Questions tagged [cone]

For questions relating to the mathematical or geometric structure known as a cone.

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6
votes
1answer
89 views

Extreme rays of a small polyhedral cone: How do I get them?

In a nutshell I have a small 2-dimensional polyhedral cone. $$C=\{(x_1,x_2): 2x_1-x_2 \leq 0, x_1+3x_2 \leq 0\}$$ I am looking for a simple, illustrative, procedure to get its extreme rays. Any ...
0
votes
2answers
123 views

Separating hyperplanes for a convex cone

Let $W$ be a fixed matrix. Define $$\operatorname{pos}W \triangleq \{t \mid Wy =t , y≥ 0\}.$$ It is called the positive hull of $W$. It represents the set of right-hand sides that can be obtained by a ...
9
votes
1answer
194 views

Intuition behind SOCP and why it sometimes can be solved more efficiently than without transforming it into a SOCP?

I was doing an assignment regarding turbine placements, where one has to write and solve a maximisation problem. As I was reading into the topic, I came across the fact that I could transform the ...
4
votes
1answer
332 views

Transforming a Quadratic constraint to SOCP

I have a problem similar to Markowitz portfolio optimization that I would like to transform into second-order cone programming. I have a linear objective function with a quadratic constraint (assuming ...
8
votes
2answers
175 views

Convex Optimization: Separation of Cones

I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as: 2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
16
votes
3answers
266 views

How to take the dual of a conic optimization problem?

Given a conic problem $$\min \{c^\top x \mid Ax \succeq_\mathit{C} b\}$$ for an arbitrary cone $C$, how can I construct the dual to the problem? Moreover, in Linear Programming one constructs the ...