# Tagged Questions

Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

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### Question on optimization problem computing its solution function

I was just presented with this in Optimization class involving an optimization problem, on which I have no clue, it reads: (P) minimize-> $f(x_1,x_2) = (x_1^2 -2x_1 + x_2^2 + 1)^{1/2}$ subject ...
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### Only one system of equation has solution

Let $A$ be a $m\times n$ real matrix. Show that only one of the following systems has solution: (I): $Ax > 0$ (II): $Ay = 0, y \geq 0, y \neq \theta$, where $\theta$ is zero vector. I ...
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### Show that system $Ax\geq 0,$ and $A^Ty=0, y\geq 0$ has solution satisfies $Ax+y>0$.

Let $A$ be a $m\times n$ real matrix. Show that system (I): $Ax\geq 0,$ (II): $A^Ty=0, y\geq 0$ has solution satisfies $Ax+y>0$. I have no idea to prove above claim. Can you give me a ...
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### Convex optimization when Hessian is non-invertible

1) Are there any extensions to Newton's method for finding minimum of a convex function when the Hessian is singular ? (I have all positive eigenvalues in the Hessian except one which is zero) I ...
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### Optimum is achieved when both variables are equal

Consider the problem $\max_{y,z:\|y\|_\infty,\|z\|_\infty \leq 1}y^TBz$, where $B$ is symmetric, positive semidefinite, $y,z\in \mathbb{R}^n$, $\|z\|_\infty=\max_{i\in\{1,\ldots,n\}}|z_i|$. It turns ...
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### how to get orthogonal rank 1 approximations?

The situation: I have $k$ matrices $A_i$, which are all real and of size $m\times n$. Now I would like to find the matrices $\tilde{A}_i$ of $A_i$ so that 1) $\tilde{A}_i$ is of rank 1 (thus a rank 1 ...
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### Finding a polynomial approximation of a PDF

I would like to find a polynomial $P(x)=\sum_{d=1}^D P_dx^d$ of degree $D$, where its derivative is larger than or equal to a given pdf $f(x)$ in $[0,1-\epsilon]$, for any $\epsilon>0$. Note that ...
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### Fenchel Duality in Prof. Bertsekas' lecture

Please see this link, p.39-41 (sufficient to answer my question), before (1.47) for detailed. For convenience, the relevant part is shown as: I am confused in two things: The ...
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### Extreme pts of a polyhedral feasible set

Consider a linear program $\min \{c^T x:Px=q,x\geq 0 \}$, where $P \in \mathbb{R}^{m \times n}$. $x\geq 0$ means each component of $x_i$ of x is nonnegative. The feasible set is $\{x:Px=q,x\geq 0\}$....
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### Some problems in finding conjugate function

Ask the following fundamental problems: How to derive the conjugate function of $g(y)$ if given "$\underset{y \geq 0}{\text{sup}}\{g(y)-y^Tx\}$"? My attempt is as following: \begin{align*} &...
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### Conjugacy relation in the primal and dual problem

The following is my derivation in the Conjugacy relation in the primal and dual problem. I am shaky in it; so hope for some advices. Consider the following problem, $f(x),g(x)$ are convex ...