On lower subdifferentiable functions. (English) Zbl 0643.49015

Martínez-Legaz, J. E.

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.

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

Keywords:

lower subdifferentiable functions; generalized conjugation; regularizations; quasiconvex envelopes; quasiconvex programming; linear time-optimal control

Citations:

Zbl 0626.00020; Zbl 0542.90083; Zbl 0195.495; Zbl 0366.90103