Numerical solution of systems of non linear equations defined by convex functions

Abstract

In this paper, the problem of numerical solution of systems of non linear equations, defined by convex differentiable functions, is studied. With this system, non-smooth (non-differentiable) unconstrained minimization problems are associated, with objective functions, which are based on the discrete l₁ - and l_∞ - norm, respectively, that is, using these two norms as proximity criteria. Also, a “differentiable” unconstrained minimization problem (least squares data fitting problem) is associated with the considered system of non linear equations, with objective function, which is based on the discrete l₂ - norm. The special case when non linear equations are defined by separable convex differentiable functions is also studied. The considered system of non linear equations is solved, in the case of l₁ - norm, by minimizing the sum of absolute values of residuals, and, in the case of l_∞ - norm, by minimizing the maximal residual, respectively. In the case of l₂ - norm, the system of non linear equations is solved by minimizing the sum of squares of the residuals. Subgradients of the objective functions of the associated unconstrained minimization problems are calculated, and the subgradient method for solving these problems is described. The gradient method for solving the corresponding “differentiable” discrete least squares problem, which is based on l₂ - norm, is also presented, and gradients of the objective functions of the corresponding “differentiable” unconstrained minimization problems are calculated.

Subject Classification: (2010):

• 41A50
• 49J35
• 49J52
• 90C59

Keywords:

• Separable optimization
• Convex optimization
• Non-differentiable optimization
• Minimax problems
• Subgradient methods
• Approximation
• Weighted discrete least squares