Decomposition Methods For A Class Of Convex Minimization Problems

In mathematical programming,We always meet many large-scale problems,Themain methods of solving theose problems include Conjugate Gradient Method,BFGSMethod,Decomposition Method and so on.Complicated optimization problem isdecomposed into several sub-problems in decomposition method. The main problemsand sub-problems are non-smooth or the optimal solution does not meet the KKTconditions in some decomposition method.Therefore,these two conditions areenormous influence on the constructed algorithm.In order to avoid the emergence ofthese problems,scholars have proposed a relaxation methhod with the augmentedLargrangian function for solving the problem,and two differentiation decompositiontechniques-the decomposition method of alternating direction methon andpre-correction multiplier method–applied in relaxation method with the augmentedLagrangian function.So,discuss the convergence of these two methods and advantagesand disadvantages of algorithms snd applied to paacticality of actual model and theseproblems become the focus of scholars' research.The main work of this paper is to promote the constraints in the originalmodel,and then applied to the pre-correction multiplier decomposition algorithm.Insome conditions,A new method to prove the pre-correction multiplier convergence ofthe algorithm and the linear convergence rate,and then compare these twodecomposition algorithms from the theory and simple examples.From theoreticalperspective,the pre-correction multiplier decomposition algorithm is faster than thealternating direction decomposition algorithms to get the optimal soltion. The pre-correction multiplier decomposition algorithms get more accurate optimal soltion thenalternating direction decomposition algorithms.In the end,Based pre-correctionmultiplier decomposition algorithm.A new method is proposed. In someconditions,convergence exiting. The present thesis is organized as follows. In Chapter1, we firstly give a Brieflyintroduces the development process of decomposition method and present some maindecomposition method, the model of unit commitment was transformed into aseparable optimization problem using the idea of decomposition method; In Chapter2,Given some concepts about the alternating direction method, and the relatedassumption conditions of convergence and convergence rate; In Chapter3, in order toovercome the shortcomings of the classic Lagrange relaxation (CLR) method andaugmented Lagrange relaxation (ALR) method for solving optimization problemsencountered, but also to better be the theory of convergence rate and convergence, thischapter makes use of the most widely used method for correction approximatemultiplier method (PCPM), and the correction approximate multiplier method (PCPM)is applied in the augmented Lagrange relaxation method to solve general linearconstrained optimization problem; and with a new method proved the correctionapproximate multiplier (PCPM) under certain conditions prove the convergence andconvergence rate., the comparison of the alternating direction method (ADI) and in thethird chapter of the correction approximate multiplier (PCPM) in theory and examples;and draws the related conclusion. Giving a class of decomposition algorithm and itsconvergence. The forth chapter, Based pre-correction multiplier decompositionalgorithm.A new method is proposed.In some conditions,convergence exiting.The fifthchapter, summarizes the full text and looking to the future.