A Modified Cauchy-Barzilai-Borwein Algorithm For Constrained Convex Optimization Problems

In the era of advanced data,people's ability in collecting data is improved and the research of data value is more in-depth.The size of data and the complexity of research are greatly enhanced,so it is very important to propose some good algorithms which can take less time and get better results.Efficient solver of constrained optimization problem is an important task in the design of optimization algorithms.In the thesis,we first proposes a modified Cauchy-Barzilai-Borwein algorithm,called MCBB algorithm,for unconstrained optimization problems.And then,a projection Cauchy-Barzilai-Borwein algorithm is proposed for the box-constrained optimization problem,which is referred to as PCBB algorithm.The convergence of all the proposed algorithm is proved and the effectiveness is demonstrated by preliminary numerical experiments.The thesis is organized as follows:Chapter 1 introduces some current optimization algorithms and the research status,background and significance of the research;Chapter 2 reviews the basic theory and the nature of the optimization problem,and introduces the Cauchy-Barzilai-Borwein algorithm and other optimization algorithms.Chapter 3 and Chapter 4 are the main parts of this thesis.In Chapter 3,we proposed a modified Cauchy-Barzilai-Borwein(MCBB)algorithm for unconstrain ed optimization problems,and proved the global convergence.The numerical experiments,compared to the Barzilai-Borwein algorithm(BB)and Cauchy-Barzilai-Borwein algorithm(CBB)algorithm,show that the proposed algorithm is more effective.At present,a relaxation MCBB algorithm is proposed based on the MCBB algorithm,which is referred as RMCBB algorithm.The numerical experiments show that the efficiency of the RMCBB algorithm is better than the MCBB algorithm.In Chapter 4,according to the characteristics of the box-constrained optimization problem,a projection Cauchy-Barzilai-Borwein is proposed,and the global convergence is proved.Numerical experiments show that the PCBB algorithm is more efficient than the external gradient method and the projection shrinkage algorithm.Chapter 5 summarizes this thesis,and presents the future research work.