The Basic Algorithm for the Constrained Zero-One Quadratic Programming Problem with k-diagonal Matrix and Its Application in the Power System

Abstract

Zero-one quadratic programming is a classical combinatorial optimization problem that has many real-world applications. However, it is well known that zero-one quadratic programming is non-deterministic polynomial-hard (NP-hard) in general. On one hand, the exact solution algorithms that can guarantee the global optimum are very time consuming. And on the other hand, the heuristic algorithms that generate the solution quickly can only provide local optimum. Due to this reason, identifying polynomially solvable subclasses of zero-one quadratic programming problems and their corresponding algorithms is a promising way to not only compromise these two sides but also offer theoretical insight into the complicated nature of the problem. By combining the basic algorithm and dynamic programming method, we propose an effective algorithm in this paper to solve the general linearly constrained zero-one quadratic programming problem with a k-diagonal matrix. In our algorithm, the value of k is changeable that covers different subclasses of the problem. The theoretical analysis and experimental results reveal that our proposed algorithm is reasonably effective and efficient. In addition, the placement of the phasor measurement units problem in the power system is adopted as an example to illustrate the potential real-world applications of this algorithm.

An innovative deterministic algorithm for optimal placement of micro phasor measurement units in radial electricity distribution systems

“…OPP to ensure complete observability of electricity distribution systems is a kind of combinatorial optimization problem, which are optimization problems for discrete variables (Gu and Chen, 2020). If zero injections are not considered, OPP is essentially the classic “dominating set” problem where a subset of buses in the system is selected so that every bus is either in the subset or in a neighbor of the subset (Li et al , 2013).…”

“…Existing deterministic algorithms can guarantee the global optimum. However, they are very time-consuming and only suitable for small-scale problems (Gu and Chen, 2020; Mabaning and Orillaza, 2016). ES is the most reliable in generating all the possible solutions and finding the global optimum one.…”

“…Metaheuristic algorithms, such as simulated annealing, genetic algorithms (GAs), neural networks and ant colony algorithms, can solve medium and large-scale problems quickly in general. However, most of them can only find the local optimum (Gu and Chen, 2020; Azizi et al , 2012). In comparison, the optimization time of the simulated annealing algorithm is very long; the tabu-search algorithm has a strong dependence on the initial solution and can only be serialized.…”

“…In comparison, the optimization time of the simulated annealing algorithm is very long; the tabu-search algorithm has a strong dependence on the initial solution and can only be serialized. GA easily falls into local optimum and has poor convergence performance; the ant-colony algorithm has a long optimization time and is prone to falling into local optimization (Gu and Chen, 2020; Azizi et al , 2012; Zhou et al , 2018).…”