Well-posedness, regularization, and viscosity solutions of minimization problems. (English) Zbl 1317.49036
Ansari, Qamrul Hasan (ed.), Nonlinear analysis. Approximation theory, optimization and applications. Contributions based on the presentations at the special session on approximation theory and optimization in the Indian Mathematical Society conference, Varanasi, India, January 12–15, 2012. New Delhi: Birkhäuser/Springer (ISBN 978-81-322-1882-1/hbk; 978-81-322-1883-8/ebook). Trends in Mathematics, 135-164 (2014).
Summary: This chapter is divided into two parts. The first part surveys some classical notions for well-posedness of minimization problems. The main aim here is to synthesize some known results in approximation theory for best approximants, restricted Chebyshev centers, and prox points from the perspective of well-posedness of these problems. The second part reviews Tikhonov regularization of ill-posed problems. This leads us to revisit the so-called viscosity methods for minimization problems using the modern approach of variational convergence. Lastly, some of these results are particularized to convex minimization problems, and also to ill-posed inverse problems.
For the entire collection see [Zbl 1300.49001].
For the entire collection see [Zbl 1300.49001].
MSC:
| 49K40 | Sensitivity, stability, well-posedness |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 49N45 | Inverse problems in optimal control |
Keywords:
well-posedness; best approximants; restricted Chebyshev centers; Tikhonov regularization; viscosity solutions; epi-convergence; hierarchial minimization; ill-posed inverse problemsReferences:
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