My teacher teached me about Sequential Quadratic Programming (SQP). It is basically about taking the second order Taylor Series (TS) expansion around your current feasible point, and then finding the minimizer of that (at the risk of finding a local minimum). Then, in the next iteration, you perturb this minimizer a little bit and take the 2nd order TS about that point instead.
My teacher also talk about the Interior Point method, specifically the "Primal Path Following" one. In this method we take the 2nd order Taylor series of the function at the beginning, and then we keep increasing the parameter $\tau$ which is in the barrier function: $G(\tau, x) = c^Tx-(1/\tau)*log(x)$, find a new $x$, and then minimize the TS approximation again.
So, so me, the Interior POInt method is just a special case of sequential quadratic programming (SQP), in which the function we are making quadratic is always of the form: $G(\tau, x) = c^Tx-(1/\tau)*log(x)$.
Is this the correct way to build intuition around this? that it is always sequtnail quadratic programming?