Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.
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381 views
If the Minkowski sum of two convex closed sets is a Euclidean ball, then can the two sets be anything other than Euclidean balls?
If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven't ...
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votes
1answer
296 views
A (mathematically) sound investment strategy
It is common wisdom in the investment community that a long-term investor saving for his future would do well to invest in high-risk/high-return assets when he is young, slowly switching his portfolio ...
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votes
5answers
407 views
Why is convexity more important than quasi-convexity in optimization?
In the mathematical optimization literature it is common to distinguish problems according to whether or not they are convex. The reason seems to be that convex problems are guaranteed to have ...
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2answers
170 views
On the convexity of element-wise norm 1 of the inverse
Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as
$$
\|A\|_1= \sum_{i,j} |A_{i,j}|.
$$
Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
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votes
1answer
250 views
invex functions and their usefulness?
An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, ...
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votes
2answers
360 views
Please explain the intuition behind the dual problem in optimization.
I've studied convex optimization pretty carefully, but don't feel that I have yet "grokked" the dual problem. Here are some questions I would like to understand more deeply/clearly/simply:
1) How ...
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votes
3answers
292 views
Why is the affine hull of the unit circle R^2?
In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $R^n$ as
$$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \ldots ...
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1answer
177 views
Minimizing l-infinity norm of complex vector
I have an $n$-dimensional complex vector space, and I want to minimize the $L_\infty$ norm of a point that is constrained to an $m$-dimensional affine subspace. That is,
Given $\mathbf{z} \in ...
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votes
0answers
92 views
Subgradient of convex minimization duality
$$\min(f_0(x))$$
$$\text{s.t. }f_i(x) \le y_i \forall i, i = 1 ,\ldots, m$$
$$f_i : \text{convex};\quad x : \text{variable}$$
It is also considered that $g(y)$ is the optimal value of the problem ...
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votes
1answer
107 views
a conjecture on norms and convex functions over polytopes
Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
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0answers
219 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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0answers
61 views
How to understand convex duality intuitively
Is there an intuitive way to understand the convex duality? If the primal problem is minimization, the dual is maximization over another set of variables - but I would love to have a geometric ...
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votes
0answers
100 views
Need advice: what should be my next step?
I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
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votes
2answers
135 views
Is the rising factorial function a convex function?
Let $(x)_p=x(x+1)\dots(x+p-1)$ be the rising factorial function.
My question is: Is $(x)_p$ a convex function or not? And how to proof?
And what is about the falling factorial function ...
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votes
2answers
71 views
What is the difference between minimum and infimum?
What is the difference between minimum and infimum?
I have a great confusion about this.
