# Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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### Can a method for a quadratically constrained quadratic programming be applied to a quadratic optimization function with linear constraints?

I want to find a time complexity for solving $$\min_x \|Ax-b\|^2_2 \quad \text{s.t.} \quad 0 \leq x \leq 1$$ In section 10.2 of Interior Point Polynomial Time Methods In Convex Programming [PDF] (...
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### Comparison of Wolfe conditions to other "weaker" conditions or facts about optimization techniques for conceptual understanding

So in the book, it states that the first wolfe condition is the following, $\begin{equation}p^Tg_k\leq-\eta_0|||p|||g_k||\end{equation}$, where $g_k=\nabla F(x_k)$. Here it states that this is a ...
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### BFGS computational complexity derivation

The update for the Hessian using BFGS is given by: $$H_{k+1}=(I-\rho_ks_ky_k^T)H_k(I-\rho_k y_ks_k^T)+\rho_ks_k s_k^T$$ where $\rho_k=\dfrac{1}{y_k^Ts_k}$. Nocedal and Wright, Numerical Optimization ...
1 vote
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### How do you represent the number of iterations in the formulation of an optimization problem?

Let's say I want to minimize some function for f(x), with respect to x, in the minimum number of iterations. How would I represent the number of iterations in the formulation of this optimization ...
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### Algorithm to minimize $(x_1+\dots+x_n)^2 + \frac{c}{2}\sum_{i=1}^n x_i^2 + \sum_{i=1}^n (\lambda_i - cz_i)x_i$ subject to $x_i\geq 0$

I am trying to minimize $(x_1+\dots+x_n)^2 + \frac{c}{2}\sum_{i=1}^n x_i^2 + \sum_{i=1}^n (\lambda_i - cz_i)x_i$ subject to $x_i\geq 0$ where $c\geq 1$ is a constant (that I can push arbitrarily ...
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### Chebyshev Nodes Interpolation

I am a bit confused as to what role the Chebyshev nodes play in the optimization of Langrage Polynomial Interpolation. Reading online for what I understand that the Chebyshev nodes gives us the ...
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### Numerically approximating projection onto an infinite-dimensional Hilbert-space

We have the following problem that we want to model numerically. We would be glad for any references, since we could not find much useful information on these kind of problems and since we do not come ...
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### What is this optimization method called?

I am using a technique to do numerical optimization of a system but I don't know what it's called. I would like to be able to look it up in literature or books, but can't without knowing what I'm ...
1 vote
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### Merit Function Line Search in Interior Point Methods - Penalty Parameter Updates

I gues the TL;DR is summarized as follows: Why do the step length and penalty parameter calculations not consider the steadily decreasing step length (in a backtracking line search) when calculating ...
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### What's the difference between a value of a dual and a solution of a dual?

In this video on total dual integrality: https://youtu.be/l8YdPRxqlXo?list=PLXsmhnDvpjORcTRFMVF3aUgyYlHsxfhNL&t=798 If the dual has an integral solution, not just an integral value ... The ...
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(sorry for the terrible title, I don't know if there is a proper name for a function like that) I have a function of several (n~100) real variables $f(x_1, x_2, \ldots, x_n)$ and I want to find the ...