Algorithm and Properties of Sparse Solution in Large-scale System and Their Application in Finance and Operation Management
With the development of information technology, data to be collected and dealt with often present characteristics such as large scale, complexity and uncertainty. Therefore, how to develop effective data processing technology to adapt to such high complexity of data has become the focus in science and engineering. As one of the most basic and important data analysis techniques, the linear regression model has been widely used. However it encounters unprecedented challenges since the date of high complexity are huge amount, high-dimensional and uncertain. This basic problem corresponds to the solving process of linear equations Ax = b. How to find a solution which satisfies the equations and of which the number of nonzero elements is least is a basic and important subject in theory of equations in applied mathematics. It also has significant applications in many scientific problems, such as some basic linear regression models in statistics, graphics and signal reconstruction in computer, decision-making and operations management in business and joint bond investment in finance.
Graduating from Stanford University and Columbia University respectively, Prof. Ge and Prof. Wen have familiar understanding and accumulation of these problems. Recently, closely cooperating with professor Yinyu Ye of Stanford University, they succeed in deeply discussing the complexity of optimal sparse solution of linear equations by means of minimizing concave functions which satisfy linear constraints. They first demonstrate the problem’s strong NP-Hard properties internationally, first put forward interior-point algorithm of the problem and demonstrate the convergence of this algorithm, followed by stochastic simulation. The results indicate that compared with other prevailing algorithms present, our method has many advantages such as low complexity, fast rate of convergence and high success rate of fully recovering sparse solution. Besides, they also have a large number of papers published and accepted in international top magazines in the field of optimization, such as Mathematical Programming, Mathematical Programming Computation, SIAM Journal on Scientific Computing and so on.