It was my father's persistent inducement that made me--a shy and clumsy boy before school--love mathematics.  Thanks to the recommendation of Yi-fa Tang, I was able to study nonlinear optimization under the supervision of  Professor Ya-xiang Yuan in Computing Center (now Institute of Computational Mathematics and Sci./Engr. Computing)  from 1992, after the four year undergraduate study in the Department of Applied Mathematics of Beijing Institute of Technology. 

My early work on nonlinear conjugate gradient (CG) methods during the graduate and Ph.D. study (1992-1997), in conjunction with Y. Yuan,  mainly concerns theoretical issues in regard to most of  the fundamental CG methods. For example, the Fletcher-Reeves, conjugate descent, and Polak-Ribi\`ere-Polyak methods. Subsequently we  studied the Beale-Powell restart method and the method of shortest residuals. A new CG method was also proposed, whose descent property and global convergence can be achieved with (not necessarily strong) Wolfe line searches. This method is known as the DY method and listed as one of the four leading CG contenders by Nazareth (1998).  The introduction of the DY method makes it possible to design efficient nonlinear CG algorithms using the Wolfe line search. The numerical experiments with Qin NI for large-scale test problems show that the DYHS hybrid method with Wolfe line searches outperforms the PRP method (the latter is generally believed as one of the most efficient CG algorithms).  Indepedently of Nazareth (1998), we proposed the concept of CG family. Several CG families are then given and studied.  We also tried to unify the global-convergence theory of CG methods.  Most of the above results has been written into the monograph:  Y.H. Dai and Y. Yuan (2000), Nonlinear Conjugate Gradient Methods, Shanghai Scientific and Technical Publishers, Shanghai (in Chinese). 

Subsequently my research interests turned to gradient methods such as the BB method and variations thereof. With J. Yuan and Y. Yuan, the BB formula was explained from the viewpoint of interpolation, and several other efficient formulae were proposed. With Wenbin Liu (at Business School of Univ of Kent at Canterbury) a ``supervisor and search engine cooperation'' (SSC) algorithm for unconstrain optimztion, and another related algorithm, were proposed. For the BB algorithm itself, $R$-linear convergence for convex quadratics of any dimension was proved, together with Li-zhi Liao (at Department of Mathematics of HongKong Baptist University). I has also proved that the BB stepsize can always be accepted by a nonmonotone line search with suitable parameters. I has recently researched many other variations of the BB method, including the periodic repeated use of the same step length, and alternating uses of different step sizes, such as by minimizing the function value and the gradient norm alternately. I has shown that some of these variations can give significant improvements in performance over the BB method, although the reasons for this are not well understood.  Together with Roger Fletcher at University of Dundee, I am trying to understand the BB method better and give a thorough study on the BB-like method. 

Since 1997 I has also looked into theoretical aspects of quasi-Newton method, and has shown that Powell's (1984) counter examples of global convergence using $8$ cyclic points can also be used to show that the BFGS method with Wolfe line searches needs not converge for general functions. More recently, I has also provided a counter example with only $6$ cyclic points by introducing a new technique of choosing parameters. Together with Michael J.D. Powell(at Cambridge University), I am studying the convergence properties of the BFGS method under a more accurate line search.

Some of my recent attention is also paid on semi conjugate gradient method for nonsymmetric line systems.  This part of research is in conjuction with Jinyun Yuan (at Department of Mathematics of Federal Univeristy of Parana),  and is partly supported by the Sino-Brazil joint research project ``Numerical Linear Algebra and Optimization".