# Tag Info

Accepted

### Involving indicator function as a constraint in a LP problem

It's possible to do with a mixed-integer linear program: just add the constraint $$-a_i x \le b + A(1-y_i)$$ where $y_1, y_2, \dots, y_M$ all take values in $\{0,1\}$, and $A$ is a large constant ...
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### Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations

I spent a few more hours on the problem, and eventually found the solution. No major breakthroughs, but a lot of algebra. First, I reversed the problem, instead maximizing the volume subject to a ...

### Minimum value of $(x^2+y^2)^2$

I think a fun way to do the problem similar to what you've done is by using polar co-ordinates. Sub $x=r \cos \theta$ and $y=r \sin \theta$. Then you want to minimise $(x^2+y^2)^2=r^4$. Note that the ...
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### Conic by three points and two tangent lines

The comment by Narasimham made me aware of a very elegant way of tackling this problem. The figure above can be interpreted as the orthogonal projection of a right cone whose axis lies in the plane. ...
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### How to use Karush-Kuhn-Tucker (KKT) conditions in inequality constrained optimization

Think about what an inequality constraint means for optimality: either the optimum is away from the boundary of the optimization domain, and so the constraint plays no role; or the optimum is on the ...
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### How is the inequality constraint $\mbox{Tr}(W) \geq c$ convex?

$\text{Tr}(W)$ is a linear function, so the constraint is convex.
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### How is the inequality constraint $\mbox{Tr}(W) \geq c$ convex?

Suppose $$Tr(X) \ge c$$ and $$Tr(Y) \ge c$$ For $\alpha \in (0,1)$, we have $$Tr(\alpha X + (1-\alpha) Y)=\alpha Tr(X)+(1-\alpha)Tr(Y) \ge c$$ Hence it is convex. Note that trace is a linear ...
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### Derivative of the solution of a linear program

This is known as sensitivity analysis. If you have a non-degenerate optimal basic feasible solution, it is relatively simple to find derivatives of the optimal BFS or the optimal objective value ...

### Constructible problems for which the solution is non-constructible?

First observe that there are constructive real numbers $a$ such that we (working in ZF) do not (currently) know whether $a$ is zero or positive. For example, let $a$ be given by the Cauchy sequence ...

### Notation: is a factor of

Well $|$ is used to describe divisibility so I think it meets our requirements. For example, $3|6$. $x$ is a factor of $y$ $\Leftrightarrow$ $x|y$.
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### Minimum value of $(x^2+y^2)^2$
$$(x^2+y^2)^2=(x^2-y^2)^2+4x^2y^2=(6-2xy)^2+4x^2y^2=2(2xy-3)^2+18\ge18$$ The equality occurs if $2xy=3$
You can rephrase $$y_{\min} \leq Ax \leq y_{\max},$$ as $$\begin{pmatrix}A \\ -A\end{pmatrix}x \leq \begin{pmatrix}y_{\max} \\ -y_{\min}\end{pmatrix},$$ which is in the format of what quadprog accepts....