Research On Parameter Control Methods For Constrained Optimization Problems

Optimization method is an important part of operations research. It has wide application to many fields, such as, natural science, social science, practical production, engineering design, and modern management, etc. Many practical problems can be reduced to optimization problems. The key to investigating optimization problems, especially constrained optimization problems, is to design efficient algorithm for solving it.At present, constrained optimization methods may be classified to two classes, one is search method which firstly asserts whether the current point is an optimal point, if the point is not, then we must choose search directions, and along the search direction, find the next iterative point which make the objective function or merit function to decrease. Such methods are generally decreasing method, such as, feasible direction methods, constrained variable metric methods, etc. Another class is sub-problems method, which approximates the optimal solution by solving a series of simple sub-problems, such as penalty function methods, trust region methods, and successive quadratic programming sub-problems, etc.The same property of two classes of methods is that they determine whether the next iterative point is " good " or " bad " by comparing the objective function value or merit function value at the current point and next iterative point. If the next iterative point is "better" than current point then the iterative point can be acceptable, else it must continue to iterate or adjust the sub-problem. The basic idea is to find iterative points which converge to optimal point and its corresponding objective function or merit function values converge to optimal value.The parameter control methods are in the contrast, which is to find a sequence of parameters that converge to optimal value and its corresponding pointsinconverge to optimal solution. The main task of traditional methods is to construct iterative points and that of parameter control methods is to find a sequence of parameters.The parameter control methods are very similar to penalty function methods, both of them are to solve constrained optimization problems by solving a series of sub-unconstrained optimization problems. But parameter control methods are different from penalty function methods. Firstly, the penalty coefficient of penalty function methods are preassigned, while the parameters of parameter control methodsare generated automatically according to some rule prescribed. Secondly, the penalty coefficient may converge to infinity in many situations when the iterative point is closely near the bound of feasible set, while the parameters are bounded if the solution set of constrained optimization is nonempty, which is available for numerical computation.The PHD thesis is systematically to study the parameter control methods for constrained optimization problems, propose several parameter control methods, analyze global convergence of the method, and report numerical results. The study shows that the parameter control methods have better numerical performance in practice than the traditional methods.The innovation of the thesis has five points. Firstly, the basic form of parameter ontrol methods is proposed, the global convergence is proved. Secondly, a series of merit functions is established, and a sequence of parameters which converges to optimal value from arbitrary real number is found. Thirdly, large scope random-cast point methods and partition interval methods are addressed. Fourthly, multi-parameter control methods, super-memory gradient methods for solving sub-unconstrained optimization problems are investigated. Fifthly, the parameter control methods for solving convex programming, especially linear programming and convex quadratic programming, are discussed and its convergenceIVis probed.