Previous |  Up |  Next


solution set of convex problems; alternative theorems; minimum norm solution; residual vector
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.
[1] L. Armijo: Minimazation of functions having Lipschitz-continuous first partial derivatives. Pacific J. Math. 16 (1966), 1-3. DOI 10.2140/pjm.1966.16.1 | MR 0191071
[2] Yu. G. Evtushenko, 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. MR 2040365 | Zbl 1044.90088
[3] A. I. Golikov, Yu. G. Evtushenko: Theorems of the alternative and their applications in numerical methods. Comput. Math. and Math. Phys. 43 (2003), 338-358. MR 1993755
[4] C. Kanzow, H. Qi, L. Qi: On the minimum norm solution of linear programs. J. Optim. Theory Appl. 116 (2003), 333-345. DOI 10.1023/A:1022457904979 | MR 1967673 | Zbl 1043.90046
[5] S. Ketabchi, E. Ansari-Piri: On the solution set of convex problems and its numerical application. J. Comput. Appl. Math. 206 (2007), 288-292. DOI 10.1016/ | MR 2337444 | Zbl 1131.90042
[6] O. L. Magasarian: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7 (1988), 21-26. DOI 10.1016/0167-6377(88)90047-8 | MR 0936347
[7] O. L. Magasarian: A Newton method for linear programming. J. Optim. Theory Appl. 121 (2004), 1-18. DOI 10.1023/B:JOTA.0000026128.34294.77 | MR 2062967
[8] O. L. Magasarian: A finite Newton method for classification. Optim. Meth. Software 17 (2002), 913-930. DOI 10.1080/1055678021000028375 | MR 1953825
Partner of
EuDML logo