Differential
Inclusions, Control and Optimization 20 (2000) 147-158

doi: 10.7151/dmdico.1009

Robert Dylewski

*Institute of Mathematics, Technical University
ul. Podgórna 50, PL-65-246 Zielona Góra, Poland*

We present the numerical behavior of a projection method for convex minimization problems which was studied by Cegielski [1]. The method is a modification of the Polyak subgradient projection method [6] and of variable target value subgradient method of Kim, Ahn and Cho [2]. In each iteration of the method an obtuse cone is constructed. The obtuse cone is generated by a linearly independent system of subgradients. The next approximation of a solution is the projection onto a translated acute cone which is dual to the constructed obtuse cone. The target value which estimates the minimal objective value is updated in each iteration. The numerical tests for some tests problems are presented in which the method of Cegielski [1] is compared with the method of Kim, Ahn and Cho [2].

**Keywords:** convex nondifferentiable minimization, projection method, subgradient
method, acute cone, obtuse cone.

**1991 Mathematics Subject Classification:** 65K05, 90C25.

[1] | A. Cegielski, A method of projection onto an acute cone with level control in
convex minimization, Mathematical Programming 85 (1999), 469-490. |

[2] | S. Kim, H. Ahn and S.-C. Cho, Variable target value subgradient method,
Mathematical Programming 49 (1991), 359-369. |

[3] | K.C. Kiwiel, Methods of Descent for Nondifferentiable Optimization,
Springer-Verlag, Berlin 1985. |

[4] | C. Lemaréchal, A.S. Nemirovskii and YU.E. Nesterov, New variants of bundle methods,
Mathematical Programming 69 (1995), 111-147. |

[5] | C. Lemaréchal and R. Mifflin, A Set of Nonsmooth Optimization Test Problems,
in: Nonsmooth Optimization, C. Lemaréchal and R. Mifflin, eds., Pergamon Press, Oxford
(1978), 151-165. |

[6] | B.T. Polyak, Minimization of unsmooth functionals, Zh. Vychisl. Mat. i Mat.
Fiz. 9 (1969), 509-521 (Russian). |

[7] | H. Schramm and J. Zowe, A version of the bundle idea for minimizing a nonsmooth
function: conceptual idea, convergence analysis, numerical results, SIAM J.
Optimization 2 (1992), 121-152. |

[8] | N.Z. Shor, Minimization Methods for Nondifferentiable Functions,
Springer-Verlag, Berlin, Heidelberg 1985. |

[9] | M.J. Todd, Some remarks on the relaxation method for linear inequalities,
Technical Report 419, Cornell University, Cornell, Ithaca 1979. |

Received 5 November 1999

Revised 7 March 2000