zbMATH — the first resource for mathematics

Optimization in Banach spaces. (English) Zbl 0669.49012
Let X and Y be real Banach spaces and \(K\subset Y\) a closed cone satisfying some additional conditions, which generates a partial ordering in Y. The authors consider the abstract optimization problem: (MP) Minimise f(x) subject to g(x)\(\geq 0\) and \(h(x)=0\), where f:X\(\to Y\), \(g:X\to Y^ m\) and \(h:X\to Y^ p\). Using so called “strict separation axiom” they establish a Fritz John and Kuhn-Tucker-type necessary condition for the existence of a solution to the problem (MP).
Reviewer: M.Todorov

49K27 Optimality conditions for problems in abstract spaces
90C48 Programming in abstract spaces
Full Text: DOI
[1] Avriel, M, Non-linear programming: analysis and methods, (1976), Prentice-Hall Englewood, NJ
[2] Guinard, M, Generalised Kuhn-Tucker conditions for mathematical problems in a Banach space, SIAM J. control, 7, 232-241, (1969) · Zbl 0182.53101
[3] Kantorovich, L.V; Akilov, G.P, Functional analysis in normed spaces, (1964), Pergamon Elmsford, NY · Zbl 0127.06104
[4] Ritter, K, Optimization theory in linear spaces, I, Math. ann., 182, 189-206, (1969) · Zbl 0177.40402
[5] Ritter, K, Optimization theory in linear spaces. part II. on systems of linear operator inequalities in partially ordered normed linear spaces, Math. ann., 183, 169-180, (1969) · Zbl 0186.18202
[6] Ritter, K, Optimization theory in linear spaces. part III. mathematical programming in partially ordered Banach spaces, Math. ann., 184, 133-154, (1970) · Zbl 0186.18203
[7] Varaiya, P, Non-linear programming in Banach space, SIAM J. appl. math., 15, 284-293, (1967) · Zbl 0171.18004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.