(301x) A Population-Based Probability Distribution Estimation Method for Dynamic Optimization

Dynamic optimization is always a challenge methodologically and computationally, especially when a system to be optimized is highly nonlinear and complex. Solving these complex problems by meta-heuristics, such as evolutionary algorithm (EA) and particle swarm optimization (PSO) that are based upon the principles of natural biological evolution or artificial life, have received considerable and increasing interest over the past decade. Recently, Xiao et al.[1] introduced a mathematical framework for solving complex process dynamic optimization problems, which is an extension of a relatively new meta-heuristic approach called Ant Colony System (ACS) algorithm. The methodology is general and has successfully solved a coating quality constrained oven energy minimization problem[1] and a cure-window-based coating quality optimization problem.[2] However, the solution search in an intrinsic discretized space may restrict the solution precision. Further, its computational efficiency is yet to be improved.

A general mathematical framework, called the population-based probability distribution estimation (PBPDE) method is developed in this work to tackle constrained complex dynamic optimization problems. Compared with the ACS approach by Xiao et al.,[1] PBPDE advances the solution identification process from search-tree-based discretized-space search to probability distribution estimation based continuous space iterative search. Solution quality and computational efficiency has been greatly improved. On the other hand, PBPDE extends the ACO algorithm by Socha[3,4] to solve dynamic constrained optimization problems. Special penalty assignment technique is applied to take care of various constraints and solution migration step is introduced to prevent premature convergence commonly occurred in constrained optimization problems. In PBPDE, initial solution population is randomly generated, and then new solutions will be constructed. The constructed better solutions will replace the old ones in the original solution population. The same procedure will be repeatedly executed until a final optimal solution is obtained. A new solution is constructed incrementally (component by component, or decision variable by decision variable) by sampling the population probability density functions (PPDFs), which are generated from the solution population in the current iteration. The essence of PBPDE is to refine the probability distribution in order to find promising areas of search spaces that contain high quality solutions. This idea is similar to the population-based probabilistic search algorithms.[5] However, PBPDE has its specialty in dealing with dynamic constrained optimization problems, penalties for infeasible solutions are assigned to the solution ranks, which consequently affect the possibility of utilizing certain search spaces. In addition, premature convergence can be prevented by integrated solution migration techniques.

To validate the efficacy of this methodology, numerical examples as well as a dynamic engineering optimization problem are solved in this work. Results for numerical examples demonstrate that the PBPDE can effectively identify a feasible solution and then converge quickly to a promising solution. The derived solutions are very close to the global optimum and much better than the solutions listed in literatures derived by GA. The coating quality constrained oven energy minimization problem, which has been solved by ACS-based algorithm,[1] is further studied. Complex system dynamics (temperature changes, extent of conversion, coating film thickness and solvent removal dynamics) are modeled and the resulting problem is a large-scale multi-stage dynamic optimization problem with a mixed set of sparse linear and nonlinear time-dependent constraints. The objective function is time-dependent and it is a sum of interdependent sub-objective functions. The optimization results clearly demonstrate the superiority of the PBPDE algorithm to the original ACS-based method in the following aspects: (a) improved solution quality, (b) dramatically increased computational efficiency, (c) solution precision improvement due to the continuous space search, and (d) applicability enhancement. If a feasible initial solution is given, PBPDE can obtain the local optimal solution derived by ACS in about only 20 seconds, while it takes ACS 20 minutes.

Above all, the principal contribution of this work is to introduce a promising new dynamic optimization approach to chemical engineering realm. This methodology demonstrates significant performance improvements compared with ACS-based dynamic optimization method. In general, it is applicable to a variety of industrial dynamic optimization problems, where product and process performance can be simultaneously considered.

[1]. Xiao J, Li J, Xu Q, Huang Y, Lou HH. ACS-based dynamic optimization for curing of polymeric coating. AIChE J. 2006; 52: 1410-1422. [2]. Xiao J, Li J, Lou HH, Huang Y. Cure-window-based proactive quality control in topcoat curing. Ind. Eng. Chem. Res. 2006; 45: 2351-2360. [3]. Socha K. ACO for continuous and mixed-variable optimization. Proceedings of Fourth International Workshop on Ant Colony Optimization and Swarm Intelligence (ANTS 2004). 2004; 3172: 25-36. [4]. Blum C, Socha K. Training feed-forward neural networks with ant colony optimization: an application to pattern classification. Fifth International Conference on Hybrid Intelligent Systems. 2005; 233-238. [5]. Pelikan M, Goldberg DE, Lobo FG. A survey of optimization by building and using probabilistic models. Computational Optimization and Applications. 2002; 21: 5-20.