Nonsmooth analysis and optimization.

Huang Liren.Thesis (Ph.D.)--Chinese University of Hong Kong, 1993.Includes bibliographical references (leaves 96).Abstract --- p.1Introduction --- p.2References --- p.5Chapter Chapter 1. --- Some elementary results in nonsmooth analysis and optimization --- p.6Chapter 1. --- "Some properties for ""lim sup"" and ""lim inf""" --- p.6Chapter 2. --- The directional derivative of the sup-type function --- p.8Chapter 3. --- Some results in nonsmooth analysis and optimization --- p.12References --- p.19Chapter Chapter 2. --- On generalized second-order derivatives and Taylor expansions in nonsmooth optimization --- p.20Chapter 1. --- Introduction --- p.20Chapter 2. --- "Dini-directional derivatives, Clark's directional derivatives and generalized second-order directional derivatives" --- p.20Chapter 3. --- On Cominetti and Correa's conjecture --- p.28Chapter 4. --- Generalized second-order Taylor expansion --- p.36Chapter 5. --- Detailed proof of Theorem 2.4.2 --- p.40Chapter 6. --- Corollaries of Theorem 2.4.2 and Theorem 2.4.3 --- p.43Chapter 7. --- Some applications in optimization --- p.46Ref erences --- p.51Chapter Chapter 3. --- Second-order necessary and sufficient conditions in nonsmooth optimization --- p.53Chapter 1. --- Introduction --- p.53Chapter 2. --- Second-order necessary and sufficient conditions without constraint --- p.56Chapter 3. --- Second-order necessary conditions with constrains --- p.66Chapter 4. --- Sufficient conditions theorem with constraints --- p.77References --- p.87Appendix --- p.89References --- p.9

Why second-order sufficient conditions are, in a way, easy -- or -- revisiting calculus for second subderivatives

In this paper, we readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on the so-called second subderivative of the objective function and of the indicator function associated with the feasible set, one easily obtains second-order sufficient optimality conditions of abstract form. In order to exploit further structure of the problem, e.g., composite terms in the objective function or feasible sets given as (images of) pre-images of closed sets under smooth transformations, to make these conditions fully explicit, we study calculus rules for the second subderivative under mild conditions. To be precise, we investigate a chain rule and a marginal function rule, which then also give a pre-image and image rule, respectively. As it turns out, the chain rule and the pre-image rule yield lower estimates desirable in order to obtain sufficient optimality conditions for free. Similar estimates for the marginal function and the image rule are valid under a comparatively mild inner calmness* assumption. Our findings are illustrated by several examples including problems from composite, disjunctive, and nonlinear second-order cone programming.Comment: 43 page

KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization

For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints (MPECs), in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper

Second-order subdifferential calculus with applications to tilt stability in optimization

The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms