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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. ...
Pantelis Sopasakis's user avatar
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 ...
YukiJ's user avatar
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13 votes
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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 ...
mtanneau's user avatar
  • 4,108
11 votes
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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 ...
Mark L. Stone's user avatar
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): ...
Mark L. Stone's user avatar
9 votes
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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 ...
Mark L. Stone's user avatar
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 "...
Mark L. Stone's user avatar
4 votes
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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 $1$...
Mark L. Stone's user avatar
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 $x_1 + 3x_2 = 0$. ...
prubin's user avatar
  • 38.8k
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,...
mtanneau's user avatar
  • 4,108
2 votes
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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 ...
mtanneau's user avatar
  • 4,108
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 ...
independentvariable's user avatar
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 ...
k88074's user avatar
  • 1,661
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 ...
jDAQ's user avatar
  • 111

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