Theories And Algorithms Of Uncertain Optimization Based On Interval

Uncertainty widely exists in practical engineering problems, and studying the theories and algorithms of uncertain optimization is significant for reliability design of industrial products and systems. Stochastic programming and fuzzy programming are two types of traditional uncertain optimization methodologies, in which a great amount of sampling information on the uncertainty is required to construct the precise probability distributions or fuzzy membership functions. Unfortunately, it often seems very difficult or sometimes expensive to obtain sufficient uncertainty information, and hence these two types of methods will encounter some limitations in applicability. The interval number optimization is a relatively newly-developed uncertain optimization method, in which interval is used to model the uncertainty of a variable. Thus the variation bounds of the uncertain variables are only required, which can be obtained through only a small amount of uncertainty information. In recent years, the interval number optimization has been attracting more and more attentions, and it is expected to become the third major uncertain optimization methodology after stochastic programming and fuzzy programming. Furthermore, the interval number optimization even shows a larger application potential for practical engineering problems than the other two methods. However, it is still at its preliminary stage for the present interval number optimization research, especially for the nonlinear interval number optimization (NINO), studies for which are just getting started. Some key technical difficulties remain, such as creation of the mathematical transformation models, effective solution of the nesting optimization problem etc.This dissertation conducts a systematical research for the NINO, and aims at contributing some useful researches and trials on mathematical programming theories and practical algorithms. Firstly, two transformation models of NINO are put forward through which the uncertain optimization problem can be transformed into a deterministic optimization problem, and this part of work is the basis of the whole dissertation. On the other hand, some computation tools in the present engineering optimization field are extended into the interval number optimization to solve the lower efficiency problem caused by the nesting optimization and whereby several efficient NINO algorithms are constructed. As a result, the following studies are carried out in this dissertation: (1) For a general uncertain optimization problem, two mathematical transformation models of NINO are suggested at the level of mathematical programming theory, respectively based on the order relation of interval number and the possibility degree of interval number. A modified construction method is suggested for the possibility degree of interval number based on the probability method, and based on it an uncertain inequality constraint can be transformed into a deterministic constraint; an approach is also suggested to transform an uncertain equality constraint into two deterministic constraints. The two transformation models employ the above same way to deal with the uncertain constraints, while different ways for uncertain objective function, namely adopt the order relation of interval number and the possibility degree of interval number to transform the uncertain objective function into deterministic objective function, respectively. Through the transformation model, a deterministic nesting optimization problem can be finally formulated, which can be solved by a suggested nesting optimization method based on a genetic algorithm (GA).(2) Two hybrid optimization algorithms respectively based on multiple networks and single network are suggested to solve the transformed nesting optimization problem, and whereby two kinds of NINO methods with high efficiency are constructed. In the multiple-networks hybrid optimization algorithm, several artificial neural network models are required and each one creates the projection relation between the design variables and the bounds of the uncertain objective function or a constraint, and the GA is adopted as optimization solver. In the single-network hybrid optimization algorithm, only one artificial neural network model is required to create the projection relation between the variables (design variables and uncertain variables) and the functional values (uncertain objective function and constraints), and the GA is also used as optimization solver for both of the inner and outer layer optimization. In the optimization process, the actual model is replaced by the efficient artificial neural network model and hence the optimization efficiency of NINO can be improved greatly.(3) Some extensions are made for the interval structural analysis method, and furthermore an efficient NINO algortithm is suggested based on the interval structural analysis method. Firstly, the interval structural analysis method is extended to compute the response bounds of structures with large uncertainty levels, based on the interval set theory and the subinterval technique. Secondly, the interval structural analysis method is introduced into the elastic wave propagation problem of composite materials, and a kind of interval numerical algorithm of elastic wave is proposed for composite laminated plates based on the hybrid numerical method. Thus the transient displacement response bounds of a composite laminated plate caused by the uncertain load and material property can be computed. Thirdly, in the NINO solving process, the interval structural analysis is adopted to compute the bounds of the uncertain objective function and constraints at each iterative step, and whereby the inner layer optimization can be eliminated successfully. Therefore, the transformed nesting optimization problem becomes a traditional single-layer optimization problem, and hence an NINO algorithm with high efficiency can be constructed.(4) Based on the sequential linear programming technique, an efficient NINP algorithm is developed. At each iterative step, linear approximation models with respect to the design variables and uncertain variables are created for the uncertain objective function and constraints using the first-order Taylor expansion, and whereby a linear interval number optimization problem is obtained; based on the interval analysis method, bounds of the uncertain objective function and constraints at the optimal design vector of the current approximation optimization problem can be achieved very efficiently, and whereby whether the currently obtained design vector is a feasible and descending point can be judged, as only a feasible and descending design can be remained to the next iterative step; several termination criteria are provided to ensure convergence of the present algorithm.(5) An efficient NINO algorithm is suggested based on an approximation model management strategy. The whole optimization process consists of a sequence of approximate optimization problems. At each iterative step, an approximate optimization problem can be created through the approximation model technique, and it can be solved by being changed as a deterministic optimization problem using a transformation model of NINO. The trust region method is then employed to manage the approximation models in the optimization process. At each iterative step, a reliability index is computed to judge the precision of the current approximation models, and whereby the design vector and trust region radius vector can be updated. Therefore, the design space can be ensured to keep closing to the actual optimal design vector.(6) An efficient NINO algorithm is suggested based on a local-densifying approximation model technique. At each iterative step, the current samples of the uncertain objective function and constraints can be densified according to the solution of current approximate optimization problem. Thus the local precision of the approximation models in the two key regions within the approximation space corresponding to the response bounds can be improved. This algorithm aims at improving the precision of the local key regions instead of the whole approximation space, and hence much less samples are required comparing with the conventional approximation optimization methods based on the uniformly distributed samples. Furthermore, the algorithm can also avoid singularity of the involved approximation models caused by the overmany samples a certain extent, as well as improvement of the optimization efficiency.