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Questions tagged [cone]

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

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6 votes
1 answer

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{...
PNoug's user avatar
  • 61
2 votes
1 answer

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 ...
Brannon's user avatar
  • 900
3 votes
1 answer

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 ...
worldsmithhelper's user avatar
3 votes
1 answer

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 $[a, 1, b] \in \text{ExpCone}$. ...
worldsmithhelper's user avatar
7 votes
1 answer

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 ...
k88074's user avatar
  • 1,671
0 votes
2 answers

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 ...
DSPinfinity's user avatar
10 votes
1 answer

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 ...
Snowflake's user avatar
  • 517
5 votes
1 answer

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 ...
Sam's user avatar
  • 161
8 votes
2 answers

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 ...
independentvariable's user avatar
16 votes
3 answers

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 ...
YukiJ's user avatar
  • 2,023