Differential stability of solutions to constrained optimization problems. (English) Zbl 0572.49012
Results concerning directional (one-sided) differentiability of the mapping of projection onto a closed, convex set in a Hilbert space are quoted and several examples are provided. These results are used for sensitivity analysis of constrained optimal control problems. An abstract convex optimal control problem subject to control constraints is considered. The cost functional and the state equation depend on a real parameter. Using the results of directional differentiability of the projection mapping it is shown that the right-derivative of the optimal solution exists and it is given by the solution of an auxiliary quadratic optimal control problem.
The abstract result is illustrated by an example of an optimal control problem for a parabolic equation. The problem of second order right- differentiability is discussed.
The abstract result is illustrated by an example of an optimal control problem for a parabolic equation. The problem of second order right- differentiability is discussed.
Reviewer: K.Malanowski
MSC:
49K40 | Sensitivity, stability, well-posedness |
49J50 | Fréchet and Gateaux differentiability in optimization |
49K20 | Optimality conditions for problems involving partial differential equations |
46C99 | Inner product spaces and their generalizations, Hilbert spaces |
52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |
46G05 | Derivatives of functions in infinite-dimensional spaces |
93B35 | Sensitivity (robustness) |
93C20 | Control/observation systems governed by partial differential equations |
93C05 | Linear systems in control theory |
Keywords:
directional (one-sided) differentiability; projection onto a closed, convex set; Hilbert space; sensitivity analysis; constrained optimal control problemsReferences:
[1] | Haraux A (1977) How to differentiate the projection on a convex set in Hilbert space: some applications to variational inequalities. J Math Soc Japan, 29:615–631 · Zbl 0387.46022 · doi:10.2969/jmsj/02940615 |
[2] | Jittorntrum K (1984) Solution point differentiability without strict complementarity in nonlinear programming. In: AV Fiacco (ed) Mathematical Programming Studies 21. North-Holland, Amsterdam · Zbl 0571.90080 |
[3] | Lions JL (1968) Sur le contrôle optimal de systèmes gouvernes par des équations aux dérivées partielles. Dunod, Paris |
[4] | Lions JL, Magenes E (1968) Problemes aux limites non homogènes et applications, vol. 2. Dunod, Paris |
[5] | Malanowski K (1984) Differential stability of solutions to convex, control constrained optimal control problems. Appl Math Optim 12:1–14 · Zbl 0561.49020 · doi:10.1007/BF01449030 |
[6] | Mignot F (1976) Controle dans les inequations variationnelles. J Functional Analysis 22:130–185 · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3 |
[7] | Rockafellar RT (1978) La theorie des sous-gradients et ses applications à l’optimisation. les Presses de l’Université de Montréal |
[8] | Sokołowski J Sensitivity analysis of control constrained optimal control problems for distributed parameter systems (to be published) · Zbl 0647.49019 |
[9] | Sokołowski J (1981) Conical differentiability of projection on convex sets–an application to sensitivity analysis of Signorini VI. Technical Report, Institute of Mathematics of the University of Genoa |
[10] | Sokołowski J, Zolesio JP (1982) Derivation par rapport au domaine dans les problemes unilateraux INRIA. Rapport de recherche 132, Rocquencourt |
[11] | Zolesio JP (1981) The material derivative (or speed) method for shape optimization. In: Haug EJ, Cea J (eds), Optimization of distributed parameter structures, vol. 2, Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands · Zbl 0517.73097 |
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