Review of Optimization Problems in Wireless Sensor Networks
Telecommunications Networks - Current Status and Future Trends
Introduction Wireless Sensor Networks (WSNs) are an interesting field of research because of their numerous applications and the possibility of integrating them into more complex network systems. The difficulties encountered in WSN design usually relate either to their stringent constraints, which include energy, bandwidth, memory and computational capabilities, or to the requirements of the particular application. As WSN design problems become more and more challenging, advances in the areas
... nces in the areas of Operations Research (OR) and Optimization are becoming increasingly useful in addressing them. This study is concerned with topics relating to network design (including coverage, topology and power control, the medium access mechanism and the duty cycle) and to routing in WSN. The optimization problems encountered in these areas are affected simultaneously by different parameters pertaining to the physical, Medium Access Control (MAC), routing and application layers of the protocol stack. The goal of this study is to identify a number of different network problems, and for each of these network problems to examine the underlying optimization problem. In each case we begin by presenting the basic version of the network problem and extend it by introducing new constraints. These constraints result mainly from technological advances and from additional requirements present in WSN applications. For all the network problems discussed here a wide range of algorithms and protocols are to be found in the literature. We cite only some of these, since we are concerned more with the network optimization problem itself, together with its different versions, than with a state of art of methods for solving it. Moreover, the cited methods have originated in a variety of disciplines, with approaches ranging from the deterministic to the opportunistic, including computational geometry, linear, nonlinear and dynamic programming, metaheuristics and heuristics, game theory, and so on. We go on to discuss the complexity inherent in different optimization problems, in order to give some hints to WSN designers facing new but similar scenarios. We try to highlight distributed solutions and information that is required to implement these schemes. For each topic the general presentation scheme is as follows: i) Present the network problem ii) Identify the relevant optimization problem 7 www.intechopen.com 154 Telecommunications Networks -Current Status and Future Trends www.intechopen.com Review of Optimization Problems in Wireless Sensor Networks 3 where A is a matrix, b and c are vectors giving respectively the right-hand terms and the cost coefficients, and x is the decision variable vector. In cases where some decision variables have integer values while others have continuous values we refer to the problem as Mixed Integer Linear Programming. If, on the other hand, the vector x contains only integer values, then we have a case of Integer Linear Programming (ILP). Note that the difficulty of the problem increases when we are dealing with ILP rather than LP, since ILP problems are commonly NP-hard. The most frequently used algorithms for solving LP problems are Simplex and Interior Points methods (Dantzig, 1963; Karmarkar, 1984) , whereas for ILP problems there are Branch-and-Bound, Branch-and-Cut and Cutting Planes methods. Besides maximizing/minimizing an objective function, LP can be adapted so that it also guarantees fairness. In this case the objective function becomes a max-min (or min-max) objective function. In WSN we may often encounter network problems modeled according to this structure. It also happens that in some networks modeled by LP the number of variables is infinite or finite but huge, making an explicit enumeration impossible. In these cases the problem is solved iteratively. At each iteration new variables that potentially would lead to better solutions are generated by a method called column generation. The problem is solved when no new variables can be generated. Finally, when the objective function, or at least one of the constraints, is a nonlinear function, the problem becomes a nonlinear programming problem. In this type of problem the nature of the objective function is very important. If it is a convex function, then the problem is a nonlinear convex programming problem, where the best-known techniques include subgradient and Lagrangian decomposition (Kuhn, 1951; Shor, 1985) . The above linear programming problems can be solved using a commercial solver such as CPLEX, Xpress-MP, etc. For nonlinear nonconvex programming the optimization becomes difficult and the solution methods less tractable. Another method worth citing is Dynamic programming (Bellman, 1957) . This is a sequential approach where the decisions are taken optimally, step-by-step, until the complete solution has been constructed. This method works for problems that can be divided into subproblems that are simpler to solve and whose solutions will produce the global solution.