Questions tagged [numerical-optimization]

Numerical methods for continuous optimization.

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32 views

A global optima: $\text{max}_{x} \frac{1}{2}\left\| X (a + b) \right\|_2^2 \ \text{s.t.} \ a^T X b \leq \delta; 0 < x \leq 1$,$X := {\rm Diag}(x)$

How to find (using any software) a global optimum for such a (non-convex) problem \begin{align} \text{maximize}_{x \in \mathbb{R}^{n \times 1}} \quad & \frac{1}{2}\left\| X (a + b) \right\|_2^2\\ ...
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1answer
31 views

Generalizing the conjugate gradient like this works?

Given $A \in \mathbb{R}^{n \times n}$, a SPD matrix, and a vector $b \in \mathbb{R}^n$, it is possible to solve the problem $$\min_x \| Ax - b\|$$ with the conjugate gradient method. Its algorithm ...
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1answer
45 views

How to optimize a function with the following constraints by using gradient descent?

I am not currently unfamiliar with a numerical optimization, so I am studying them. What I am wondering is that I'd like to optimize a certain function with the following constraints by using gradient ...
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15 views

What does “reduced-Hessian” mean? [on hold]

What does "reduced-Hessian" mean? I came across this concept in the book named "numerical optimization". Thanks a lot.
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22 views

Prove that superlinear convergence of gradients implies superlinear convergence of sequence itself

I'm stuck on proving a result that is not specifically proven in a research paper. https://people.maths.ox.ac.uk/cartis/papers/ARCpI.pdf It is the proof of Corollary 4.8. I'm trying to show that $(4....
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0answers
40 views

Convergence of complex sequence with gradient descent

I am looking to solve the following otimization: $\underset{{{{\bf x} \in \mathcal{X}}}}{\text{min}} \; f({\bf x})$ where ${\bf x} \in \mathbb{C}^N$ and $f({\bf x}) \in \mathbb{R}$, $f(x)$ is a ...
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11 views

Stopping criterion

I've seen this stopping criterion for iterative optimization algorithms such as Newton-Ralphson, Gradient Descent, etc. However, I do not remember its name nor where I saw it. It seems that this ...
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19 views

Proof of the convergence of a convex function with newton method

I need to prove the following: Assuming that $f$ is convex, under the assumption that $f$ is convex and $x^{(0)} \geq x_\star$, the algorithm always delivers a converging sequence: Theorem: Let's ...
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1answer
46 views

Maximize constrained log-sum

Given constants $c>0$ and $\beta_i \in [0, \infty)^d$, for $i=1,..,n$, I want to (numerically?) solve the following problem: $\max_{x \in [0, \infty)^d} \sum_{i=1}^n \log(\beta_i^T x), \text{ ...
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1answer
12 views

Initial feasible solution for barrier method

From this example page 9 It said set initial feasible solution at 2 here's barrier function: $$T(x)=\frac{100}{x}+\frac{1}{r}(\frac{-1}{x-5})$$ after derivative: $$\frac{\delta T}{\delta x}=\frac{1}{...
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73 views

Semi-infinite programming discretization theorem from the book ,,Theory, Methods, and Applications'

I think this theorem from Kortanek and Hettich book ,,Semi-Infinite Programming: Theory, Methods, and Applications'' is false. (P) is a semi-infinite programming problem in the following manner: $$\...
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1answer
37 views

Variational inference: Does the natural gradient follow geodesics locally?

Amari's natural gradient descent is a well-known optimisation algorithm from information geometry that is well-suited for finding optima of functionals on statistical manifolds. It consists of ...
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2answers
49 views

How to solve this quadratic program using the penalty method?

example: $$\min\frac{1}{2}((x_1-3)^2+(x_2-2)^2)$$ s.t.$$-x_1+x_2{\le}0$$ $$x_1+x_2{\le}1$$ $$-x_2{\le}0$$ and we start with $~x^0=[3,2]^T~$ its violate the condition : $$q(x,c)=\frac{1}{2}((x_1-3)^2+(...
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2answers
113 views

Efficient algorithm for determining whether value of convex optimization program is below some value?

Let $X$ be a convex subset of $\mathbb{R}^N$, let $c \in \mathbb{R}^N$. I want to know whether $$ \min_{x \in X} x^\top c < 0. $$ Obviously, I can (efficiently, with standard software) evaluate ...
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17 views

Is there a way to identify singular points in a spectrahedron without finding the entire set?

I'm a physicist working in a non-linear optimization problem that reduces to a semidefinite program (SDP). In the simple low dimensional cases we workout in the research we could identify singular ...
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1answer
132 views

finding a solution to a matrix inequality

I would like to find a 19x19 matrix V such that the following inequality holds: $$V^TAV<K$$ and where all entries of V are positive and the sum of entires in a row of V are equal to 1. Also < is ...
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13 views

Stochastic Optimization with Piecewise Function

I have a stochastic optimization problem where my objective is a piecewise function: $$ \underset{x}{\text{min}} \: \sum_{i=1}^{N} E(g(Y, x_{i})) $$ where $Y \sim N(\mu, \sigma^2)$ is a random ...
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22 views

How to numerically solving a spectral optimisation problem?

Consider the following one-dimensional eigenvalue problem \begin{align*} -\frac{d}{dx}\left(\sigma(x)\frac{du}{dx}\right) & = \lambda u \ \ \textrm{ in $(0,L)$} \\ u(0) = u(L) & = 0, \end{...
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19 views

Comparisons of Solving Speed of QP & SOCP

For two same scale optimization problem, quadratic programming (QP) and second-order cone programming (SOCP), which one is faster to solve? As far as I know, the computational complexity of QP and ...
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57 views

How can I apply backpropagation with matrix algebra? - Deep learning

Deep learning and backpropagation is taught out very badly and is often looks like a mess, according to me. So I want to start with a simple example about how to use backpropagation: Assume that we ...
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12 views

Specific number of Halton Points

I'm starting to study many article about numerical calculus and I see the definition of Halton Points (HP). But in this article sometimes one author use 289 HP, and sometimes another author use 1089 ...
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42 views

What is the arithmetic cost (computational complexity) of an SDP with nonlinear matrix inequality constraint

In the book ''Interior-Point Polynomial Algorithms in Convex Programming'' (Nesterov and Nemirovskii) section 6.4, there is a computational complexity result for the general positive semi-definite ...
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1answer
50 views

Why the solution does not converge in this optimization problem?

I want to use the "projected Gradient decent algorithm" to solve this optimization problem but I do not know why it does not converge. I appreciate if anybody can help me to find the mistake. Given $$...
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61 views

Adding identity to invert matrix

I'm looking into algorithm implementation which is essentially linear regression: $$\|Ax - b\| \rightarrow \min$$ Matrix A and vector b are estimated using data, then we do $A^{-1}b$ to find x. But ...
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30 views

Extracting diagonal of $J^TJ$ via automatic differentiation like techniques

First of all please, let me know if this question is more suited for scicomp.stackexchange.com or or.stackexchange.com, and ...
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28 views

Can we always find a proper step size?

In convex optimization, if we know the gradient of a function $f(x)$, then is it true that we could always find a way to determine a proper step size in the gradient descent method? When I say "proper"...
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1answer
17 views

Optimal approximation of nonlinear probability density function by piecewise constant density

Given a nonlinear probability density function F, the problem is to estimate F using histogram over a partition with N intervals. I have tried to realise this with MATLAB function fmincon, but it ...
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36 views

LMI-based versus standard form semidefinite programs

In the context of semidefinite programming (SDP), under what conditions is it preferable to formulate and solve an LMI-based SDP rather than an equivalent standard form SDP? I have been told that ...
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1answer
44 views

Positive semidefinite inequality

Let $\mathbf{A}\in\mathbb{C}^{m\times m}$, and $\mathbf{B}\in\mathbb{H}^{m\times m}$ be an $m$-dimension Hermitian matrix, solve $\theta$ that satisfies the condition \begin{equation*} e^{\jmath\theta}...
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1answer
52 views

What is known about optimizing a function whose evaluation cost is variable?

Let's imagine I have a function $f(x)$ whose evaluation cost (monetary or in time) is not constant: for instance, evaluating $f(x)$ costs $c(x)\in\mathbb{R}$ for a known function $c$. I am not ...
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1answer
100 views

Quadratic programming on a small embedded device — can I do the hard work on my PC first?

I have the following constrained quadratic program in $x$ $$\begin{array}{ll} \text{minimize} & \frac{1}{2}x^TQx + x^Tc\\ \text{subject to} & b_{\min} \leq Ax \leq b_{\max}\\ & x_{\min} \...
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18 views

Optimizing a system of equations to minimize purchasing cost

I am new to this (posting on stack exchange and higher level math in general) so please correct me where I am in error. I am trying to minimize the cost to my company for purchasing a certain product. ...
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17 views

Formula for Area of a Triangle - nodal basis function

Let T be a triangle with corners $P_1, P_2, P_3$ and the nodal basis function $\lambda_1, \lambda_2, \lambda_3$ and $\alpha, \beta, \in \mathbb{N}_0$. I want to show that $$ \int_{T}^{} \lambda_1^\...
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1answer
38 views

Why is this inequality true in the proof of the convergence of Newton's method?

From Convex Optimization by Boyd & Vandenberghe: Let $f$ be a twice continuously differentiable convex function that is strongly convex with constant $m$, i.e., $\nabla^2 f(x) \succeq m I$ for $...
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24 views

What are possible ways to update the Hessian if the calculated gradient is negative in BFGS algorithm (Quasi-Newton method)?

When applying the globalized BFGS algorithm (Quasi-Newton Method, optimization, minimization) to approximate the minimum of a function using the Quasi-Newton-Method, sometimes one can get a negative ...
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136 views

Verify if my idea is correct

Hi guys I have been reading Numerical optimization by Nocedal. While I was on the chapter on quasi- Newton methods it was mentioned that $g_{k+1}^Ts_k=0$ where $g_k = Ax_k -b^Tx_k$ if we use exact ...
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46 views

When is the simplex method slower than the ellipsoid algorithm?

In an undergrad class on linear programming, we learned about the simplex and ellipsoid methods for solving linear programs (LPs). I know that the simplex method is generally faster than the ellipsoid ...
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13 views

multi-objective optimization test function explanation

i took this image from a paper that describes a multi-objective optimization algorithm where UF1 is a multi-objective function to optimize. can you explain to me what J1, J2 variables and the second ...
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43 views

$\lambda_{max}$ in trust region method: Leveberg-Marquardt algorithm

I am learning levenberg-marquardt algorithm and in the process implementing the same. I am comfortable with the $Jacobian$, $Hessian$ and step size computation. For trust region implementation, I have ...
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54 views

Block Separability in ADMM

I am reading Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers, where Boyd claimed that ADMM is an algorithm that is intended to blend the ...
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1answer
31 views

Find a suitable zero equation to solve the optimization problem $\min_{x \in \mathbb{R}^N} f(x)$

Suppose we have an optimization problem for this general form of $f: \mathbb{R}^N \rightarrow \mathbb{R}$ $$\min_{x \in \mathbb{R}^N} f(x)$$ and this problem is solvable. How could I construct a ...
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1answer
25 views

Minimizing $\ell_p$ norm on a flat - what type of convex programming subproblem is this?

Suppose that we are in a finite-dimensional real vector space $\Bbb R^n$, and we are on a flat $F \subset \Bbb R^n$ (aka, an affine translation of a subspace of $\Bbb R^n$). We want to minimize some $\...
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47 views

Understanding the steps of Karmarkar's algorithm

I am working through Karmarkar's seminal paper [0] and came across something I didn't quite understand. In section 2.3, Description of the Algorithm, he explains how to calculate the next point. The ...
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24 views

Estimiate Fourier transform coefficients of optimial $L1$ function?

Let there be a loss function $h$ that takes an $L1$ function as input. Assume that $h$ has a finite global minima reached by a single $L1$ function $f^\ast$. In other words... $$f^\ast = \...
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1answer
24 views

Gradient-based interpretation of the simplex algorithm

The simplex algorithm iterates from vertex to vertex of the convex polytope that bounds the feasible region of the constrained optimization problem, such that each iteration of the algorithm moves ...
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1answer
23 views

Showing a Chebyshev set

I want to show that $\{1,e^{ix},...,e^{(n-1)x} \}$ is a Chebyshev Set on $(0,2\pi]$. Now I know that $\{1,x,...,x^n \}$ is one and that $e^{ix}$ is injective on $(0,2\pi]$. But how do I show that if I ...
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0answers
79 views

Computation of function with sup

I am trying to compute the value of the following function $R:\mathbb{R}^m\mapsto\mathbb{R}$ $$R_n(\theta)=\sup_{\lambda\in\mathbb{R}^m}\left\{-\frac{1}{n}\sum_{i=1}^{n}\sup_{x\in\mathbb{R}^m}\lbrace ...
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1answer
30 views

Divergence criteria of Secant method on $\arctan(x)$?

I want to make sure I understand when the secant method will not converge as compared to the Newton's method. When I look at $\arctan(x)$ and try to determine the initial guesses for which it will ...
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27 views

Globalized BFGS (Quasi-Newton method) condition

I didn't find any information on the internet about the globalized $\textit{BFGS}$ method. Wikipedia only talks about the normal BFGS, without this ...
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1answer
49 views

“Rectangular” Cholesky decomposition of lower dimension

Given a symmetric PSD matrix $A \in \mathbb R^{n \times n}$, we can Cholesky-decompose it into $LL^T$, where $L \in \mathbb R^{n \times n}$ is lower triangular. However, we can also consider ...