppose I'm able to obtain values of $f(0),f(1),f(2),\ldots,f(m)$ with $f$ defined as follows: $$f(s)=\sum_{i=1}^n \exp(-h_i s)h_i$$ Is there a numerical procedure to obtain $h_1,\ldots,h_n$? We know ...
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### Optimization on $X^TX$ with all positive entries

Suppose I have an algorithm $\mathit{A}$ that can solve the optimization problem over certain functions $g: Sym(n) \to \mathbb{R}$. Now given $g$, let $f: \mathbb{R}^{k ,n} \to \mathbb{R}$ be another ...
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