Development Of The Theory Of Integral Approach For Global Optimization And Its Parallel Algorithm

The problem considered in this dissertation is how to characterize and find the global minimum value and global minimizers of an objective function in R~n and functional space.Let X be a topological space and f : X→R a real-valued function. Consider the following minimization problem:c~* = (?) f(ar) (0.0.5)In general, minimizers of (0.0.5) may not exist. Under the assumption(A): f is lower semi continuous, X is inf-compact.minimizers of (0.0.5) exist.The problem of minimizing a function has been investigated since the seventeenth century with the concepts of derivative and Lagrangian multiplier. The gradient-based approach to optimization is the mainstream of that research. However , the requirement of differentiability restricts its application to many practical problems. Moreover, it can only be utilized to characterize and find a local solution of a general optimization problem.In this dissertation, we apply the integral global optimization technique to investigate a minimization problem with discontinuous objective function. Integral global optimization technique is a very powerful yet flexible tool to treat various optimization problems.We first recall basic concepts of robust sets, functions and the integral approach to global minimization. With the integral approach to global optimization , several new optimality conditions for global minimization are studied. Using m-mean value condition one can design algorithms for finding unconstrained global minimizers. With u-variance condition, one can set stopping criterion. A class of discontinuous penalty functions is proposed to solve constrained minimization problems with the integral approach to global optimization. Optimality conditions of a penalized minimization problem and a non sequential algorithm is proposed. New optimality conditions of the integral global minimization are applied to characterize global minimum in functional space as a sequence of approximating solutions in finite-dimensional spaces. A variable measure algorithm is used to find such solutions. For a constrained problem, a discontinuous penalty method is proposed to convert it to unconstrained ones. The integral algorithm can be implemented by a properly designed Monte Carlo method. Numerical examples are given to illustrate the effectiveness of the algorithm.The integral global minimization algorithm is also implemented on parallel computer, the results show the great advantage of the integral approach.