[Scilab-Dev] Quadratic programming: Fold equality constraints into objective function?

[Ran Gutin]

Hi, the quadratic programming solver qpsolve in Scilab has an argument for
specifying equality constraints, while also requiring that the quadratic
form $Q$ be positive definite. I've done some manipulations (similar to the
ones on Wikipedia [1]), and the equality constraints can be "folded into"
$Q$, so that it can go from not being positive definite to being positive
definite. Could the requirement that $Q$ be positive definite be relaxed?

[1] - https://en.wikipedia.org/wiki/Quadratic_programming#
Equality_constraints
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The way to "eliminate" the equality constraints $Ax = b$ is to solve $Ax =
b$ by finding a vector $x_0$ and a matrix $K$ such that $Ax_0=b$ and $K$ is
the kernel of $A$ (this is what the `linsolve` function does). You can then
do the substitution $x = x_0 + K u$ where $u$ is an arbitrary column vector
with compatible dimensions. The substitution is done inside both the
objective function and the inequality constraints. The quadratic program
then tries to optimize in terms of the variable $u$. Since $u$ has fewer
dimensions than $x$, this results in a quadratic program with fewer
dimensions. Once the optimal $u^*$ is found, compute $x^*$ by $x^* = x_0 +
K u^*$.
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