Vera Kovačević-Vujčić

General variable neighborhood search for the continuous optimization

We suggest a new heuristic for solving unconstrained continuous optimization problems. It is based on a generalized version of the variable neighborhood search metaheuristic. Different neighborhoods and distributions, induced from different metrics are ranked and used to get random points in the shaking step. We also propose VNS for solving constrained optimization problems. The constraints are handled using exterior point penalty functions within an algorithm that combines sequential and exact penalty transformations. The extensive computer analysis that includes the comparison with genetic algorithm and some other approaches on standard test functions are given. With our approach we obtain encouraging results.

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

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.

Computing the metric dimension of graphs by genetic algorithms

Abstract In this paper we consider the NP-hard problem of determining the metric dimension of graphs. We propose a genetic algorithm (GA) that uses the binary encoding and the standard genetic operators adapted to the problem. The feasibility is enforced by repairing the individuals. The overall performance of the GA implementation is improved by a caching technique. Since the metric dimension problem up to now has been considered only theoretically, standard test instances for this problem do not exist. For that reason, we present the results of the computational experience on several sets of test instances for other NP-hard problems: pseudo boolean, crew scheduling and graph coloring. Testing on instances with up to 1534 nodes shows that GA relatively quickly obtains approximate solutions. For smaller instances, GA solutions are compared with CPLEX results. We have also modified our implementation to handle theoretically challenging large-scale classes of hypercubes and Hamming graphs. In this case the presented approach reaches optimal or best known solutions for hypercubes up to 131072 nodes and Hamming graphs up to 4913 nodes.

TABU search methodology in global optimization

Abstract This paper investigates the application of TABU search methodology in global optimization. A general multilevel TABU search algorithm is proposed. The algorithm is applied to the problem of finding constrained global minima of a piecewise smooth function of the form ф(x) = max {ϕ 1 (x), …, ϕ m (x)} subject to box constraints. The tests are performed on a special class of problems of this type arising from the synthesis of radar polyphase codes. It is shown that problems of this type are NP-hard.

Computing minimal doubly resolving sets of graphs

In this paper we consider the minimal doubly resolving sets problem (MDRSP) of graphs. We prove that the problem is NP-hard and give its integer linear programming formulation. The problem is solved by a genetic algorithm (GA) that uses binary encoding and standard genetic operators adapted to the problem. Experimental results include three sets of ORLIB test instances: crew scheduling, pseudo-boolean and graph coloring. GA is also tested on theoretically challenging large-scale instances of hypercubes and Hamming graphs. Optimality of GA solutions on smaller size instances has been verified by total enumeration. For several larger instances optimality follows from the existing theoretical results. The GA results for MDRSP of hypercubes are used by a dynamic programming approach to obtain upper bounds for the metric dimension of large hypercubes up to 2^9^0 nodes, that cannot be directly handled by the computer.

Minimal doubly resolving sets and the strong metric dimension of some convex polytopes

Abstract In this paper we consider two similar optimization problems on graphs: the strong metric dimension problem and the problem of determining minimal doubly resolving sets. We prove some properties of strong resolving sets and give an integer linear programming formulation of the strong metric dimension problem. These results are used to derive explicit expressions in terms of the dimension n, for the strong metric dimension of two classes of convex polytopes D n and T n . On the other hand, we prove that minimal doubly resolving sets of D n and T n have constant cardinality for n > 7 .

Semidefinite Programming Methods for the Symmetric Traveling Salesman Problem

In this paper the symmetric traveling salesman problem (STSP) is modeled as a problem of discrete semidefinite programming. A class of semidefinite relaxations of STSP model is defined and two variants of a branch-and-bound technique based on this class of relaxations are proposed. The results of preliminary numerical experiments with randomly generated problems are reported.

Stabilization of Interior-Point Methods for Linear Programming

The paper studies numerical stability problems arising in the application of interior-point methods to primal degenerate linear programs. A stabilization procedure based on Gaussian elimination is proposed and it is shown that it stabilizes all path following methods, original and modified Dikins method, Karmarkars method, etc.

Variable neighborhood search for metric dimension and minimal doubly resolving set problems

In this paper, two similar NP-hard optimization problems on graphs are considered: the metric dimension problem and the problem of determining a doubly resolving set with the minimum cardinality. Both are present in many diverse areas, including network discovery and verification, robot navigation, and chemistry. For each problem, a new mathematical programming formulation is proposed. For solving more realistic large-size instances, a variable neighborhood search based heuristic is designed. An extensive experimental comparison on five different types of instances indicates that the VNS approach consistently outperforms a genetic algorithm, the only existing heuristic in the literature designed for solving those problems.

GLOB — A new VNS-based Software for Global Optimization

We describe an application of Variable Neighbourhood Search (VNS) methodology to continuous global optimization problems with box constraints. A general VNS algorithm is implemented within the software package GLOB. The tests are performed on some standard test functions and on a class of NP-hard global optimization problems arising in practice. The computational results show the potential of the new software.