Questions tagged [cone]
For questions relating to the mathematical or geometric structure known as a cone.
10 questions
6
votes
1
answer
267
views
Is it possible to express these constraints with basic cones?
I have the following optimization problem:
\begin{align}\min&\quad x\\
\text{s.t.}&\quad x=\max_{i} \{x_{i}\}\\
&\quad x_{i}y_{i}=z_{i}\\
&\quad x_{i}, y_{i}, z_{i}\geqslant0
\end{...
- 61
2
votes
1
answer
152
views
general approach to iterating extreme rays of solution cone
Suppose I'm at an optimal solution of an LP relaxation in a MILP branch-and-bound descent. I want to add an additional cut of my own devices. To compute this cut I need the extreme rays of the cone ...
- 970
3
votes
1
answer
306
views
Practical, Short example of Mixed Integer Conic Program
Mixed Integer Conic Programs is a family of Mixed Integer Programs which are convex in all non integer variables. I am giving presentation on Mixed Integer technology. A large part of the presentation ...
- 4,049
3
votes
1
answer
350
views
Express equality constraint involving exponentials cones
The exponential cone is define such that
$(x, y, z) \in \text{ExpCone: if } y \exp(x / y) \leq z \land y > 0.$ The inequality $\exp(a) \leq b$ can be expressed as $[a, 1, b] \in \text{ExpCone}$.
...
- 4,049
7
votes
1
answer
1k
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 ...
- 1,731
0
votes
2
answers
184
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 ...
- 419
10
votes
1
answer
933
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 ...
- 527
5
votes
1
answer
907
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 ...
- 161
8
votes
2
answers
474
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 ...
- 4,030
16
votes
3
answers
574
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 ...
- 2,043