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

Minimizing and stationary sequences.

by Wong Oi Ping.Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.Includes bibliographical references (leaves 77-79).Abstracts in English and Chinese.Chapter 1 --- LP-minimizing and Stationary Sequences --- p.8Chapter 1.1 --- Residual function --- p.8Chapter 1.2 --- Minimizing sequences --- p.14Chapter 1.3 --- Stationary sequences --- p.17Chapter 1.4 --- On the equivalence of minimizing and stationary se- quence --- p.21Chapter 1.5 --- Complementarity conditions --- p.25Chapter 1.6 --- Subdifferential-based stationary sequence --- p.29Chapter 1.7 --- Convergence of an Iterative Algorithm --- p.32Chapter 2 --- Minimizing And Stationary Sequences In Nonsmooth Optimization --- p.38Chapter 2.1 --- Subdifferential --- p.38Chapter 2.2 --- Stationary and minimizing sequences --- p.40Chapter 2.3 --- C-convex and BC-convex function --- p.43Chapter 2.4 --- Minimizing sequences in terms of sublevel sets --- p.44Chapter 2.5 --- Critical function --- p.48Chapter 3 --- Optimization Conditions --- p.52Chapter 3.1 --- Introduction --- p.52Chapter 3.2 --- Second-order necessary and sufficient conditions with- out constraint --- p.55Chapter 3.3 --- The Lagrange and G-functions in constrained problems --- p.63Chapter 3.4 --- Second-order necessary conditions for constrained prob- lems --- p.73Chapter 3.5 --- Sufficient conditions for constrained problems --- p.74Bibliograph

Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn-Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fr\'echet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.Comment: 52 pages. Keywords: optimal control, tumor growth models, singular potentials, optimality conditions, second-order analysi

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