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Linear programming is the study of optimizing a linear function over a set of linear inequalities. The Simplex Method, Ellipsoid Method and Interior Point Method are popular algorithms to solve linear programs.

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How do you call a linear programming problem when the solution should be "constrained" to a ...

If you are willing to replace $a_i > 0$ by $a_i \ge 0$, then this becomes a quadratic program. Indeed, it can be formulated as \begin{align*} \text{Minimize}\quad & \frac12 a^\top Q a + q^\top a, \\ …
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