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**On lower subdifferentiable functions.**
*(English)*
Zbl 0643.49015

Trends in mathematical optimization, 4th French-German Conf., Irsee/FRG 1986, ISNM 84, 197-232 (1988).

[For the entire collection see Zbl 0626.00020.]

This paper is mainly devoted to the study of lower subdifferentiable functions, in the sense of F. Plastria [J. Optimization Theory Appl. 46, 37-53 (1985; Zbl 0542.90083)], from the viewpoint of the generalized conjugation theory of J. J. Moreau [J. Math. Pures Appl., IX. Sér. 49, 109-154 (1970; Zbl 0195.495)]. For the coupling function \(Q: R^ n\times (R^ n\times R)\to R\) given by \(Q(x,(x^*,k))=\min \{<x,x^*>,k\}\) (where \(<.,.>\) denotes the Euclidean scalar product), the projection onto \(R^ n\) of the Q- subdifferential, in the sense of E. J. Balder [SIAM J. Control Optimization 15, 329-343 (1977; Zbl 0366.90103)], coincides with the lower subdifferential. Among other results, it is proved that the second Q-conjugate of any function f coincides with the minimum of its lower semicontinuous quasiconvex hull and the supremum of those \(\lambda\in R\) for which there exists a nonconstant affine function minorizing f on \(f^{-1}([-\infty,\lambda))\); it also coincides with the supremum of all Lipschitz quasiconvex minorants of f.

Some other conjugation operators are studied, for which the corresponding second conjugates yield Hölder (in particular, Lipschitz) regularizations, lower semicontinuous hulls and Lipschitz quasiconvex envelopes. Finally, applications of lower subdifferentiability to quasiconvex programming and to linear time-optimal control theory are examined.

This paper is mainly devoted to the study of lower subdifferentiable functions, in the sense of F. Plastria [J. Optimization Theory Appl. 46, 37-53 (1985; Zbl 0542.90083)], from the viewpoint of the generalized conjugation theory of J. J. Moreau [J. Math. Pures Appl., IX. Sér. 49, 109-154 (1970; Zbl 0195.495)]. For the coupling function \(Q: R^ n\times (R^ n\times R)\to R\) given by \(Q(x,(x^*,k))=\min \{<x,x^*>,k\}\) (where \(<.,.>\) denotes the Euclidean scalar product), the projection onto \(R^ n\) of the Q- subdifferential, in the sense of E. J. Balder [SIAM J. Control Optimization 15, 329-343 (1977; Zbl 0366.90103)], coincides with the lower subdifferential. Among other results, it is proved that the second Q-conjugate of any function f coincides with the minimum of its lower semicontinuous quasiconvex hull and the supremum of those \(\lambda\in R\) for which there exists a nonconstant affine function minorizing f on \(f^{-1}([-\infty,\lambda))\); it also coincides with the supremum of all Lipschitz quasiconvex minorants of f.

Some other conjugation operators are studied, for which the corresponding second conjugates yield Hölder (in particular, Lipschitz) regularizations, lower semicontinuous hulls and Lipschitz quasiconvex envelopes. Finally, applications of lower subdifferentiability to quasiconvex programming and to linear time-optimal control theory are examined.

Reviewer: J.E.Martínez-Legaz

### MSC:

49J52 | Nonsmooth analysis |

49N15 | Duality theory (optimization) |

90C30 | Nonlinear programming |

26B25 | Convexity of real functions of several variables, generalizations |

26B35 | Special properties of functions of several variables, Hölder conditions, etc. |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49K15 | Optimality conditions for problems involving ordinary differential equations |

93B99 | Controllability, observability, and system structure |

90C25 | Convex programming |