The characterization of the solution set of a convex constrained problem is a well-known attempt. In this paper, we focus on the minimum norm solution of a specific constrained convex nonlinear problem and reformulate this problem as an unconstrained minimization problem by using the alternative theorem.The objective function of this problem is piecewise quadratic, convex, and once differentiable. To minimize this function, we will provide a new Newton-type method with global convergence properties.

solution set of convex problems, alternative theorems, minimum norm solution, residual vector

90C05, 90C51

- L. Armijo: Minimazation of functions having Lipschitz-continuous first partial derivatives. Pacific J. Math. 16 (1966), 1-3. CrossRef
- Yu. G. Evtushenko and A. I. Golikov: New perspective on the theorems of alternative. In: High Performance Algorithms and Software for Nonlinear Optimization, Kluwer Academic Publishers B.V., 2003, pp. 227-241. CrossRef
- A. I. Golikov and Yu. G. Evtushenko: Theorems of the alternative and their applications in numerical methods. Comput. Math. and Math. Phys. 43 (2003), 338-358. CrossRef
- C. Kanzow, H. Qi and L. Qi: On the minimum norm solution of linear programs. J. Optim. Theory Appl. 116 (2003), 333-345. CrossRef
- S. Ketabchi and E. Ansari-Piri: On the solution set of convex problems and its numerical application. J. Comput. Appl. Math. 206 (2007), 288-292. CrossRef
- O. L. Magasarian: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7 (1988), 21-26. CrossRef
- O. L. Magasarian: A Newton method for linear programming. J. Optim. Theory Appl. 121 (2004), 1-18. CrossRef
- O. L. Magasarian: A finite Newton method for classification. Optim. Meth. Software 17 (2002), 913-930. CrossRef