We changed our privacy policy. Read more.

Questions tagged [lagrange-multipliers]

The method of Lagrange multipliers finds critical points (including maxima and minima) of a differentiable function subject to differentiable constraints.

Filter by
Sorted by
Tagged with
0
votes
0answers
7 views

RSM and steepest ascent for new $x'$ coordinates

In Myers, Montgomery, & Anderson-Cook, C. M. (2016)$^1$ the authors provide (p. 235) a general methodology to find new $x'$ coordinates in the path of steepest ascent based on constrained ...
2
votes
0answers
32 views

Gradient descent finds local minima for a problem that can be formulated as a convex problem

I am trying to find $$ \min_W \|Y-XW \|_F^2$$ $$s.t. \exists ij, W_{ij}\geq0 $$ where X is input data and Y is the output data we try to fit to. This is a convex optimization problem that can be ...
0
votes
0answers
16 views

Extracting the value of the constraint given a known lambda parameter in LASSO

The LASSO problem is expressed as \begin{equation} \hat{\theta}\in \arg\min_{\theta\in\mathbb{R}^d} \left\{\frac{1}{2n}\lVert y-X\theta\rVert_2^2+\lambda_n\lVert\theta\rVert_1\right\} \end{equation} ...
2
votes
0answers
49 views

Rewriting Lagrange Multiplier with non-linear optimization

NOTE: if you have access to the book mentioned below , please skip to the part of this post marked with (*), as I will now provide a short summary of the relevant background information for my ...
2
votes
0answers
24 views

convex optimisation formulation of SVM

I am currently learning about Support Vector Machine's (SVM) from the CS229 Stanford Class. In page 16 of the notes, they transformed $$max_{\hat{\gamma}, w, b} \frac{\hat{\gamma}}{||w||} \\ s.t \,\,\,...
1
vote
0answers
48 views

How to derive the $\ nR^2 $ form of the LM test statistic?

Several lagrange multiplier (LM) test statistics =$\ nR^2 $~$ \chi^2 $, where the LM test statistic is generated from regressing the square of residuals on some function in an auxiliary equation. The ...
0
votes
0answers
17 views

Lagrange multiplier test statistic math

I'm trying to understand the how the Lagrange multiplier test statistic written like this: $$LM = \hat{d'}_1 (\hat{\phi}_{11}-\hat{\phi}_{12}\hat{\phi}_{22}^{-1}\hat{\phi}_{21})^{-1}\hat{d}_1 \...
1
vote
0answers
19 views

Optimization problem for SVM with hard margin

I am trying to implement the SVM with hard margin from scratch. I have reduced the problem to finding the solution from the following optimisation problem $$ \operatorname{max}\limits_{\alpha}L(\alpha)...
1
vote
0answers
10 views

Score test using restricted mle

I am doing a study on Rao score test and still confused about the score vector. How am I supposed to calculate the score vector? Can I just use (ˆθn - θn) for the score vector? As I understood that ...
1
vote
1answer
29 views

Rewrite $\frac{1}{2}||x-u||_2^2$ subject to $||x||_1\le c$ to lagrangian form with multiplier $\lambda \ge 0$

Rewrite $\frac{1}{2}||x-u||_2^2$ subject to $||x||_1\le c$ to lagrangian form with multiplier $\lambda \ge 0$ So I'm pretty new to converting constraint functions to Lagrangian form, but I read that ...
0
votes
1answer
23 views

Non-negative Constraints in Soft-Margin SVM Lagrange Equation

I was reading the A Tutorial on Support Vector Machines for Pattern Recognition as a supplemental for my Intro to ML class and I wasn't sure why $ a_i \geq 0 $ and $ \lambda_i \geq 0 $ cannot be ...
0
votes
1answer
23 views

Finding the MLEs for pi in a contingency table proof

Reference: https://data.princeton.edu/wws509/notes/c5.pdf In a two-dimensional table, (the multinomial model - section 5.1.2 in the link above), I was wondering how they get $$\hat\pi_{i.} = \frac{y_{...
2
votes
1answer
61 views

Logarithms in Karush-Kuhn-Tucker conditions

I have the following optimization (and standarised) problem: $$minimize\ -(x\ ln(x)\ +\ y\ ln(y))$$ $$subject\ to\ \ \ \ \ \ \ \ \ \ \ \ \ x+y-1<=0$$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...
2
votes
1answer
96 views

Maximum Entropy Discrete Distribution

In Pattern Recognition and Machine Learning the author uses Lagrange multipliers to find the discrete distribution with maximum entropy. Entropy is defined by; $$H=-\sum_i p(x_i)\ln(p(x_i))$$ and the ...
1
vote
0answers
32 views

What is $\alpha$ in SVM Lagrangian fucntions

Is it a column matrix or a row matrix? I saw on a CMU slide that $\alpha$ is defined as $\alpha=(\alpha_1, \alpha_2, ..., \alpha_n)^T>=0$? I dont see much of a formal definition for the exact form ...
1
vote
1answer
42 views

Convert the following expression w.r.t to the whole dataset instead of element of the dataset?

I am in the process of expressing the w in LSSVM with data points and constants. After I resolve the KKT conditions for the LSSVM I got $$w = \sum ^N _{i=1} \alpha_i x_i$$ Is it possible to convert ...
1
vote
1answer
153 views

Minimize $f(A,B)$ s.t. $\text{exp}(A)^T \text{exp}(B)=J_K$

I have a function $f(A,B)$ that maps a pair of two (tall) matrices, $A$ and $B$, to a scalar cost that I want to minimize. $A$ and $B$ both have $K$ columns. I also want to impose a set of equality ...
1
vote
0answers
13 views

Lagrange Multiplier Test (Score Test) reliability at low likelihood

The Lagrange Multiplier Test (Score Test) in the multivariate case has the following formula: $$ LM = [\underline{s}(\underline{\theta_0})]^T*[Var(\underline{\theta_0})]^{-1}*\underline{s}(\underline{\...
0
votes
0answers
53 views

What is the maximum entropy distribution with a mean of 0, stdev of 1, skew of 1 (instead of zero), and kurtosis of 1 (instead of 3)?

As we know, there are an infinite number of distributions with mean of zero and variance of one. One of the special things about the normal distribution is that is has the maximum entropy. (https://...
2
votes
2answers
61 views

How to solve MNP (minimum norm) problem in SVM?

I'm reading an article, which says that MNP (minimum norm problem) can be solved as SVM. In the minimum norm problem, we're given a set of points in $R^d$ and need to find a point in convex hull of ...
0
votes
0answers
128 views

Minimizing Mean Square Error

Suppose we have a random sample $\textbf{X}=(X_1,...,X_n)$ from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & ...
2
votes
0answers
50 views

What are Lagrange multiplier and Dual decomposition methods?

What are the Lagrange multiplier and Dual decomposition methods? How are these methods used in Machine Learning(ML) for Optimization?
1
vote
1answer
147 views

Are Lagrange multipliers of regularization terms learned or set?

Background knowledge I knew about regularization in machine learning: A regularization term is added to the objective function to prevent overfitting. Usually, one would like to restrict the variance ...
4
votes
0answers
150 views

How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression?

How to prove the equivalence between constrained form and Lagrange form for lasso and ridge regression? Given lasso (constrained form): $$\underset{\beta}{\min}{(\frac{1}{2N}||y-x\beta||_2^2)} \...
1
vote
2answers
2k views

Calculating the value of $b^{*}$ in an SVM

In Andrew Ng's notes on SVMs, he claims that once we solve the dual problem and get $\alpha^*$ we can calculate $w^*$ and consequently calculate $b^*$ from the primal to get equation (11) (see notes) ...
0
votes
0answers
26 views

If the Lagrangian formulation also has constraints, what is then the simplification?

Consider the following constrained optimization problem. $$\begin{array}{ll} \text{minimize} & f(w)\\ \text{subject to} & g_{i}(w) \leq 0\\ & h_{j}(w) = 0\end{array}$$ The equivalent ...
1
vote
0answers
59 views

Optimal proposal for the Metropolis-Hastings algorithm using Tierney's theorem

Let $(E,\mathcal E,\lambda)$ be a measure space $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\lambda p\in(0,\infty)$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ $q:E^2\to[0,\...
1
vote
0answers
486 views

Solving constrained optimization problems with Adam

The adam algorithm has been very successful for solving non-convex optimization problems that appear in deep learning. Are there ways to extend adam to solve constrained optimization problems? Among ...
1
vote
1answer
37 views

why in SVM we have different indices for dot product?

I am confused by Lagrangian method in SVM, I can not understand why we use different indices in dot product. Suppose with using Lagrangian W is : $ W_{i}=\sum_{i}L_{i}y_{i}x_{i} $ In SVM ...
1
vote
0answers
83 views

why we use Lagrangian method in SVM?

I am wondering why we are using Lagrangian method in SVM? if we have just 3 features and 1000 rows, then objective function would be $w_{1}^{2}+w_{2}^{2}+w_{3}^{2}$ but if we use Lagrangian then we ...
1
vote
0answers
19 views

how we can constraints in dual of lagrangian?

I am confused about the general rule of Lagrangian multiplier. (which usully use for SVM). I could not find a good book or paper that explain it completely and clearly. Suppose we have $$minimize: ...
3
votes
0answers
62 views

Maximum entropy probability distribution over non-negative support and finite mean?

I'm trying to derive which univariate probability distribution maximizes entropy, assuming finite mean $\mu$ and non-negative support $[0, \infty)$. I know that the answer is the exponential ...
3
votes
1answer
1k views

How to choose between dual gradient descent and the method of Lagrangian multipliers?

For an optimization problem $$ \max f(x)\\\ s.t. g(x)\le 0 $$ The Lagrangian is $$ \mathcal L(x, \lambda)=f(x)-\lambda g(x) $$ Dual gradient descent solves it by (according to Page 43 of this lecture,...
1
vote
0answers
33 views

Wide variety of results in Breusch-Pagan Tests on simulated data?

I wanted to see how the results of the BP-test would come out. This stemmed from a debate with someone who claimed that the BP-test would start rejecting the null hypothesis of homoskedasticity even ...
1
vote
0answers
562 views

How do we know we're maximizing the Lagrangian objective function in PCA?

In Principal Component Analysis, we start with $m$ observations $x_1,\dots,x_m$, each of which is an $n$-dimensional vector. Assume we have centered the data; that is, we have subtracted the variable ...
0
votes
1answer
84 views
1
vote
0answers
26 views

Finding three coefficients (weight/ratio) by minimising variance

I am finding the value of a,b and c for minimising the variance of the following equation (four variables are correlated): $$ Var(\Delta V)=V ar[\Delta S-a\Delta F_1 - b\Delta F_2 -c\Delta F_3] $$ ...
4
votes
1answer
1k views

Gradient descent in SVM

I am studying about SVM now. Then I came across the problem. The dual optimization problem is as follows: \begin{align*} &\max_\alpha~~~~~ W(\alpha) = \sum_{i=1}^{n} \alpha_i -\frac{1}{2}\...
3
votes
0answers
269 views

Support Vector Machines: a beginner's question about the underlying math

I'm new to Support Vector Machines and I've been trying to get into the underlying math (instead of just using Scikit Learn or something like that). I understand the math behind it up to the point ...
5
votes
3answers
3k views

SVM: Why alpha for non support vector is zero and why most vectors have zero alpha?

In the optimization problem in SVM to compute the margin, we use Lagrange multipliers to insert the constraint: $$L(w,b,\alpha)= \frac{1}{\lambda}|w| - \sum \alpha (y_i(w*x_i+b) -1)$$ Now we want to ...
1
vote
0answers
656 views

Lagrange multipliers and mle

Could someone give a method that works for the following question? I am new to the topic and cannot understand why there is no constant $\lambda$ in the first two equations (which would arise from the ...
17
votes
4answers
8k views

The proof of equivalent formulas of ridge regression

I have read the most popular books in statistical learning 1- The elements of statistical learning. 2- An introduction to statistical learning. Both mention that ridge regression has two formulas ...
0
votes
0answers
214 views

How do we know the value of the regularization parameter satisfies the gradient equations required by Lagrange Multipliers?

I've take multiple machine learning classes and I am always told been told say when we do regularization on the training error $\mathcal E(W) = \frac{1}{n} \sum^n_{i=1}Loss(f(x_i),y_i)$: $$ \text{ (1)...
1
vote
0answers
317 views

Determine lagrange multiplier LASSO

I used the function l1ce from the lasso2 package in R, since I need to solve a minimization problem with a lasso constraint. min f(x) s.t. ||beta|| where beta are the parameters. The reason it ...
14
votes
1answer
1k views

LASSO relationship between $\lambda$ and $t$

My understanding of LASSO regression is that the regression coefficients are selected to solve the minimisation problem: $$\min_\beta \|y - X \beta\|_2^2 \ \\s.t. \|\beta\|_1 \leq t$$ In practice ...
3
votes
1answer
358 views

Using technique of Lagrange Multiplier

How can we use the technique of Lagrange multipliers to find a new vector of parameters $w$ which solves the optimization problem: minimize J(w) = $\frac{1}{2} || w -u ||^2$ such that: $w^T (x − y)...
0
votes
1answer
699 views

R built-in Breusch-Pagan Test getting different results from manual Calculation

I was using the following code to calculate the bp-test (eaef.csv can be found here). ...
0
votes
0answers
838 views

ARCH effect - Conflicting Results from Lagrange Multiplier Test and Ljung Box Test?

I want to test for presence of conditional heteroskedasticity in a vector of ARIMA(0,0,0) residuals. For this purpose, I would like to see if both ARCH Lagrange Multiplier Test (on levels of ...
3
votes
0answers
26 views

Scaling prediction from VAR model subject to a equality constraint

I have a forecasting problem and already built a decently working VAR model which provides forecasts as $\hat{Y}_{iT}$, for $i = 1,..n$ and $T$ is forecast time period. But now I have an additional ...
0
votes
0answers
1k views

Number of lags to use in Engle's ARCH-LM test [duplicate]

How does one decide on number of lags to use in this test? Can one just justify the number based on previous papers? To clarify, I am forecasting conditional variance, and have generated my log ...