For a linear system of equations constrained by inequalities, is A x ≤ b =&gt; y T A x ≤ y T b acceptable? Or does that not generally hold.( y T being the transpose of y).

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For a linear system of equations constrained by inequalities, is   A  x  ≤  b  =&amp;gt;      y    T    A  x  ≤      y    T    b acceptable? Or does that not generally hold.(      y    T   being the transpose of   y).

Brennen Bishop · Accepted Answer

Assuming that   x and   y are column vectors, and that the entries of   y are non-negative, then yes,   A  x  ≤  b implies         y    T    A  x  ≤        y    T    b.This amounts to taking a linear combination of inequalities where the weights are non-negative. Incidentally, this can be used to prove the &quot;Weak Duality Theorem&quot;, i.e., that the value of the objective function of a maximization problem is bounded above by the value of the corresponding objective function of the dual minimization problem.

tilfaen4a · Accepted Answer

What do you mean by &quot;transform&quot;? Do you mean:   A  x  ≤  b is equivalent to       y    T    A  x  ≤      y    T    b, or   A  x  ≤  b implies       y    T    A  x  ≤      y    T    b

For a linear system of equations constrained by inequalities, is A x ≤

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