ENHANCED VARIANTS OF DIFFERENTIAL EVOLUTION ALGORITHM AND THEIR APPLICATIONS

Abstract

Most of the real life optimization problems arising in various fields of science and engineering can be modeled as global optimization problems. In such problems it is desired and is often necessary to determine a global optimal solution rather than a local optimal solution. Determining the global optimal solution of a nonlinear optimization problem is considered to be more difficult as compared to the problem of determining a local optimal solution. The various approaches available for solving the global optimization problems can be broadly categorized as deterministic and probabilistic approaches. Deterministic approaches extensively use the analytical properties such as continuity, convexity, differentiability etc of the objective and the constraints to locate a neighborhood of the global optimum. Most of these techniques are designed to solve a particular class of optimization problem. Consequently, these techniques are not generic in nature. On the other hand stochastic methods, utilize randomness in an efficient way to explore the set over which the objective function is to be optimized. Stochastic methods performed well in the case of the most of the realistic problems over which these have been applied. Among stochastic approaches, Evolutionary Algorithms (EA) or Nature Inspired Algorithms (NIA) are found to be very promising search techniques for solving global complex optimization problems. Some popular EA/NIA includes Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO) and Differential Evolution (DE) etc. The focus of the present study is on DE, which has emerged as a powerful optimization tool for solving complex global optimization problems. Comparative studies have confirmed that DE outperforms many other optimizers. Practical experiences however show that DE is not completely flawless. It is vulnerable to problems like slow and/ or premature convergence, is sometimes unable to locate global optima or gets stuck in local optima. Also, like most of the other population based EA/NIA, the performance of DE deteriorates with the increase in the size of the problem. These shortcomings of DE become more persistent in case of multimodal or noisy functions. i This study concentrates on development of efficient variants of DE 'algorithm for solving global optimization problems. Initially, three variants of DE are proposed named as: Differential Evolution with Cauchy mutation (CDE), Differential Evolution with mixed mutation strategy (MSDE) and Synergetic Differential Evolution (SDE). The first variant CDE maintains a failure counter (FC) to keep a tab on the performance of the algorithm by scanning or monitoring the individuals and makes use of Cauchy mutation operator in an effective manner which helps in escaping the individual entrapped in local minima. The second variant MSDE uses the concept of evolutionary game theory to integrate basic DE mutation and quadratic' interpolation based mutation to generate a new solution. This is contrast to the basic DE where a single mutation operator is used throughout the algorithm. The third version which is SDE is based on the rule of synergy that the combined effect s,. is always beneficial than the individualistic effect. In SDE three algorithmic components are fused together. These concepts are three recent modifications in DE (1) opposition based learning (OBL) (2) tournament method for mutation and (3) single population structure. These features have a specific role which helps in improving the performance of DE. First of all the performance of these algorithms are analyzed on a set of unconstrained benchmark problems. For this purpose, a comprehensive set of well-known complex benchmark functions is employed to experimentally compare and analyze the three proposed algorithms. The test suite consists of 25 traditional/ classical problems and 7 nontraditional shifted benchmark problems. All the algorithms are also compared with some of the recent modified versions of DE and the results showed that the proposed variants are either superior or at par with the competing algorithms. The algorithms are also analyzed statistically with the help of non parametric tests. It is observed that although all the proposed variants achieve solutions with good accuracy, maintain stable convergence characteristics and are simple to implement within a satisfactory computation time; MSDE and SDE are better than CDE. MSDE and SDE after suitable modifications are further applied to solve the constrained optimization problems (COP). For this purpose a comprehensive set of 24 constrained benchmark problems is considered over which the proposed algorithms are analyzed and are 11 ompared with other modified DE versions for solving COP. The numerical and statistical esults indicated that both the algorithms performed quite satisfactorily over the considered test uite of COP. However, taking into account the simple structure of SDE algorithm, it is further riodified for solving multi objective optimization problems (MOPs). For analyzing the )erformance of SDE on MOPs, 9 unconstrained, bi-objective MOPS are taken from literature Lnd the obtained results are also compared with some of the contemporary algorithms for solving MOPs. The efficiency of SDE was observed numerically and statistically for dealing with multiple objectives as well. Finally, the proposed SDE algorithm is applied for solving two real life problems; (1) frim Loss Problem (TLP) arising in paper industry, which is a highly constrained non-linear, non convex integer programming problem and (2) Image Thresholding Problem which arises frequently in the area of Image Processing. The complex mathematical model of both the problems makes it a challenging task for an optimization algorithm to obtain the global solution and it was observed that SDE was able to deal with both the real life problems in quite an efficient manner giving quality solution while maintaining a reasonably good convergence speed. ff