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I still cannot understand why the topic is true.

The following is from convex optimization, S. Boyd

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My claim is the following:

  • The feasible region for $Gx \preceq h$ is a polyhedron, which is easy to understand and can be easily drawn on the paper.
  • The feasible region for $Ax=b$ is the intersection of all affine set defined by $a_i^Tx = b_i$ for each $i$, where $a_i, \forall i$ are rows of $A$. So there are three possibilities: 1. No solution. Then the feasible set of LP is empty. 2. One solution. Then the feasible set of LP is "at most" a point. 3. $\infty$ solution. Then the feasible set of LP is "at most" a line.

Why is the feasible set a polyhedron?

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The feasible region for $Ax=b$ is an intersection of affine sets, therefore also an affine set. However there are no further restrictions on the dimension of the solution space for $Ax=b$. It may very well be two- or higher-dimensional.

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  • $\begingroup$ But there is no guarantee that the feasible set on the lower dimension will be a polyhedron. $\endgroup$ – sleeve chen Sep 4 '16 at 0:25
  • $\begingroup$ The feasible set is not necessarily bounded, but it is always a polyhedron of some dimension. $\endgroup$ – Anon Sep 4 '16 at 0:58

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