# Sequential Optimization

I am looking for the name of a sequential optimization, if that technique makes indeed any sense and exists.

Given the solution $x^*$ to a non-linear non-convex problem \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & \mathbf{f(x)}\\ & \text{subject to} & A_1\mathbf {x} \leq \mathbf {b}_1 \end{aligned} \end{equation*} with $f(x)$ non-linear and non-convex, is it reasonable to look at the quadratic problem \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & \frac{1}{2}\mathbf {x} ^{\mathrm{T} }Q\mathbf{x} +\mathbf{c}^{\mathrm {T} }\mathbf {x} \\ & \text{subject to} & A_2\mathbf {x} \leq \mathbf {b}_2 \\ & &\lVert \mathbf{x-x^*} \rVert_2 \leq \epsilon \end{aligned} \end{equation*} and if so, is there a name for looking for the solution of one optimization problem that is in some sense (not necessarily in the sense of the Euclidean norm) close to the solution of another. The practical background is that I would like to use different solvers for each problem. I understand that the solution $\overline{x}$ of the quadratic problem is not a global solution. Is there a name under which this has been studied?

• Are you just referencing Sequential Quadratic Programming, cite: Griva, Nash and Sofer: Linear and Non-Linear Optimization p573. Note this algorithm implemented in SciPy as method "SLSQP", sequential least squares quadratic programming. – Attack68 Aug 2 '18 at 23:53

• Gradient descent ($Q = \frac{1}{\alpha} I$) or Newton's method ($Q = \nabla^2 f$) can both be interpreted as minimizing successive quadratic approximations of a function.