The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

Abstract

We study the method for solving a kind of nonsmooth optimization problems with 1 -norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems with wide application background in engineering technology. Transformated by the absolute value equations, this kind of nonsmooth optimization problem is rewritten as a general unconstrained optimization problem, and the transformed problem is solved by a smoothing FR conjugate gradient method. Finally, the numerical experiments show the effectiveness of the given smoothing FR conjugate gradient method.

Modified Three‐Term Conjugate Gradient Method and Its Applications

“…Now, we give some numerical experiments of Algorithm 9, which are also considered in [19,21,22,27,28]. The numerical results of all examples indicate that the modified three-term conjugate gradient method is also effective for solving the 1 -norm minimization problem (40).…”

“…where ∈ × ( ≪ ), ∈ , and > 0 is a parameter to trade off both terms for minimization. This problem is widely used in compressed sensing, signal reconstruction, and some related problems [18][19][20][21][22][27][28][29]. In this subsection, we translate (40) into the absolute value equation problem based on the equivalence between the linear complementary problem and the absolute value equation problem [30] and then use Algorithm 9 to solve it.…”

“…Example 15. Consider a typical compressed sensing problem with the form as (40), which is also considered in [21,22,27,28]. We choose = 2 4 , = 2 6 , = 0.5, = 0.4, = 0.5, = 10 −6 , = 5, and = 2.…”

“…So, in this paper, we propose a new modified threeterm conjugate gradient method with the Armijo line search. The proposed method is used to solve M-tensor systems [16,17] and a kind of nonsmooth optimization problems with 1 -norm [18][19][20][21][22].…”

A new smoothing modified three-term conjugate gradient method for l 1 $l_{1}$ -norm minimization problem

“…Many researchers have studied the numerical algorithms, which can be used to solve problem ( 1 ) with large-scale data such as fixed point method [ 1 ], gradient projection method for sparse reconstruction [ 2 ], interior-point continuation method [ 3 , 4 ], iterative shrinkage thresholds algorithms in [ 5 , 6 ], linearized Bregman method [ 7 , 8 ], alternating direction algorithms [ 9 ], nonsmooth equations-based method [ 10 ] and some related methods [ 11 , 12 ]. Just recently, a smoothing gradient method has been given for solving problem ( 1 ) based on the new transformed absolute value equations in [ 14 , 15 ]. The transformation is based on the equivalence between a linear complementarity problem and an absolute value equation problem [ 17 , 18 ].…”