Abstract
First-order and second-order necessary and sufficient optimality conditions are given for infinite-dimensional programming problems with constraints defined by arbitrary closed convex cones. The necessary conditions are immediate generalizations of those known for the finite-dimensional case. However, this does not hold for the sufficient conditions as illustrated by a counterexample. Here, to go from finite to infinite dimensions, causes an essential change in the proof-techniques and the results. We present modified sufficient conditions of first-order and of second-order which are based on a strengthening of the usual assumptions on the derivative of the objective function and on the second derivative of the Lagrangian.
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Maurer, H., Zowe, J. First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Mathematical Programming 16, 98–110 (1979). https://doi.org/10.1007/BF01582096
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DOI: https://doi.org/10.1007/BF01582096