# Tagged Questions

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### Minimization Problems & Collective Behavior

I am having some trouble with this problem: http://imgur.com/MX1aPzT Could anybody help me understand the concepts behind the question and how I should go about it? I have been reading through some ...
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### Maximize $x^2+y^2+z^2$ on $x^2+y^2+4z^2 = 1$

Hi this is a lagrangian optimization problem. Essentially as the title says, the question is asking us to maximize (if possible) $x^2+y^2+z^2$ on $x^2+y^2+4z^2=1$. I started by the standard ...
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### 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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### Optimizing single element of vector wrt. second order cone constraints

Can anybody put me on the right path to solving the following problem analytically: Given a vector $\bf{x}=(x_1,...,x_n)^T$, how do I find the bounds for a single element subject to second order cone ...
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### 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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### 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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### Maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$

I'm trying to find the maximum of $(1-q_1)(1-q_2)\ldots(1-q_n)$ where $n\ge 2$, on a the set $\{(q_1,\ldots , q_n) :q_1^2+q_2^2+\ldots+q_n^2=1 \ q_i\ge 0 \}$ (With the condition $q_i\ge0$ this is just ...
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### Extrema of two variable function

Find extrema of $f(x,y)=x^2-xy+y^2$ from set $M=\{ [x,y] \in \mathbb{R}^2;|x|+|y|\le1\}$ I am solving this kind of problems for the first time and I am not sure what I am doing, what I have got ...
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### Optimization using Lagrange Multipliers for conditions with different codomain

I'm trying to maximize the trace of $X^TAX$ subject to the columns of $X$ being orthonormal, where $A$ is a diagonal matrix and X is not necessarily square, but does not have more columns than rows. ...
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### Using LaGrange multipliers to solve for minimum

I am having troubles with one part of this homework problem. Hopefully somebody can help me out: Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the given constraint ...
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### Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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### Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
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