A new polynomially solvable class of quadratic optimization problems with box constraints

Abstract

We consider the quadratic optimization problem max x ∈C x T Qx + q T x, where C ⊆ R n is a box and r ≔ rank(Q) is assumed to be O(1) (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary Q and q. The idea is based on a reduction of the problem to enumeration of faces of a certain zonotope in dimension O(r). This paper generalizes previous results where Q had been assumed to be positive semidefinite and no linear term was allowed in the objective function. Positive definiteness was a strong restriction and it is now relaxed. Generally, the problem is NP-hard; this paper describes a new polynomially solvable class of instances, larger than those known previously.

Warm-starting quantum optimization

“…By using a grid scan at depth one, we mitigated the effect of local optima. Further work could exploit other possible warm-starts, e.g., based on polynomially-solvable special cases [60,61], where one could for example consider low-rank approximations of Σ, as well as analysis of the convergence properties when using a modified mixer that does not have the initial state as eigenstate.…”

“…Considering that any mixed-integer linear program can be encoded in a QUBO [58], QUBO is NP-Hard. Indeed, even checking local optimality is NP-Hard [59], and hence only very special cases [60,61] can be solved in polynomial time.…”