Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem

Abstract

The unconstrained binary quadratic programming problem (BQP) is known to be NP-hard and has many practical applications. This paper presents a simulated annealing (SA)-based heuristic for the BQP. The new SA heuristic for the BQP is based on a simple (1-opt) local search heuristic and designed with a simple cooling schedule, but the multiple annealing processes are adopted. To show practical performances of the SA, we test on publicly available benchmark instances of large size ranging from 500 to 2500 variables and compare them with other heuristics such as multi-start local search, the previous SA, tabu search, and genetic algorithm incorporating the 1-opt local search. Computational results indicate that our SA leads to high-quality solutions with short times and is more effective than the competitors particularly for the largest benchmark set. Furthermore, the values of new best-known solutions found by the SA for several large instances are also reported.

Introduction

The objective of the unconstrained binary quadratic programming problem (BQP) is to find, given a symmetric rational n×n matrix Q=(q_ij), a binary vector of length n that maximizes the following quantity:

$$\text{f(x)=x}{}^{\mathrm{t}}\text{Qx=∑}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{∑}{}_{\mathrm{j=1}}{}^{\mathrm{n}}\text{q}{}_{\mathrm{ij}}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{j}}\text{,}\text{x}{}_{\mathrm{i}}\text{∈{0,1}}\text{∀i=1,…,n.}$$

The BQP is an NP-hard problem and has a large number of important applications, e.g., machine scheduling [2], traffic message management problem [9], CAD problem [16], capital budgeting and financial analysis problem [19], molecular conformation problem [25]. Furthermore, it has been known that the BQP is equivalent to many classical combinatorial optimization problems such as maximum cut, maximum clique, maximum vertex packing, minimum vertex cover, maximum independent set, and maximum weight independent set problems [4], [6], [12], [23], [24].

To solve the BQP, several exact methods such as branch and bound or branch and cut have been developed by researchers Pardalos and Rodgers [22], Barahona et al. [4], Billionnet and Sutter [7], and Helmberg and Rendl [11]. However, due to the computational complexity of the problem, at the present time they are only capable of solving the small-size instances. For larger instances, such methods would become prohibitively expensive to apply, whereas high-performance heuristic algorithms might find high-quality solutions with short times.

Recently, several heuristic approaches have been proposed for the BQP. Glover et al. [10] have proposed a tabu search heuristic algorithm to find near-optimal solutions within reasonable running times for instances with up to 500 variables. As another heuristic approach, Lodi et al. [18] have proposed an evolutionary heuristic and tested it on the same problem set studied by Glover et al. Simulated annealing (SA) approach has been already proposed by Alkhamis et al. [3]. Unfortunately, they have dealt with only small-problem instances with up to 100 variables, but some suggestions on setting parameters of the SA may be useful. Beasley [6] has presented a tabu search and SA for larger instances with up to 2500 variables provided as new benchmark test problems of the OR-Library [5]. Merz and Freisleben [20] have showed impressive results obtained by a genetic local search algorithm. Furthermore, they reported the new best-known solution values for several instances of the benchmark.

These heuristic algorithms described above belong to the meta-heuristics and often are based on a local search. Moreover, useful ideas derived from natural analogies are incorporated to improve the performance of the local search and to find high-quality solutions within reasonable times. The SA is one of the most well-known meta-heuristics for solving difficult optimization problems. From a simple point of view, the SA is generally composed of a process of local search and another process of avoiding entrapment in poor local optima by using a physical annealing mechanism that depends on a cooling scheduling of temperature. In the local search process for the BQP, 1-opt local search, which iteratively searches solutions that can be reached by changing one element in the current solution, has been adopted for a basic implementation of the SA, and in the other process, many forms have been investigated to efficiently solve the problems.

In this paper, we present a SA-based heuristic that is based on the 1-opt local search and is designed with a simple cooling scheduling, but the multiple annealing process starting from different temperatures is adopted, for the BQP. In order to show the practical performances of SA, we test on publicly available problem instances ranging from 500 to 2500 variables in the OR-Library [5] and compare them with the recent results of other existing heuristics that have been investigated on the same instances. Computational results indicate that our SA for the large-scale instances is highly promising in a trade-off between solutions obtained and running times in comparison with the previous existing heuristics such as other SA, tabu search, and genetic local search algorithms incorporating the 1-opt local search. We also give the new cost values of best-known solutions found by our SA for several large instances.

In Section 2, we describe the outlines of the local search heuristic and standard SA. Section 3 includes implementation details of our SA heuristic with the reannealing process. In Section 4, empirical performances of the SA are evaluated for the large instances and compared with the other heuristic approaches. Section 5 contains concluding remarks.