×

Mesh independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems. (English) Zbl 1154.65057

An optimal control problem with pure state constraints and a second-order linear elliptic differential equation is considered. The Lagrange multiplier associated with the pointwise almost everywhere state constraints is a Borel measure. Consequently, numerical methods are difficult to realize. A Lavrentiev-type regularization of pointwise state constrains is proposed. An alternative path follows a generalized Moreau-Yosida-type regularization. This gives a mixed control-state constrained control problem. The regulized problem is solved efficiently by a semismooth Newton method. The method is mesh independent and it is superlinear convergent. The paper contains a report on numerical test runs including a comparison with a short-step path-following interior-point method and a coarse-to-fine mesh sweep. Certain convergence and smoothness properties of the solution are proved, even if the Lavrentiev parameter vanishes.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M37 Numerical methods based on nonlinear programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1287/moor.5.1.43 · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[2] Robinson, Math Programming Stud 19 pp 200– (1982) · Zbl 0495.90077 · doi:10.1007/BFb0120989
[3] DOI: 10.1007/BF01581275 · Zbl 0780.90090 · doi:10.1007/BF01581275
[4] DOI: 10.1137/0324078 · Zbl 0606.49017 · doi:10.1137/0324078
[5] DOI: 10.1016/S0167-6911(02)00262-1 · Zbl 1134.49310 · doi:10.1016/S0167-6911(02)00262-1
[6] DOI: 10.1023/A:1015489608037 · Zbl 1015.49026 · doi:10.1023/A:1015489608037
[7] DOI: 10.1137/S1052623498343131 · Zbl 1001.49034 · doi:10.1137/S1052623498343131
[8] Bank, Interior methods for a class of elliptic variational inequalities pp 218– (2003)
[9] DOI: 10.1023/A:1020576801966 · Zbl 1033.65044 · doi:10.1023/A:1020576801966
[10] Alt, Discretization and mesh-independence of Newton’s method for generalized equations pp 1– (1998)
[11] DOI: 10.1137/0723011 · Zbl 0591.65043 · doi:10.1137/0723011
[12] Adams, Sobolev spaces (1975)
[13] DOI: 10.1137/0315061 · Zbl 0376.90081 · doi:10.1137/0315061
[14] Luenberger, Optimization by vector space methods (1969) · Zbl 0176.12701
[15] Kummer, Discuss Math Differ Incl 20 pp 209– (2000) · Zbl 1016.90058 · doi:10.7151/dmdico.1013
[16] DOI: 10.1007/s10107-004-0540-9 · Zbl 1079.65065 · doi:10.1007/s10107-004-0540-9
[17] DOI: 10.1137/050637480 · Zbl 1121.49030 · doi:10.1137/050637480
[18] DOI: 10.1137/S1052623401383558 · Zbl 1080.90074 · doi:10.1137/S1052623401383558
[19] Gilbarg, Elliptic Partial Differential Equations of Second Order (1977) · doi:10.1007/978-3-642-96379-7
[20] DOI: 10.1137/S0036142999356719 · Zbl 0979.65046 · doi:10.1137/S0036142999356719
[21] Wright, Primal-dual Interior-Point Methods (1997) · Zbl 0863.65031 · doi:10.1137/1.9781611971453
[22] Troltzsch, Optimale Steuerung partieller Differentialgleichungen (2005) · doi:10.1007/978-3-322-96844-9
[23] Robinson, Math Programming Stud 10 pp 128– (1979) · Zbl 0404.90093 · doi:10.1007/BFb0120850
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.