Gradient-based Algorithm For A Class Optimization Problem With Nuclear Norm

Optimization problem with the nuclear norm is widely applicable in many fields such assignal processing, pattern recognition, machine learning, system control and so on. Since thenuclear norm is nonsmoothing, the classical gradient-based algorithm(Newton method, conju-gate gradient method, gradient projection method, etc.) is difficult to solve this class problem. Inthis dissertation, we consider using the gradient-based algorithm to handle a class optimizationproblem with the nuclear norm.In order to use the gradient-based algorithm effectively, it should smooth the nuclear norm,we use the technique which is similar to smooth the absolute value function. In Chapter2, everysingular value of a matrix can be approximated by a monadic smoothing function, respectively,and the sum of all the monadic smoothing function is the approximative function of the nuclearnorm. Furthermore, the approximative function is close enough to the nuclear norm.In Chapter3, we use the conjugate gradient method to solve the problem, and the optimalsolution of the approximative problem is also close enough to the optimal solution of the primaryproblem. With the purpose of improving the efficiency of the algorithm, we modify the conju-gate gradient method mentioned, and the global convergence of modified conjugate gradientmethod with the weak Wolfe-Powell line search rule is established under suitable conditions.The result of our experiment is showed in Chapter4. The result presents that the modifiedconjugate gradient method is very effective for solving the problem with the nuclear norm andthe performance of the modified conjugate gradient method is better than the conventional con-jugate gradient method.