15
votes
How to take the dual of a conic optimization problem?
$\newcommand{\Rbar}{\overline{\mathbb{R}}}\newcommand{\R}{\mathbb{R}}\newcommand{\minimize}{\operatorname{Minimize}}$Another way to derive the dual for any convex problem is to use Fenchel duality.
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13
votes
Accepted
How to take the dual of a conic optimization problem?
In Linear Programming (LP) one chooses a vector $\lambda \geq 0$ to obtain $\lambda^\top Ax \geq \lambda^\top b$ and whenever we find such a $\lambda \geq 0$ with $A^\top\lambda =c$ we obtain a lower ...
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12
votes
Accepted
Intuition behind SOCP and why it sometimes can be solved more efficiently than without transforming it into a SOCP?
Whether a given formulation is faster / more stable than another depends on the software you use to solve either.
What is the intuition behind it that a SOCP formulation can be solved more ...
- 3,998
11
votes
Accepted
Express equality constraint involving exponentials cones
Q: "How do i write $\text{exp}(a) = b$ using cone programming?"
A You don't.
$\text{exp}(a) = b$ is a nonlinear equality constraint, and is therefore non-convex.
$\text{exp}(a) \le b$ is ...
- 12.4k
9
votes
How to take the dual of a conic optimization problem?
Succinct and (freely) accessible references, which include general "theory". Also, solved examples,. for instance for Second Order Cones (SOCP) and Linear Semidefinite cones (LMI, i.e., Linear SDP):
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- 12.4k
9
votes
Accepted
Practical, Short example of Mixed Integer Conic Program
Your opening sentence could be more accurately written as
Mixed Integer Conic Programs are a family of Mixed Integer Programs whose continuous relaxations are (convex) conic programs.
One easy to ...
- 12.4k
5
votes
Is it possible to express these constraints with basic cones?
Let $x_i = \frac{z_i}{y_i}$.
Then, presuming $x_i$ does not appear in any of the other constraints, this appears to be a Generalized linear-fractional programming problem per section 4.3.2 "...
- 12.4k
4
votes
Accepted
Transforming a Quadratic constraint to SOCP
Revamp of my answer given the example now provided.
Let there be $n$ VaR factors. Let $R$ = $n$ by $n$ matrix of correlations (the 2nd matrix in your example) of the VaR factors.
Let $W$ = $n$ by $1$...
- 12.4k
4
votes
Extreme rays of a small polyhedral cone: How do I get them?
For your simple (2 variable, 2 side) cone, you are on the right track. An extreme ray will be defined by $n-1$ binding constraints, which in this case means either $2x_1 - x_2= 0$ or $x_1 + 3x_2 = 0$. ...
- 36.4k
4
votes
Separating hyperplanes for a convex cone
How can we prove that the number of possible separating hyperplanes (separating $p$ and $\operatorname{pos}W$) is finite based on the fact that $\operatorname{pos}W$ is finitely generated?
In general,...
- 3,998
2
votes
Accepted
general approach to iterating extreme rays of solution cone
The cone you are describing is often referred to as a basis cone (for instance, in Sec 2.3 of this paper, where the concept is used to derive cuts too). Note that you have such a cone for every ...
- 3,998
2
votes
Accepted
Convex Optimization: Separation of Cones
Ok, after seeing the wrong attempt below which has been edited multiple times, I believe it is time to close this question. I will just leave my attempt:
Assume $K^* \neq (\operatorname{int}K)^*$, so ...
- 3,818
1
vote
Accepted
Separating hyperplanes for a convex cone
Perhaps the argument is the following. Observe that $\operatorname{pos}W$ is a polyhedral cone (every finitely generated cone is polyhedral). That it is finitely generated can be seen from the fact ...
- 1,543
1
vote
Convex Optimization: Separation of Cones
A new approach focusing only on $(\boldsymbol{\operatorname{int}}K)^* = K^*$, since that seems to be the biggest problem to you.
From section 2.6 of Convex Optimization (Boyd, Vanderberghe) we have ...
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