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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.

##### 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