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Directional differentiability of the optimal value function in convex semi-infinite programming

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Abstract

In this paper, directional differentiability properties of the optimal value function of a parameterized semi-infinite programming problem are studied. It is shown that if the unperturbed semi-infinite programming problem is convex, then the corresponding optimal value function is directionally differentiable under mild regularity assumptions. A max-min formula for the directional derivatives, well-known in the finite convex case, is given.

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References

  1. J.F. Bonnans, A.D. Ioffe and A. Shapiro, “Expansion of exact and approximate solutions in nonlinear programming,” in: W. Oettli and D. Pallaschke, eds.,Advances in Optimization, Proceedings of the 6th French—German Conference on Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 382 (Springer, Berlin, 1992) pp. 103–117.

    Google Scholar 

  2. J. Gauvin and R. Janin, “Directional behaviour of optimal solutions in nonlinear mathematical programming,”Mathematics of Operations Research 13 (1988) 629–649.

    Google Scholar 

  3. E.G. Gol'shtein,Theory of Convex Programming, Translations of Mathematical Monographs, Vol. 36 (American Mathematical Society, Providence, RI, 1972).

    Google Scholar 

  4. F. Lempio and H. Maurer, “Differential stability in infinite-dimensional nonlinear programming,”Applied Mathematics and Optimization 6 (1980) 139–152.

    Google Scholar 

  5. H. Maurer and J. Zowe, “First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems,”Mathematical Programming 16 (1979) 98–110.

    Google Scholar 

  6. J.P. Penot, “On regularity conditions in mathematical programming,”Mathematical Programming Study 19 (1982) 167–199.

    Google Scholar 

  7. S.M. Robinson, “First order conditions for general nonlinear optimization,”SIAM Journal on Applied Mathematics 30 (1976) 597–607.

    Google Scholar 

  8. S.M. Robinson, Stability theorems for systems of inequalities, Part II: differentiable nonlinear systems,”SIAM Journal on Numerical Analysis 13 (1976) 497–513.

    Google Scholar 

  9. R.Y. Rubinstein and A. Shapiro,Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Method (Wiley, New York, 1993).

    Google Scholar 

  10. A. Shapiro, “On Lipschitzian stability of optimal solutions of parametrized semi-infinite programs,”Mathematics of Operations Research 19 (1994) 743–752.

    Google Scholar 

  11. A. Shapiro and J.F. Bonnans, “Sensitivity analysis of parametrized programs under cone constraints,”SIAM Journal on Control and Optimization 30 (1992) 1409–1422.

    Google Scholar 

  12. P. Zencke and R. Hettich, “Directional derivatives for the value-function in semi-infinite programming,”Mathematical Programming 38 (1987) 323–340.

    Google Scholar 

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, 30332-0205, Atlanta, GA, USA

    Alexander Shapiro

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  1. Alexander Shapiro
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Shapiro, A. Directional differentiability of the optimal value function in convex semi-infinite programming. Mathematical Programming 70, 149–157 (1995). https://doi.org/10.1007/BF01585933

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  • DOI: https://doi.org/10.1007/BF01585933

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