Abstract

In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed variable optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular, we define a general algorithm model for the solution of this class of problems, that draws inspiration from the approach recently proposed by Audet and Dennis [SIAM J. Optim., 11 (2001), pp. 573--594], and is based on the strategy of combining in a suitable way a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove global convergence of the algorithm model without specifying the local continuous search, but only identifying some reasonable requirements. Moreover, we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally, we report numerical results obtained by the proposed algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

MSC codes

  1. 65K05
  2. 90C30
  3. 90C56
  4. 90C11

Keywords

  1. nonlinear optimization
  2. mixed variable programming
  3. derivative-free methods

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References

1.
M. A. Abramson, Pattern Search Algorithms for Mixed Variable General Constrained Optimization Problems, Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, 2002.
2.
Charles Audet, J. Dennis, Jr., Pattern search algorithms for mixed variable programming, SIAM J. Optim., 11 (2000/01), 573–594
3.
Charles Audet, J. Dennis, Jr., Analysis of generalized pattern searches, SIAM J. Optim., 13 (2002), 0–0889–903 (electronic) (2003)
4.
Claude Berge, Topological spaces, Dover Publications Inc., 1997xiv+270, Including a treatment of multi‐valued functions, vector spaces and convexity; Translated from the French original by E. M. Patterson Reprint of the 1963 translation
5.
O. Chubar, P. Elleaume, and J. Chavanne, A 3d magnetostatics computer code for insertion devices, J. Synchrotron Radiation, 5 (1998), pp. 481–484.
6.
P. Elleaume, O. Chubar, and J. Chavanne, Computing 3 d magnetic field from insertion devices, in Proceedings of the PAC97 Conference, Vancouver, BC, Canada, 1997, pp. 3509–3511.
7.
Robert Lewis, Virginia Torczon, Pattern search algorithms for bound constrained minimization, SIAM J. Optim., 9 (1999), 1082–1099, Dedicated to John E. Dennis, Jr., on his 60th birthday
8.
Robert Lewis, Virginia Torczon, Pattern search methods for linearly constrained minimization, SIAM J. Optim., 10 (2000), 917–941
9.
G. Liuzzi, S. Lucidi, G. Placidi, and A. Sotgiu, A Magnetic Resonance Device Designed via Global Optimization Techniques, Technical report, 09‐02, Department of Computer and Systems Science “Antonio Ruberti,” University of Rome “La Sapienza,” 2002.
10.
S. Lucidi, M. Sciandrone, P. Tseng, Objective‐derivative‐free methods for constrained optimization, Math. Program., 92 (2002), 37–59
11.
S. Lucidi, V. Piccialli, A derivative‐based algorithm for a particular class of mixed variable optimization problems, Optim. Methods Softw., 19 (2004), 371–387, The First International Conference on Optimization Methods and Software. Part II
12.
V. Piccialli, Methods for Solving Mixed Variable Programming Problems, Ph.D. thesis, Department of Computer and Systems Science “Antonio Ruberti,” University of Rome “La Sapienza,” 2003.

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Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1057 - 1084
ISSN (online): 1095-7189

History

Published online: 28 July 2006

MSC codes

  1. 65K05
  2. 90C30
  3. 90C56
  4. 90C11

Keywords

  1. nonlinear optimization
  2. mixed variable programming
  3. derivative-free methods

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