Subgradient projection algorithms for constrained nonsmooth optimization. II: Nonlinear constraints. (English) Zbl 0817.90117
Summary: [For part I see the author and Y. Li, Numer. Math., Nanjing 12, No. 3, 270-283 (1990; Zbl 0735.65037).]
A kind of subgradient projection algorithms is established for minimizing a locally Lipschitz continuous function subject to nonlinearly smooth constraints, which is based on the idea to get a feasible and strictly descent direction by combining the \(\varepsilon\)-projection direction that attempts to satisfy the Kuhn-Tucker conditions with one corrected direction produced by a linear programming subproblem. The algorithm avoids the zigzagging phenomenon and converges to Kuhn-Tucker points, due to using the c.d.f. maps of Polak and Mayne (1985), \(\varepsilon\)-active constraints and \(\varepsilon\)-adjusted rules.
A kind of subgradient projection algorithms is established for minimizing a locally Lipschitz continuous function subject to nonlinearly smooth constraints, which is based on the idea to get a feasible and strictly descent direction by combining the \(\varepsilon\)-projection direction that attempts to satisfy the Kuhn-Tucker conditions with one corrected direction produced by a linear programming subproblem. The algorithm avoids the zigzagging phenomenon and converges to Kuhn-Tucker points, due to using the c.d.f. maps of Polak and Mayne (1985), \(\varepsilon\)-active constraints and \(\varepsilon\)-adjusted rules.
MSC:
| 90C30 | Nonlinear programming |
| 65K05 | Numerical mathematical programming methods |
| 49J52 | Nonsmooth analysis |
Keywords:
constrained nonsmooth optimization; subgradient projection algorithms; nonlinearly smooth constraints; \(\varepsilon\)-projection direction; Kuhn- Tucker conditions; \(\varepsilon\)-active constraints; \(\varepsilon\)- adjusted rulesCitations:
Zbl 0735.65037References:
| [1] | Berge C., Topological Spaces (1963) |
| [2] | Clarke F., Optimization and Nonsmooth Analysis (1983) · Zbl 0582.49001 |
| [3] | Cullum J., Math. Prog. Study 3 pp 35– (1975) |
| [4] | Lemarechal, C., Strodiot, J.J. and Bihain, A. 1981.On a bundle algorithm for nonsmooth optimization, Edited by: Mangasarian, O.L., Meyer, R.R. and Robinson, S.M. 245–281. New York: Academic Press. Nonlinear Programming 3 |
| [5] | Yingiu Lin., Numer. Anal. J. Chinese Univ 13 pp 270– (1990) |
| [6] | DOI: 10.1016/0141-0296(84)90056-7 · doi:10.1016/0141-0296(84)90056-7 |
| [7] | DOI: 10.1080/00207168408803425 · Zbl 0571.90069 · doi:10.1080/00207168408803425 |
| [8] | DOI: 10.1007/BF02591738 · Zbl 0641.90061 · doi:10.1007/BF02591738 |
| [9] | DOI: 10.1137/1029002 · doi:10.1137/1029002 |
| [10] | DOI: 10.1137/0323031 · Zbl 0569.90073 · doi:10.1137/0323031 |
| [11] | DOI: 10.1016/0005-1098(82)90087-5 · Zbl 0532.49017 · doi:10.1016/0005-1098(82)90087-5 |
| [12] | DOI: 10.1137/0321010 · Zbl 0503.49021 · doi:10.1137/0321010 |
| [13] | Rosen J.B., J. SIAM 9 pp 514– (1961) |
| [14] | DOI: 10.1137/0311036 · Zbl 0259.90035 · doi:10.1137/0311036 |
| [15] | DOI: 10.1016/0021-9045(84)90098-4 · Zbl 0546.65040 · doi:10.1016/0021-9045(84)90098-4 |
| [16] | DOI: 10.1016/0021-9045(86)90031-6 · Zbl 0618.90076 · doi:10.1016/0021-9045(86)90031-6 |
| [17] | DOI: 10.1007/BF02594782 · Zbl 0495.90067 · doi:10.1007/BF02594782 |
| [18] | Xianjian Ye. Some subgradient projection algorithms for constrained nonsmooth optimization problems Nanjing University Chinese 1988 M. S. thesis |
| [19] | Xianjian Ye. Constraint qualifications for nonsmooth optimization to appear |
| [20] | Xianjian Ye., Math. Biquarterly 8 pp 56– (1991) |
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