Theory And Algorithm Of Layout Optimization Characterized By Topological Structure

This dissertation studies the packing problem characterized by topologieal structure with the layout optimization of an artificial satellite module in the background. The contents include the layout optimization models of the difform graph elements, the optimality conditions , the optimality functions, the optimization algorithms for the subproblems, the algorithms for the non-overlap constraints and the unproved genetic algorithm. The main results, obtained in this dissertation, may be summarized as follows:1. The two-dimensional layout optimization problem about placing the rectangular graph elements on a round board is investigated and the semi-infinite optimization model is presented. The definition of the adjoining graph elements is improved, such that the one-to-one correspondence between the nonisomorphic layout schemes and the graphs is established. The finite subproblems of the semi-infinite optimization are obtained by virtue of the theories presented in reference [108] and [109], where the subproblems are solved on every equivalent class of isomorphic layout. The objective function in model is proved to be continuous and it follows that the optimal solutions of the subproblems exist. The corresponding relaxed subproblem for each subproblem is constructed and the relation of the locally optimal solutions between the subproblem and its relaxed subproblem are discussed. It is proved that the global optimal solution can be obtained through solving the relaxed subproblems.2. The constraints of the relaxed subproblem combine to make a max function @(Y). This max function @(Y) is proved to be continuous and the directional derivative and subgradient are presented by virtue of the properties of max function. At the same time, it is proved that the subgradient is outer semi-continuous.3. A min-max problem which is locally equivalent to the relaxed subproblem is constructed,(MMP) min F(Y)It is discussed that the relation of the locally minimizer between the relaxed subproblem and the corresponding min-max problem. The first-order necessary conditions for the relaxed subproblem and the sufficient condition for the correspondent min-max problem are proved. Because the constructed min-max problem requires the advance knowledge of a locally minimizer of the relaxed subproblem, the first-order necessary conditions can't be directly regarded as the terminal criteria of algorithms. To overcome this difficulty, afunction F(z, Y) is constructed where for all Y € R4n, F(Y, Y) = F(Y), if Y is a locally optimal solution of the relaxed subproblem. The optimality function is presented by using the first-order convex approximation to the function F(Y,Y + h). The optimality function is nonnegative continuous and the first-order necessary condition is satisfied at a point if and only if this point is a zero of the optimality function.4. According to the practical instance of the apparatus modules of artificial satellite, the layout optimization models of the difform graph elements are discussed. Firstly, the expressions of the circle, the triangle and the ordinary convex polygon graph elements are presented. It is not only that the optimization models of one kind of graph elements such as the circles or triangles are established, but also the optimization models of two kinds of graph elements are presented, which include the combination of the rectangles and the circles, or the triangles. To obtain the better models, the capacity function and the static non-equilibrium quantity are regarded as the objective functions of the models respectively. For the layout problem of the rectangular and triangle graph elements, where the capacity function is regarded as the objective function, the optimality conditions of its corresponding relaxed subproblem are studied.5. The optimization algorithms for solving the packing problem are studied. For the particularity of the layout problem, the paper changes the form of the optimality function and constructs the algorithm for solving the relaxed subproblem of...