Remarks to an equivalent formulation of Ekeland’s variational principle. (English) Zbl 0815.49012
Summary: A recent theorem of W. Takahashi is pointed out as an equivalent formulation of Ekelands variational principle. This gives rise to study functions having sets of weak sharp minima in a generalized sense. Connections to the proximal point algorithm in the convex case leads to the basic and still open question: How to use Ekelands principle numerically.
MSC:
| 49J52 | Nonsmooth analysis |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
| 90C26 | Nonconvex programming, global optimization |
References:
| [1] | Takahashi W., Fixed point theory and applications (1991) |
| [2] | Daneš J., Commutationes Mathematicate Universitatis Carolinae 26 (1985) |
| [3] | DOI: 10.1016/0362-546X(86)90069-6 · Zbl 0612.49011 · doi:10.1016/0362-546X(86)90069-6 |
| [4] | DOI: 10.1090/S0273-0979-1979-14595-6 · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6 |
| [5] | Ekeland I., Methods of Nonconvex Analysis (1990) · Zbl 0707.70003 |
| [6] | Burke J. V. Ferris M. C. Weak Sharp Minima In Mathematical Programming Computer Science Department, Uni.of Wisconsin-Madison 1991 Computer Science Techinical Report #1050 |
| [7] | DOI: 10.1007/BF01594944 · Zbl 0741.90051 · doi:10.1007/BF01594944 |
| [8] | DOI: 10.1137/0329022 · Zbl 0737.90047 · doi:10.1137/0329022 |
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