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I was wondering if anyone can help me with this (probably basic) question. I want to know how the following feasible region looks like if we have thousands of variables. The constraints are linear. The region is convex and closed. Does the region have a scientific or mathematical name and specific properties (especially for finding projections of vectors on this region)?

\begin{array}{ll} & Ax = b \\ & Bx \le d \\ &x \ge 0. \end{array}

Thanks a lot for your time and help.

Mehrzad
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  • Welcome to Math.SE! I don't understand what you are asking for specifically. If there are thousands of variables, then it isn't possible to visualize, right? Anyway this would just be a (thousands-dimensional) polyhedron right? Are you referring to LP? https://en.wikipedia.org/wiki/Linear_programming Any effort that could be put into clarifying this question would be appreciated. – Chill2Macht Dec 09 '18 at 01:10
  • Thanks for your reply. I am not familiar with mathematical terms, so I am looking to understand if the region has a special characteristic that I can use to find projections much easier (find an answer to this question https://math.stackexchange.com/questions/3030617/projection-on-convex-sets-with-equality-and-inequality-constraints). – Mehrzad Dec 09 '18 at 02:06
  • As Chill2Macht notes, it’s a polyhedron. Optimizing linear and quadratic objective functions over polyhedra can be done very efficiently, even for thousands of variables. I suspect there is no better way, unless you know more about $A$, $B$, etc. – David M. Dec 12 '18 at 05:10
  • Thanks for your answer. Yes. I can solve it using CPLEX, but I need to find this projection over thousands of iterations too, so I am looking for a really fast performing way. In fact, computational time is really important in my case. I know what are A and B, but what specific structure should I look for in A and B? any resource (book, paper or webpage) that explains how the projection becomes easier like multiplying matrices instead of optimization and in which cases (what should be A and B) will really help too. Thanks in advance. – Mehrzad Dec 13 '18 at 00:44

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The mathematical name for this region is convex hull.

Now, how does this region look like ? It is not representable if you have thousands of variables, as we cannot see in $4D$ and above. Each variable is a dimension, so you can only draw the region if there are less than $3$ variables.

In $3D$ it would look like this.

Kuifje
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  • Thanks a lot for your answer. Does this set include the inner part of the convex hull in this figure "https://commons.wikimedia.org/wiki/File:Truncated_cuboctahedron,_ball-and-stick,_triangles.png#/media/File:Truncated_cuboctahedron,_ball-and-stick,_triangles.png" or does this set only include the boundaries of this convex hull (since there are equality constraints, and all variables should satisfy some type of equality constraints)? – Mehrzad Dec 09 '18 at 19:27