Subgradient Methods in Deterministic and Stochastic Optimization.

Abstract

February 27. 1984 Introduction. Research supported by this two-year grant in the period from January, 1982, through December, 1984, has resulted in a total of 11 technical articles and two doctoral theses. These range over several areas of mathematical optimization theory but share the common theme of the development and application of subgradient methods and duality to problems in mathematical programming. Fundamental advances in concept have been made, and in the case of stochastic problems, new techniques of solution have been initiated that may revolutionize the subject. The publications are grouped under the following headings, which will be discussed individually: 1. Stochastic programming (4 papers). 2. Subgradient theory (3 papers, 1 thesis). 3. Nonlinear programming (4 papers). 4. Optimal control (1 thesis). >_AIR IpORC' CE

Characterizations and Classification of Paraconvex Multimaps

“…Related to the needs of applications in variational analysis a wide literature has been devoted to extensions of convexity and/or smoothness for sets, functions and multifunctions. For real valued functions a robust extension of both convexity and smoothness is given by LC k -functions, or lower -C k (k ∈ N) functions, introduced (for k ≥ 2) by T. Rockafellar (1982) in finite dimension, then extended to infinite dimensional Hilbert space by J. P. Penot (1996). The lower -C 1 functions, due to J. E. Springarn (1981), are characterized by a submonotonicity property of subdifferentials.…”

“…Then, J. P. Vial (1983) introduced in finite dimension the weakly convex functions and characterized them as difference of two convex functions (dc-functions). But Rockafellar (1982) and Penot (1996) showed that for all k ≥ 2 the lower -C k functions are lower -C 2 and coincide actually with weakly convex and locally Lipschitz functions. For the role of these classes in optimization we refer also to Georgiev (1997).…”

Rectifiable Curves in Proximally Smooth Sets

“…In turn, such classes of functions can be equivalently described in geometric terms via the geometric properties of their epigraphs. A notable example is the class of lower-C 2 functions studied by R. T. Rockafellar [Roc81] and the class of proximally smooth sets [CSW95]. We recall that a closed set A of a normed space is R-proximally smooth if the distance function dist(•, A) is continuously differentiable on the open R-neighborhood of the form {x : 0 < dist(x, A) < R}.…”

A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian

“…then for any fixed (d, r) we have that x → U (x, d, r) is prox-regular and subdifferentially continuous (see [25,Example 2.9]), Clarke regular (see [28]), semismooth (see [21]) and hence submonotone and directionally upper semicontinuous (see [27] and [30]) and g is fully amenable at F (x) when f i ∈ C 2 [28, Example 10.23]. It is thus a very well-behaved function indeed from the standpoint of nonsmooth analysis.…”

“…Observe that in this case, if f α ∈ C 2 (IR n ) and Λ is finite, then f is locally Lipschitz, prox-regular and subdifferentially continuous (see [25, Example 2.9]). Also f is regular function (see [27] regarding lower C 2 functions) and semismooth (see [21] regarding suprema of semi-smooth functions). Hence by the results of [30] we must have ∂f submonotone and hence directionally continuous.…”