First and foremost, I would like to express my sincere gratitude to Prof. Xiaojun Chen, my chief supervisor, for her motherly patient care and generous support throughout my PhD career. She pushed me to go ahead, taught me to value my time, trained me how to do presentations and many more things than I could have expected from a supervisor. Special gratitude to her for supporting me to apply for the PhD outgoing program so that I could visit University of California, Los Angeles during my PhD period. A big thanks goes to Dr. An Li, husband of Prof. Chen, for treating us delicious food and good wine during Christmas/Spring Festival dinners and other important days, which colored my memory of the past three years.

Special thanks are due to Prof. Xin Liu (Academy of Mathematics and System Sciences, Chinese Academy of Sciences) for leading me into the field of matrix optimization and showing me the charm of matrix. I also appreciate that he invited me to visit him and recommended me to Prof. Wotao Yin, which provided me with the opportunities to learn from him and Prof. Yin. He is a great collaborator and friend. He spent a tremendous amount of time discussing with me on my research which dragged me back from dilemma again and again. Without his help and guidance, it is impossible for me to finish my first PhD project, which led to the first part of this thesis.

I would also like to thank Prof. Ya-xiang Yuan (Academy of Mathematics and System Sciences, Chinese Academy of Sciences) for his insightful comments and suggestions on the first part of this thesis. Furthermore, I really appreciate the opportunities to visit his research group in Beijing with support from his grants. I learned a lot from those precious visiting experiences. And it was really happy and unforgettable to hike and play cards with Prof. Yuan and his group.

My sincere acknowledgements go to Prof. Wotao Yin (University of California, Los Angeles) for giving me the opportunity to visit his research group. His vision and passion have urged me a lot. He also spent plenty of time discussing with me, from which I learned how to think and gradually approach to the target when facing a new topic.

I am very grateful to Prof. Xiaoping Xue and Prof. Wei Bian from Harbin Institute of Technology, my former supervisor and academic sister in Harbin, respectively. Without their support and help, it would be impossible for me to be here to conduct my PhD study. Also, Prof. Wei Bian provides me with a lot of help on my research, including proofreading of my draft, offering me her MATLAB code and so forth.

Prof. Philippe Toint (University of Namur) visited my supervisor Prof. Xiaojun Chen for three weeks at the beginning of 2016, during which we started a new project and hence we spent a lot of time together on discussions. The second part of this thesis is mainly derived from our joint work. My special appreciations are due to him for his insightful ideas, patient guidance and generous help.

I would also like to acknowledge Prof. Zhaosong Lu (Simon Fraser University) and Prof. Fengmin Xu (Xi’an Jiaotong University) for their kind help and special care. Prof. Lu is a well-read scholar. I learned a lot from several discussions with him both in our group meetings and in private. It was great kind of Prof. Xu to give me insightful suggestions on my future career. I really appreciate it.

Over the past three years, it is my great honor to have the opportunities to visit Chinese Academy of Sciences and University of California, Los Angeles. I was lucky to meet and make many new friends during those visits, including Dr. Qian Dong, Mr. Ran Gu, Mr. Cheng Chen, Mr. Bin Gao, Mr. Liang Zhao in Beijing and Dr. Ming Yan, Dr. Zhiming Peng, Mr. Zhenbo Lu in Los Angeles. Specially, my appreciations go to Mr. Cheng Chen and Mr. Zhenbo Lu for their kind help when I was in Beijing and Los Angeles, respectively.

My sincere and deep appreciations are due to my academic brothers and sisters: Prof. Congpei An, Dr. Yanfang Zhang, Dr. Zaikun Zhang, Prof. Hailin Sun, Dr. Yang Zhou, Ms. Qiyu Wang, Mr. Lei Yang, Mr. Bo Wen, Mr. Yun Shi and Mr. Guidong Liu for their constant and substantial support.

My special thanks go to my former and current officemates for their company and help throughout my stay. The list is very long and dull, but deserves to be here: Dr. Chuanxin Bian, Dr. Haiyang Jin, Ms. Xianping Wu, Mr. Haibin Chen, Ms. Qun Wang, Mr. Shujun Wang in DE 405, Dr. Zhibao Li, Dr. Boshi Tian, Mr. Jin Yang, Ms. Qiujin Peng, Ms. Yan Wei in HJ 609, and Ms. Yue Shi, Ms. Meiling Hao, Ms. Chenchen Zu, Ms. Huili Zhang, Ms. Kaihui Liu, Ms. Chen Ouyang, Ms. Fei Fang and Mr. Qingzheng Wang in TU 834. Here I would particularly like to single out Dr. Chuanxin Bian and Dr. Yang Zhou, who gave me generous help when I was fresh in Hong Kong.

Last but far from least, my deepest gratitude and love belong to my mother Shulan Wu, my father Nengsong Wang, my brother Yong Wang and all my family for their unconditional love and consistent support through my life.