Variable neighborhood search

Variable neighborhood search (VNS), proposed by Mladenović, Hansen, 1997, is a metaheuristic method for solving a set of combinatorial optimization and global optimization problems.It explores distant neighborhoods of the current incumbent solution, and moves from there to a new one if and only if an improvement was made. The local search method is applied repeatedly to get from solutions in the neighborhood to local optima.VNS was designed for approximating solutions of discrete and continuous optimization problems and according to these, it is aimed for solving linear program problems, integer program problems, mixed integer program problems, nonlinear program problems, etc. Variable neighborhood search (VNS), proposed by Mladenović, Hansen, 1997, is a metaheuristic method for solving a set of combinatorial optimization and global optimization problems.It explores distant neighborhoods of the current incumbent solution, and moves from there to a new one if and only if an improvement was made. The local search method is applied repeatedly to get from solutions in the neighborhood to local optima.VNS was designed for approximating solutions of discrete and continuous optimization problems and according to these, it is aimed for solving linear program problems, integer program problems, mixed integer program problems, nonlinear program problems, etc. VNS systematically changes the neighborhood in two phases: firstly, descent to find a local optimum and finally, a perturbation phase to get out of the corresponding valley. Applications are rapidly increasing in number and pertain to many fields: location theory, cluster analysis, scheduling, vehicle routing, network design, lot-sizing, artificial intelligence, engineering, pooling problems, biology, phylogeny, reliability, geometry, telecommunication design, etc. There are several books important for understanding VNS, such as: Handbook of Metaheuristics, 2010, Handbook of Metaheuristics, 2003 and Search methodologies, 2005.Earlier work that motivated this approach can be found in Recent surveys on VNS methodology as well as numerous applications can be found in 4OR, 2008 and Annals of OR, 2010. Define one deterministic optimization problem with min { f ( x ) | x ∈ X , X ⊆ S } {displaystyle min {{f(x)|xin X,Xsubseteq S}}} , (1) where S, X, x, and f are the solution space, the feasible set, a feasible solution, and a real-valued objective function, respectively. If S is a finite but large set, a combinatorial optimization problem is defined. If S = R n {displaystyle {S=R^{n}}} , there is continuous optimization model. A solution x ∗ ∈ X {displaystyle {x^{*}in X}} is optimal if