As was pointed out in Section 1.1, the canonical form of the sparse recovery problem (Eq. 2) is NP-hard [14] and cannot be solved efficiently as-is.
And instead of multiplying the distances, we now sum them and we turn the problem into a top- k shortest path problem, which can be solved efficiently by extending algorithms, such as Dijkstra algorithm (Dijkstra, 1959).
They are easy to implement as they can be solved efficiently by standard linear algebra components and outperform the traditional clustering algorithms such as the k-means algorithm.
It is observed that the above problem belongs to the linear programming and thus can be solved efficiently by linear programming methods such as the simplex method [34].
They can be solved efficiently by advanced optimization solvers, such as the IBM ILOG CPLEX Optimizer, making them suitable for robust online application.
This is significant because QUBO problems are considered among the main contenders for NP-hard problems that could be solved efficiently on a quantum computer such as D-Wave.
We note that the relaxed SDP problem (7) is a common convex optimization problem which can be solved efficiently by the existing solvers such as SDPT3 [18] and SeDuMi [19].
For a fixed scalar variable, we represent this criterion in terms of linear matrix inequalities (LMIs), which can be solved efficiently via existing numerical algorithms such as interior point algorithms [13, 14].
When first-stage variables are mixed-integer and second-stage variables are continuous, these MILPs can be solved efficiently by classical decomposition methods, such as Dantzig/Wolfe decomposition (DWD), Lagrangian decomposition, and Benders decomposition (BD), or a cross decomposition strategy that combines some of the classical decomposition methods.
As a result, the optimization problem in (27) can be solved efficiently by well-known linear programming methods, such as simplex methods or interior point methods [27].
This formulation can be applied to areas such as document clustering [26] and can be solved efficiently for very large m and n [30].
