Inexact Alternating Direction Method And Fixed-point Algorithm For The Sparest Solution Of Absolute Value Equations

The sparse solution problem has become the focus of research in the field of optimization.Since this kind of problem has been widely used in many fields,especially in the fields of statistics,signal and image processing,many scholars are looking for the fast method to solve the problem,and there are many good methods to solve the sparse solution of linear equations.However,there are few studies on the sparse solution of absolute value equations.As more and more problems can be transformed into the problem of absolute value equations,the effective method to find the sparse solution of the absolute value equation is concerned by the scholars in the optimization field.In this paper,we use the following two methods to find the sparse solution of the absolute value equations respectively.They are inexact alternating direction method?inexact ADM?and fixed point algorithm.When inexact ADM is used to find the sparse solution of the absolute value equationsAx-x?28?b,the problem is relaxed to a?₁-norm minimization problem,and it is further relaxed to a constrained optimization problem.Finally,the constrained optimization problem is transformed to the Lagrange problem,and we use the inexact ADM to solve the above optimization problem.The optimal solution formulas of the related sub-optimization problem are derived,so the calculation speed is improved much.When the fixed point algorithm is used to find the sparse solution of the absolute value equationsAx-x?28?b,the problem is also relaxed to a?₁-norm minimization problem.By exterior penalty function method,it is further relaxed to an unconstrained optimization problem.By the Taylor formula,the approximate optimization problem of the above unconstrained optimization problem is obtained.The fixed-point algorithm is used to solve the approximate optimization problem.Numerical experiments about the above two methods show that they are very effective in finding the sparse solution of absolute value equations.