# Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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This concerns the convergence of Newton's method in unconstrained optimization. But the question can be taken out of this context. Suppose $f$ is twice differentiable and that the Hessian $\... 0answers 33 views ### Numerically compact representation of high dimensional phase space maps I am studying the behavior of a transmission electron microscope lens system numerically. The lens system has only one approximate mirror symmetry. Essentially, I am studying the function taking in 5 ... 0answers 104 views ### How to generate a large PSD matrix$A \in \mathbb{R}^{n \times n}$, where$\mathcal{O}(n) \sim 10^3$I would like to generate a large PSD matrix, i.e.,$A \in \mathbb{R}^{n \times n}$, where$\mathcal{O}(n) \sim 10^3$. The entries of the matrix should be randomly generated using a standard function ... 2answers 66 views ### Nesterov's momentum derivation On page 75 in Sutskever's thesis http://www.cs.utoronto.ca/~ilya/pubs/ilya_sutskever_phd_thesis.pdf In equation (7.5) setting$a_0=1$,$a_{t+1} = (1+\sqrt{4 a_t^2 + 1})/2 $The author said, "to ... 1answer 41 views ### What solver can I use to solve the following optimization problem? Let$\xi_{1},\xi_{2},\ldots,\xi_{N}\in\mathbb{R}^{m}$,$\mu,\varepsilon\in\mathbb{R}$with$\varepsilon>0$(These are not variables, they are constant). We consider the following optimization ... 0answers 319 views ### How to solve this QCQP efficiently? I'd like to solve the following quadratically constrained quadratic program (QCQP) \begin{equation}\label{bijective} \begin{split} \min_{x} \quad &x^{T}Ax\\ \mathrm{s.t.}\quad &... 0answers 35 views ### How to characterize objective function in order to choose optimization method? What are some good quantitative metrics to describe an objective function that may help choose which optimization method will work best? E.g. if there are a lot of local minima we know that we ... 0answers 77 views ### Is there a stochastic analogue of Zangwill's global convergence theorem for deterministic descent algorithms? Zangwill's well-known global convergence theorem (Zangwill, W. I. 1969. Nonlinear Programming: a Unified Approach. Englewood-Cliffs, N.J.: Prentice-Hall) provides sufficient conditions under which ... 0answers 122 views ### When does the nuclear norm fail to minimize rank? Given$F$and$G, I'm solving the following problem: \begin{align} \min_{t} & \quad \mbox{rank}(A) \\ \text{s.t.}& \quad A = \text{diag}(t) F - G \\ \end{align} I used the nuclear norm,\|...
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When using stochastic subgradient descent, the solution $f_{best}(x_k)= \min \{f(x_1),f(x_2),....f(x_k) \}$, i.e., the best "point" over all the steps. As I understand, I should evaluate the function ...
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### Learn Numerical methods in Python

Please suggest me some texts that I could use to learn Numerical Methods in Python (irrespective of 2.7 or 3.x). I would like to improve my coding skills with Python along with refreshing my math in ...
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### Scaling/nondimensionalization for numerical optimization

I have a numerical optimization problem that I am trying to scale appropriately, in order to allow for the solver to achieve faster and more accurate results. I found a paper here that had a short ...
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### Writing an accurate SDP solver in Matlab

As part of a research project I'm supposed to write an semidefinite programming solver in Matlab (similar to SDTP3, MOSEK, SEDUM, etc) except it needs to be able to solve to many significant digits ...
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### How to solve this optimization problem analytically (parameter selection problem)?

My problem is as follows: \begin{align} \underset{\boldsymbol{x}}\max \quad & \boldsymbol r^T\boldsymbol x-\boldsymbol t^T\boldsymbol x \\ \text{s.t.} \quad & \boldsymbol1^T\boldsymbol x = N \...
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### Minimizing sum of two functions, minimize one and maximize the other.

I'm new in optimization and I have a problem involving minimizing the sum of two functions f(x) and g(x). The objective function must be in such a way that f(x) has to be minimized and g(x) maximized, ...
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### A nonlinear optimization problem: numerical solution fails

I am working on a practical problem which requires a theoretical analysis. Specifically, the problem is: Given a list of items, each enclosed in a rectangular box and with a given weight, and given ...
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### How to show that the method of steepest descent does not converge in a finite number of steps?

I have a function, $$f(\mathbf{x})=x_1^2+4x_2^2-4x_1-8x_2,$$ which can also be expressed as $$f(\mathbf{x})=(x_1-2)^2+4(x_2-1)^2-8.$$ I've deduced the minimizer $\mathbf{x^*}$ as $(2,1)$ with $f^*=-8$...
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### numerical optimization algorithm with approximate Gradient and Hessian only!

Suppose I have a energy functional E depending on X, where X is a N-dimensional real value vector and N could be very large (~=2000). I assume that there exists (at least) a (local) minimizer for E. ...
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### Understanding a bound in numerical optimization

I'm reading through the deepest descent method, and I'm struggling to understand a specific bound. I'll try to be as much clear as I can, but the notation used is a bit messy. It is basically ...
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