Underdetermined system with inequality constraints A x = b ,where A ∈ R m × n with m &lt; n, subject to 0 ⪯ x ⪯ c .1. I would like to know if there is any way to express the feasible set for this problem analytically.2. Is there any way to obtain any of feasible solutions in closed form?

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Underdetermined system with inequality constraints  A  x  =  b  ,where   A  ∈        R          m      ×      n       with   m  &amp;lt;  n, subject to  0  ⪯  x  ⪯  c  .1. I would like to know if there is any way to express the feasible set for this problem analytically.2. Is there any way to obtain any of feasible solutions in closed form?

Jovan Wong · Accepted Answer

In general, the answer is no to both questions.Of course, you could always try a finite number of test points, including   x  =  0,   x  =  c, to see if they happen to satisfy   A  x  =  b; and you can try the minimum-norm solution   x  =      A    T    (  A      A    T        )          −      1        b, to see if it happens to satisfy   0  ⪯  x  ⪯  c. If any of these tests hold, then you&#039;ve found a closed form solution.But again, in general, you will not be able to. It&#039;s a very simple convex optimization problem to solve numerically, though. In fact, it is a linear program, unless you choose a nonlinear function (like the norm of   x) as an objective function. But there&#039;s no reason to do that; I&#039;d just minimize       ∑    i        x    i   in this case, if I didn&#039;t have any other preference. Technically, your objective could just be &quot;0&quot; as well, in which case the point it selects will truly be solver dependent.

Jamiya Weber · Accepted Answer

Thank you for your answer! I wonder if there is any condition that can tell us whether the feasible set is empty or not.

Underdetermined system with inequality constraints A x = b , where A &#x2

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