For questions on Lagrange multipliers, a strategy to solve constrained optimisation problems.

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Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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87 views

Use Lagrange's method to find the maximum value

Use Lagrange's method to find the maximum value of $\langle A\mathbf{x},\mathbf{x}\rangle$ subject to condition $\langle \mathbf{x},\mathbf{x}\rangle=1$ and $\langle \mathbf{u}_1,\mathbf{x}\rangle =0$ ...
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37 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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35 views

Inequality constraints in Lagrangian

I was reading Lagrangian multipliers . In the above text I can't understand why $\lambda \ge 0 $ for $g(x)\ge0$ and vice versa . Can anyone give me the explanation to this ?
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30 views

Question about Lagrange multipliers and a basic example

I am trying to understand the Lagrange multipliers from reading the Wikipedia page. Then I tried to apply this to the problem of finding a discrete probability distribution which is the solution of ...
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79 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
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118 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
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79 views

Lagrange multipliers in the context of the calculus of variations

Suppose we wanted to extremise the function (of a finite number of variables) $f$ subject to the constraint $g = 0$. The Lagrange multiplier approach is to extremise without constraint the function ...
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26 views

Position-dependent Lagrange multipliers for functionals

I'm trying to extremise the functional $$ S = \int L(Q_i) \, dx\,dy $$ Where the $Q_i$ are functions of $x$ and $y$, subject to the constraint $$ \vec{\nabla} \cdot \vec{Q} = A(x,y) \,.$$ My ...
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87 views

Calculus of variations: Isoperimetric and holonomic constraints.

A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...
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128 views

Lagrange multipliers for matrix calculus problem

I have what is probably an elementary matrix calculus question. Let $P$ be a linear operator and $||\cdot||$ be the ordinary unsquared vector norm. Suppose one wants to minimize $||Px||$ subject to ...
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77 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
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486 views

Optimization with symmetric matrix constraint

Consider the following optimization problem: ''Minimize some objective $f(A)$ over all matrices $A$ s.t. $A \mathbf{1} = \mathbf{1}$ and $A = A^T$.'' I wonder in which ways one can handle the ...
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56 views

Trouble with geometrical application of Lagrange multiplier

Let $A\in\mathbb R^{n\times n}$ be positive-definite and $\langle Ax,x\rangle=1$ be the equation of an ellipsoid $M\subset\mathbb R^n$. Use Lagrange multipliers to prove that the greatest distance of ...
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17 views

Doubts Lagrange Multipliers with random variables

Guys I really need your help... Days ago I found and solved an optimization problem with Lagrange Multipliers . I'll try to explain the general framework: There is a finite probability space, $N$ ...
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24 views

What is the meaning of positive lagrange multiplier?

I'm handling a maximization problem with a constraint and there is a sentence "but, the lagrange multiplier is positive". I can't understand why that sentence is needed. Is there any difference ...
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32 views

maximum and minimum values of a function

HI! I am currently working on some calc3 online homework problems and this one is giving me a bit of tough time. I found the gradient of f to be <16x,10y> and the gradient of g to be <4,20>. I ...
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19 views

Lagrangian method for Numerical Optimization

I know of a technique, but I don't know of its name and I don't have any real literature on the technique. On the wikipedia page for Lagrange multipliers a method is provided to convert a Lagrangian ...
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25 views

Regularized least squares

In Image Restoration, a true image f (in vector form)can be related to degraded data y through a linear model of the form $$y = Hf + n$$ where H is 2d blurring matrix and n denotes noise vector and ...
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19 views

Lagrangian with Nonholonomic Velocity Constraints

I have a Lagrangian $L$ dependent on four generalized coordinates $[\theta(t), \phi(t), l(t), h(t)]$. And I have two differential non-holonomic constraints given by: $$ Eq1: \dot l(t) = -c\dot \phi/ ...
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Lagrange optimimzation with a Diff.eq constraint

sorry in advance if my English isn't perfect here, it is not my first language( or second, for that matter...) im having some issues with understanding some of the details. the question is as ...
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16 views

How do I find the maximum and minimum values of xy on an off-center ellipse?

What's the maximum and minimum of f(x,y)=xy with the constraint $$\dfrac{(x-x_o)^2 }{A^2} + \dfrac{(y-y_o)^2 }{B^2} = 1$$ Using lagrangian multipliers is simple when the center of the ellipse is ...
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29 views

How can I find the Min and max of this question?

I have been trying for the past 2 hours on this question and cannot seem to figure out the answer. So far I have gotten the 'green' bits correct. Someone Help please
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59 views

a question about Lagrange multiplier?

Q)Given $x_1+x_2+...+x_n=a$ where $a>0$, find the extremum value of $f(x_1,x_2,...,x_n)=x_1^k+x_2^k+...+x_n^k$ Also, find the range of $k$ in which the extremum value of $f$ is a maximum ...
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59 views

Strong Duality and Duals of linear programming problem

I have the following problem: $ max_{x,y} \ x + y $ subject to $ 2x + y \leq 1 $ $ x + 3y \leq 3 $ $ x,y \geq 0 $ How to find the dual of this problem using the Lagrangian? I have done the ...
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72 views

Methods to minimise multilinear functions with trilinear, quad-linear and higher-linear terms?

My goal is to minimize functions such as $$f_1(\mathbf{p})=p_1p_3p_7+p_1p_4p_7+p_2p_3p_7+p_2p_4p_7-p_1p_3p_5p_6-p_1p_4p_5p_6-p_2p_3p_5p_6-p_2p_4p_5p_6$$ and ...
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51 views

Lagrange multiplier on inequalities

Let's say we have an inequality, then we change it into a function so for example we have: $f(x,y) = x^2-xy-2$ Also we have the constraint $g(x,y) = x+y - 1 = 0$ Using Lagrange multiplier, I get ...
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108 views

What do I need to know to understand Lagrange multipliers?

I've seen Lagrange multipliers used as a powerful method for tackling inequalities and some IMO problems, and I'm aware that it's a part of calculus. I'm currently taking BC Calculus in high school, ...
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146 views

K.K.T. conditions, Lagrangian gradient not defined for zero.

When I write the K.K.T. conditions for the problem I have, I get the following expression for the gradient of the Lagrangian: $$\frac{\partial \mathcal{L}}{\partial x} = - \frac{\sqrt{x} + ...
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59 views

Closed form for Lagrange dual

Can Lagrange dual always be computed in closed form? Can you give me a simple example where the dual is not analytically computable?
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15 views

proof of log-sum giving maximum value given equality constraint

How to prove the following equation: $$ -\log\sum_{k=1}^K f_k=\min_{\bf{u}}-\sum_{k=1}^K u_k \log(f_k) +\sum_{k=1}^K u_k \log(u_k)\\ s.t.\ u_k \in (0,1), \sum_k u_k=1 $$ using Lagrangian multiplier? ...
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23 views

When does a polynomial have finitely many critical points on a level set of another polynomial?

Suppose I have two polynomial functions $f$ and $g$ and I am interested in the critical points that $f$ has on a level set of $g$, i.e. $\{x\in \mathbb R^n : g(x)=a_1\}$ for some $a_1\in \mathbb R$ . ...
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21 views

Find the extreme value of $f(x,y,z)=x(y+z)$ on the curve of intersection of the right circular cylinder $x^2+y^2=1$ and the hyperbolic cylinder $xz=1$

Using the concept of Lagrange multipliers, we can treat the circular cylinder as $G$ and assign it $\lambda$, and treat the hyperbolic cylinder as $H$ and assign it $\mu$ Then $\Delta$f=$\lambda G + ...
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14 views

Determine $P_2 = f(0.7)$ when Neville's method is used to approximate $f(0.5)$

Let $f(x) = \ln(x + 1)$. Neville's method is used to approximate $f(0.5)$, giving the following table. $$x_0 = 0 - P_0 = 0$$ $$x_1 = 0.4 - P_1 = 2/8 - P_{0,1} = 3/5$$ $$x_2 = 0.7 - P_{2=?}- ...
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33 views

Help getting a closed-form solution to a maximisation problem

I'm working through a maximisation problem that I can't seem to get a closed-form solution to. It may be the case that there is no closed-form solution, but I would like a second opinion, since I've ...
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12 views

Finding extremes on set with one constraint

I have $f(x,y)=x*y*e^{-x^2-y^2}$ and I have set $A=\{[x,y]\in \mathbb{R}^2,x^2+2y^2\ge2\}$. I have to find extremas on set A. How do I do it? It is first time when I am encountering problem with only ...
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Trouble with Largrange Multipliers and expectation.

I am reading the following argument: Maximize $E[Log(X(T))]$ subject to $E[Z(T) X(T)]=x$, where $X(T),Z(T)$ are random variables, $x$ is a constant, and E is expected value (You can read this as ...
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40 views

KKT Conditions and Convexity

min $x^2 -xy +y^2 -5x+6y$ subject to $1 \leq y$, $y^3 \leq 2x$, and $x \leq 8$ Write out the KKT conditions for this problem. Show that $(x,y) = (4,2)$ is a KKT point, and is therefore a global ...
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18 views

Lagrange multiplier with linear constraints

See the figure below, We need to optimize $f_1$ under the constraint $f_2 \le B$. Clearly maximum and minimum are reached at the boundary points $(0, B)$ and $(B, 0)$ respectively. If we try a ...
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26 views

Can someone explain Method of Lagrangian multipliers

Can someone explain Method of Lagrangian multipliers to a beginner? I need some knowledge about solving problems using this method. If someone can provide the basic details in a simple manner along ...
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55 views

Lagrange dual method and KKT condition

Consider the following optimization problem \begin{equation}\notag \begin{split} \max & x^2+y^2 \\ \mathrm{s.t.} & x^2 \leq 1 \\ & 0\leq y\leq 2 \end{split} \end{equation} Obviously, the ...
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33 views

Lagrangian Method Proof

Suppose $f(\mathbf x)$, $g(\mathbf x)$ are smooth functions where $\mathbf x^*$ is a constrained local minimizer of $f(\mathbf x)$ subject to $g(\mathbf x)=0$. If $\nabla g(\mathbf x^*) \neq 0$ and ...
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28 views

Convex Minimization Problem with double sum

Given fixed natural number $n$ and two real numbers $A$ and $B$. I'd like to find $c_{12},\dots c_{(n-1)n}$, i.e., ${n\choose2}$ real numbers, such that $\sum_{1\le i<j\le n}^nc_{ij}=1$ which ...
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53 views

Solve multivariable nonlinear equation with mixed constraints?

How do I solve a multivariable ($x_1$ to $x_m$) nonlinear (in this case a quadratic) objective function with mixed (equality (my Sum) and inequality (bounded variables) in this case linear ...
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39 views

Lagrange multiplier uses

I'm currently self studying calculus of variations and stumbled upon something called a Lagrange multiplier, used in solving isoperimetric problems and the like. Not knowing what they are I backtrack ...
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36 views

constrained minimization in N dimensions

I am looking to create an algorithm to minimize an N dimensional problem. I am unsure how to write it in its generic form, so I will show it in 1, 2 and 3 dimensions Minimize $ \sum_{i} x_i\left [ ...
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37 views

Which Lagrangian multiplier is correct?

$$\max_{x_1,x_2} f(x_1,x_2) \text{ subject to }g(x_1,x_2)=0.$$ So, for critical points, form Lagrangian$$\mathcal L(x_1,x_2,\lambda)\equiv f(x_1,x_2)-\lambda g(x_1,x_2).$$ and then, ...
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22 views

how to get the dual of that optimization problem

max $1^\top x$ such that $x^\top M x = 0$ and $x_i^2 = x_i$ for all i For the above problem how can I derive the dual form. My main problem is to choose matrix notation or the element-wise notation ...
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24 views

how to check slater condition for a constrained optimization problem?

Given any optimization problem that you suppose to solve with Lagrange by thrusting strong duality, you need to be sure the Slater Conditions. And I guess there is no algorithmic way to solve for all ...
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65 views

What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot ...