I'm solving a geometric constrained optimization problem. The variables in the optimization are the $x-y$ components for a set of vectors (given by $v_i.x,v_i.y$ for vector $i$). The objective function is quadratic in these variables.
However, I need to constrain the SUM of the magnitudes of a subset of the vectors.
Specifically, suppose this subset consists of $\{v_1,v_2,\dots,v_n\}$
I need the solution to satisfy
$$\|v_1\|_2 + \|v_2\|_2 + \dots + \|v_n\|_2 < L$$
If it was just a single vector $v$ I could square both sides to get a quadratic constraint and frame the problem as a Quadratically Constrained Quadratic Program
$$\|v\|_2^2=(v.x)^2 + (v.y)^2 < L^2$$
However, I have multiple vectors. So is there any way to express the constraint in such a way that I could apply a technique more specific than general non-linear constrained optimization? Could I formulate it as a second order cone program?
Or given that my objective function can be minimized analytically, would it make sense to solve the problem by
- Ignoring the constraint and obtaining an optimum value $x^*$ for the objective function
- Projecting $x^*$ onto the constraint manifold numerically to get a final solution that satisfies the constraints?