Questions tagged [lagrange-multipliers]

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

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Strange constraint in optimization problem

I am trying to solve the following QP : $$ \min_{x} \quad (1/2)\|y - Ax \|^2 _2 + \lambda \|Dx\|_1$$ where $y \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times m}$, $D \in \mathbb{R}^{l\times m}$. Assume :...
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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://...
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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 ...
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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 & ...
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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?
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Lagrange Multiplier Test Meaning?

I'm using this in R from the car package https://www.rdocumentation.org/packages/car/versions/3.0-7/topics/linearHypothesis ...
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1answer
25 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 ...
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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)} \...
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158 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) ...
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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 ...
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Spatial Lag or spatial Error Model? Deciding by using the Lagrange multiplier diagnostics

Honestly, my knowledge of geostatistics is limited. My assumptions are as follows: If I want to choose between a Spatial Lag Model (SLM) and a Spatial Error Model (SEM), I can use the Lagrange ...
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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,\...
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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 ...
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Lagrange multiplier test in Mixed Level Model

I want to estimate a mixed level model with AR(1) errors and then conduct a Lagrange Multiplier test. The mixed model allows for rich covariance structures but it does not allow for AR(1) errors. Can ...
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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 ...
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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 ...
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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: ...
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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 ...
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487 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,...
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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 ...
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395 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 ...
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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): $$ Var(\Delta V)=V ar[\Delta S-a\Delta F_1 - b\Delta F_2 -c\Delta F_3] $$ ...
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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}\...
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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 ...
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SVM: Why alpha for non support vector is zero?

In the optimization problem in SVM to compute the margin, we use Lagrange multipliers to insert the constraint: $L(w,b,\alpha)= \frac{1}{2}|w| - \sum \alpha (y_i(w*x_i+b) -1) $ Now we want to compute ...
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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 ...
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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 ...
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162 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)...
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256 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 ...
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732 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 ...
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198 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)...
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522 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). ...
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703 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 ...
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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 ...
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658 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 ...
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1answer
453 views

Can I use a spatially lagged Dependent Variable while using Spatial Error Model?

I have following issue: I run spatial diagnostics on dependencies for my Log-Log Transformed regression model. LM Tests (including Robust) are highly significant. Since I am using GeoDa, I cannot ...
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SVM - Reason for using Lagrangian Dual [duplicate]

Objective To confirm if the understanding is correct regarding the reason why Lagrangian Dual is used in SVM. Background While Machine Learning, gradient descent is used at regression, logistic ...
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895 views

KKT in a nutshell graphically

Objective Confirm if the understanding of KKT is correct or not. Seek for further explanation and confirmations on KKT. Background Trying to understand KKT conditions, especially the complementary ...
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1answer
2k views

Lagrangian multiplier: role of the constraint sign

I am beginner learning Lagrange multipliers with wiki article. Consider: maximize $f(x,y)$ subject to $g(x,y) = 0$ I understand that to maximize I must follow the gradient $\nabla {_{x, y}}^{}f$. I ...
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Why are the Lagrange multipliers sparse for SVMs?

I've read that for the Maximal Margin Classifier SVM, after solving the dual problem, most of the lagrange multipliers turn out to be zeros. Only the ones corresponding to the support vectors turn out ...