Abstract
In the following sections we shall consider optimization problems of the form
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Literature
For 5.1: An introduction to the differential calculus in normed linear spaces is contained in several texts on optimization, e. g. LUENBERGER [53], IOFFE-TICHOMIROV [37].
For 5.2: The fundamental paper on the Theorem of LYUSTERNIK is LYUSTERNIK [55]; c.f. alpo LYUSTERNIK-SOBOLEV [56]. Theorem 5.2.3, the generalization of the open mapping theorem, is proved in ZOWE-KURCYUSZ [81]. Lemma 3.1 of that paper also gives the main result of this section, namely Theorem 5.2.5, and refers the reader for the proof to ROBINSON [67]. However ROBINSON uses the theory of convex processes, so the proof given here is surely easier. A similar “elementary” proof of the Theorem of LYUSTERNIK is also given in BROKATE-KRABNER [10]. In this connection one should also mention LEMPIO [51].
For 5.3: The literature on necessary optimality conditions of first order resp. Lagrange multiplier rules of KUHN-TUCKER and F. JOHN type is extremely extensive. Among the historical papers we shall only mention besides that of LYUSTERNIK [55] for optimization problems with equations as constraints those of KARUSH [40], JOHN [38] and KUHN-TUCKER [48] for finite dimensional programs with inequalities as side conditions. A thorough exposition of necessary optimality conditions in finite dimensional optimization problems can also be found in MANGASARIAN [57], HESTENES [33]; BAZARAA-SHETTY [3]. A small selection of literature on necessary optimality conditions of first order for not necessarily finite dimensional problems is given by LUENBERGER [53], IOFFE-TICHOMIROV P7], PONSTEIN [65], KIRSCH-WARTH-WERNER [42], GIRSANOV [27], NEUSTADT [62], C0L0- NIUS [15], BEN TAL-ZOWE [4], PENOT [63].
For 5.4: For applications of abstract necessary optimality conditions to problems of the calculus of variations and of control theory we used LUENBERGER [53], IOFFE-TICHOMIROV [37] and KIRSCH-WARTH-WERNER [42]. How one gets from the local to the global Pontryagin maximum principle can be found in GIR- SANOV [27].
For 5.5: Theorem 5.5.2 is due to MAURER-ZOWE [59]. Other important papers in this connection are e. g. HOFFMANN-KORN- STAEDT [35], BEN TAL-ZOWE [4], LINNEMANN [52]. Necessary and sufficient optimality conditions of second order for finite dimensional optimization problems are contained in FIACCO- McCORMICK [22], LUENBERGER [54].
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© 1984 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Werner, J. (1984). Necessary Optimality Conditions. In: Optimization Theory and Applications. Advanced Lectures in Mathematics. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-84035-6_5
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DOI: https://doi.org/10.1007/978-3-322-84035-6_5
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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