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A linear optimization problem can be transformed from a optimization problem to a feasibility problem by using the dual program:

A solution of $\min \{c^Tx : Ax \geq b\}$ is equivalent to finding a solution to $\{ Ax \geq b, y^TA=c^T, y\geq 0\}$

Is this also possible in case of a quadratic convex objective (H is positive semidefinite) with linear bounds?

How can i rewrite the minimization problem $\min \{x^TH x + d^Tx : l \leq x \leq u\}$ to an equivalent feasibility problem?

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First note that convex quadratic program is not linear so you should not expect a linear counterpart duality sys for it , but using KKT condition you can characterize optimal solution by solving and quadratic inequalities .

KKT

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