Numerical behavior of the method of projection onto an acute cone with level control in convex minimization

Robert Dylewski

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2000)

  • Volume: 20, Issue: 2, page 147-158
  • ISSN: 1509-9407

Abstract

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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].

How to cite

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Robert Dylewski. "Numerical behavior of the method of projection onto an acute cone with level control in convex minimization." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 20.2 (2000): 147-158. <http://eudml.org/doc/271502>.

@article{RobertDylewski2000,
abstract = {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].},
author = {Robert Dylewski},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {convex nondifferentiable minimization; projection method; subgradient method; acute cone; obtuse cone},
language = {eng},
number = {2},
pages = {147-158},
title = {Numerical behavior of the method of projection onto an acute cone with level control in convex minimization},
url = {http://eudml.org/doc/271502},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Robert Dylewski
TI - Numerical behavior of the method of projection onto an acute cone with level control in convex minimization
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2000
VL - 20
IS - 2
SP - 147
EP - 158
AB - 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].
LA - eng
KW - convex nondifferentiable minimization; projection method; subgradient method; acute cone; obtuse cone
UR - http://eudml.org/doc/271502
ER -

References

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  1. [1] A. Cegielski, A method of projection onto an acute cone with level control in convex minimization, Mathematical Programming 85 (1999), 469-490. Zbl0973.90057
  2. [2] S. Kim, H. Ahn and S.-C. Cho, Variable target value subgradient method, Mathematical Programming 49 (1991), 359-369. Zbl0825.90754
  3. [3] K.C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Springer-Verlag, Berlin 1985. 
  4. [4] C. Lemaréchal, A.S. Nemirovskii and YU.E. Nesterov, New variants of bundle methods, Mathematical Programming 69 (1995), 111-147. Zbl0857.90102
  5. [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. [6] B.T. Polyak, Minimization of unsmooth functionals, Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969), 509-521 (Russian). 
  7. [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. Zbl0761.90090
  8. [8] N.Z. Shor, Minimization Methods for Nondifferentiable Functions, Springer-Verlag, Berlin, Heidelberg 1985. 
  9. [9] M.J. Todd, Some remarks on the relaxation method for linear inequalities, Technical Report 419, Cornell University, Cornell, Ithaca 1979. 

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