# Potential methods for solving quadratic optmization problem

I am trying to solve a non-convex optimization problem with the help of sequential quadratic programming.

I need to develop an algorithm inside SQP to solve this subproblem. What potential methods (conjugate gradient, etc) can be used to solve this, keeping in mind the Hessian $$H$$ is not positive semidefinite?

Use an off-the-shelf non-convex QP solver. There is a choice of solving to local optimality or to global optimality. Solving to local optimality is generally much faster, and may (but not necessarily) provide better overall algorithm progress (with starting point for the QP solve being the current iterate (incumbent)). That is because a local minimum is more likely to be close to the starting point, and therefore be in a region where the QP objective function is likely to be a better model of the objective of the original problem. If the QP solves for the step to the new iterate (which your formulation apparently does based on $$\Delta x$$ ), then a starting point of the vector of zeros can be used for the QP solve.