Solving spread spectrum radar polyphase code design problem by tabu search and variable neighbourhood search

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Abstract

A basic variable neighbourhood search (VNS) heuristic is applied for the first time to continuous min–max global optimization problems. The method is tested on a class of NP-hard global optimization problems arising from the synthesis of radar polyphase codes that has already been successfully treated by tabu search. The computational results show that VNS in average outperforms tabu search.

Introduction

Problems of optimal design are natural field of application for global optimization algorithms. Such an engineering problem arises in the spread spectrum radar polyphase code design. When designing a radar system that uses pulse compression, great attention must be given to the choice of the appropriate waveform. Many methods of radar pulse modulation that make pulse compression possible are known. Polyphase codes are attractive as they offer lower side-lobes in the compressed signal and easier implementation of digital processing techniques. Dukić and Dobrosavljević [2] propose a new method for polyphase pulse compression code synthesis, which is based on the properties of the aperiodic autocorrelation function and the assumption of coherent radar pulse processing in the receiver.

The problem under consideration is modelled as a min–max nonlinear nonconvex optimization problem in continuous variables and with numerous local optima. It can be expressed as follows:globalminx∈Xf(x)=max1(x),…,ϕ2m(x)},X={(x1,…,xn)∈Rn|0⩽xj⩽2π,j=1,…,n},where m=2n−1 andϕ2i−1(x)=∑nj=icosjk=|2i−j−1|+1xk,i=1,…,n,ϕ2i(x)=0.5+∑nj=i+1cosjk=|2i−j|+1xk,i=1,…,n−1,ϕm+i(x)=−ϕi(x),i=1,…,m.Here the objective is to minimize the module of the biggest among the samples of the so-called autocorrelation function which is related to the complex envelope of the compressed radar pulse at the optimal receiver output, while the variables represent symmetrized phase differences. It has been proved that problem (1) is NP-hard [10]. The objective function of this problem for the case n=2 is illustrated in Fig. 1.

Problem (1) belongs to the class of continuous min–max global optimization problems. They are characterized by the fact that the objective function is piecewise smooth. A number of methods for global optimization have been proposed, both deterministic and nondeterministic (for a comprehensive bibliography see [14], [15]). Nevertheless, there are no efficient mathematical approaches to solve the problem in general. Neither does there exist widely available and efficient software, which works on problems of higher dimensions (for a survey see [11], [13]).

The existing heuristic approaches for solving global optimization problems include Monte-Carlo methods (MCs), multi-level random search methods, adaptive Simulated Annealing and Genetic algorithms, clustering methods, tabu search, etc. A multi-level variant of tabu search heuristic, called the multi-level tabu search (MLTS), for solving global optimization problems was developed in [4], [9], [10] and successfully applied to (1). In this paper recently proposed variable neighbourhood search (VNS) metaheuristic is for the first time applied to continuous min–max global optimization problems. The VNS is tested against the MLTS on instances of problem (1) with n=2,…,20. We find that VNS in average outperforms MLTS.

The organization of the paper is as follows: Section 2 describes a tabu search approach for solving problems of the type (1). In Section 3 we describe the basic VNS approach. Section 4 presents our implementation of VNS for solving min–max problems subject to box constraints. Computational experiments with various instances of problem (1) are reported in Section 5, while Section 6 summarizes conclusions of the paper.

Section snippets

Multi-level tabu search

The MLTS heuristic [4], [10] arises from the original tabu search methodology [5], which was first proposed to solve combinatorial optimization problems. The method gave encouraging results on a series of standard lower dimensional test problems and the first serious challenge was the Spread Spectrum Radar Polyphase Code Design problem (1). Dukić and Dobrosavljević analyzed (1) using a multistart simulated annealing approach and the results for n⩽16 reported in [3] show that radar pulse

A basic variable neighbourhood search heuristic

In this section we describe in general terms the basic rules of the VNS metaheuristic.

The VNS is a recently proposed metaheuristic for solving combinatorial problems (see [12] and for a recent survey see [7], [8]). The basic idea is to use more than one neighbourhood structure and to proceed to a systematic change of them within a local search. The algorithm remains in the same solution until another solution better than the incumbent is found and then jumps there. So it is not a trajectory

VNS for solving the min–max global optimization problem

Two different easy VNS versions for solving Spread Spectrum Radar Polyphase Code Design problem are developed in this section. They differ only in step (2b) of the basic VNS, i.e., in local search routine. The neighbourhoods Nk(x) (k=1,…,kmax) used in both versions are hyper-cubes around the incumbent x with different sizes.

The first proposed local search uses the same set of directions as MLTS heuristic. In [4], [10] it was shown that MLTS with one level and a small step can itself be used as

Computational results

The power of the VNS algorithms, described in Section 4 was tested on problem (1) for dimensions n=2,…,20. Heuristics were implemented in C programming language, and for the sake of comparison, the MLTS heuristic has been implemented in C as well (previous results reported in [4], [10] were obtained from Fortran implementation of MLTS). Moreover, in [4], [10] the stopping criteria were the maximal number of iterations and the maximal number of iterations between two improvements.

In order to

Conclusion

The main disadvantage of the described tabu search procedure for solving the Spread Spectrum Radar Polyphase Code Design problem is the use of more than five parameters. In this paper, the basic VNS heuristic suggested for that purpose uses two or three parameters only. The neighbourhood structure is defined by the choice of a set of directions around the current solution and a parameter, which is equal to the distance travelled along a direction. Directions selected are those of steepest

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