| Title:
|
Primal interior point method for minimization of generalized minimax functions (English) |
| Author:
|
Lukšan, Ladislav |
| Author:
|
Matonoha, Ctirad |
| Author:
|
Vlček, Jan |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
697-721 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we propose a primal interior-point method for large sparse generalized minimax optimization. After a short introduction, where the problem is stated, we introduce the basic equations of the Newton method applied to the KKT conditions and propose a primal interior-point method. (i. e. interior point method that uses explicitly computed approximations of Lagrange multipliers instead of their updates). Next we describe the basic algorithm and give more details concerning its implementation covering numerical differentiation, variable metric updates, and a barrier parameter decrease. Using standard weak assumptions, we prove that this algorithm is globally convergent if a bounded barrier is used. Then, using stronger assumptions, we prove that it is globally convergent also for the logarithmic barrier. Finally, we present results of computational experiments confirming the efficiency of the primal interior point method for special cases of generalized minimax problems. (English) |
| Keyword:
|
unconstrained optimization |
| Keyword:
|
large-scale optimization |
| Keyword:
|
nonsmooth optimization |
| Keyword:
|
generalized minimax optimization |
| Keyword:
|
interior-point methods |
| Keyword:
|
modified Newton methods |
| Keyword:
|
variable metric methods |
| Keyword:
|
global convergence |
| Keyword:
|
computational experiments |
| MSC:
|
49K35 |
| MSC:
|
90C06 |
| MSC:
|
90C47 |
| MSC:
|
90C51 |
| idZBL:
|
Zbl 1204.49022 |
| idMR:
|
MR2722096 |
| . |
| Date available:
|
2010-10-22T05:28:15Z |
| Last updated:
|
2013-09-21 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140779 |
| . |
| Reference:
|
[1] Bertsekas, D. P.: Nondifferentiable optimization via approximation.In: Nondifferentiable Optimization (M. L. Balinski and P.Wolfe, eds.), Math. Programming Stud. 3 (1975), 1–25. Zbl 0383.49025, MR 0440906 |
| Reference:
|
[2] Coleman, T. F., Garbow, B. S., Moré, J. J.: Software for estimating sparse Hessian matrices.ACM Trans. Math. Software 11 (1985), 363–377. MR 0828562, 10.1145/6187.6190 |
| Reference:
|
[3] Coleman, T. F., Moré, J. J.: Estimation of sparse Hessian matrices and graph coloring problems.Math. Programming 28 (1984), 243–270. MR 0736293, 10.1007/BF02612334 |
| Reference:
|
[4] Demyanov, V. F., Malozemov, V. N.: Introduction to Minimax.Dover Publications, 1990. MR 1088479 |
| Reference:
|
[5] Fletcher, R.: Practical Methods of Optimization.Second edition. Wiley, New York 1987. Zbl 0905.65002, MR 0955799 |
| Reference:
|
[6] Gill, P. E., Murray, W.: Newton type methods for unconstrained and linearly constrained optimization.Math. Programming 7 (1974), 311–350. Zbl 0297.90082, MR 0356503, 10.1007/BF01585529 |
| Reference:
|
[7] Griewank, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation.SIAM, Philadelphia 2000. Zbl 0958.65028, MR 1753583 |
| Reference:
|
[8] Griewank, A., Toint, P. L.: Partitioned variable metric updates for large-scale structured optimization problems.Numer. Math. 39 (1982), 119–137. MR 0664541, 10.1007/BF01399316 |
| Reference:
|
[9] Le, D.: Three new rapidly convergent algorithms for finding a zero of a function.SIAM J. Sci. Stat. Computat. 6 (1985), 193–208. Zbl 0561.65033, MR 0773291, 10.1137/0906016 |
| Reference:
|
[10] Le, D.: An efficient derivative-free method for solving nonlinear equations.ACM Trans. Math. Software 11 (1985), 250–262. Zbl 0581.65033, MR 0814340, 10.1145/214408.214416 |
| Reference:
|
[11] Lukšan, L., Matonoha, C., Vlček, J.: Interior point method for nonlinear nonconvex optimization.Numer. Linear Algebra Appl. 11 (2004), 431–453. MR 2067814, 10.1002/nla.354 |
| Reference:
|
[12] Lukšan, L., Matonoha, C., Vlček, J.: On Lagrange multipliers of trust-region subproblems.BIT Numer. Math. 48 (2008), 763–768. Zbl 1203.65092, MR 2465702, 10.1007/s10543-008-0197-5 |
| Reference:
|
[13] Lukšan, L., Matonoha, C., Vlček, J.: Primal interior-point method for large sparse minimax optimization.Kybernetika 45 (2009), 841–864. Zbl 1198.90394, MR 2599116 |
| Reference:
|
[14] Lukšan, L., Matonoha, C., Vlček, J.: Trust-region interior-point method for large sparse $l_1$ optimization.Optimization Methods and Software 22 (2007), 737–753. Zbl 1193.90197, MR 2354765, 10.1080/10556780601114204 |
| Reference:
|
[15] Lukšan, L., Spedicato, E.: Variable metric methods for unconstrained optimization and nonlinear least squares.J. Comput. Appl. Math. 124 (2000), 61–93. MR 1803294, 10.1016/S0377-0427(00)00420-9 |
| Reference:
|
[16] Lukšan, L., Vlček, J.: Sparse and Partially Separable Test Problems for Unconstrained and Equality Constrained Optimization.Technical Report V-767, ICS AS ČR, Prague 1998. |
| Reference:
|
[17] Lukšan, L., Vlček, J.: Variable metric method for minimization of partially separable nonsmooth functions.Pacific J. Optim. 2 (2006), 59–70. MR 2548209 |
| Reference:
|
[18] Pillo, G. Di, Grippo, L., Lucidi, S.: Smooth transformation of the generalized minimax problem.J. Optim. Theory Appl. 95 (1997), 1–24. Zbl 0890.90165, MR 1477348, 10.1023/A:1022627226891 |
| Reference:
|
[19] Vanderbei, J., Shanno, D. F.: An interior point algorithm for nonconvex nonlinear programming.Comput. Optim. Appl. 13 (1999), 231–252. Zbl 1040.90564, MR 1704122, 10.1023/A:1008677427361 |
| Reference:
|
[20] Xu, S.: Smoothing methods for minimax problems.Computational Optim. Appl. 20 (2001), 267–279. MR 1857058, 10.1023/A:1011211101714 |
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