Nonsmooth Optimization and Related Topics

Transcription

1 Nonsmooth Optimization and Related Topics Edited by F. H. Clarke University of Montreal Montreal, Quebec, Canada V. F. Dem'yanov Leningrad State University Leningrad, USSR I and F. Giannessi University of Pisa Pisa, Italy Plenum Press New York and London

2 CONTENTS 1 Scalar and vector generalized convexity (E. Castagnoli and P. Mazzoleni) 1. Introduction 1 2. Generalized concavity for scalar functions 2 3. Generalized concavity for vector functions 3 4. The weak Pareto ordering 7 5. Weak convex sets 8 6. Weakly concave functions A differentiable setting Conclusions 20 2 Compactness and boundedness for a class of concave-convex functions {E.Cavazzuti and N.Pacchiarotti) 1. Introduction. Notations and preliminary definitions Some properties about T-closed functions Properties of the e-h-limit of a sequence of T-closed functions Compactness results 32 3 Applications of proximal subgradients (F.H. Clarke) 1. Introduction An intersection formula A regularity theorem in the Calculus of Variations 44 4 New functionals in Calculus of Variations (E. De Giorgi and L. Ambrosio) 1. Definition of the functional Thec[sissesGBV(n,WL h ;E),GSBV{n,M. k ;E) Semicontinuity problems An example from the static theory of liquid crystals Representation of functionals in GBV(n,H k )\GSBV(n,M k ) Quasi-variational inequalities and applications to equilibrium problems with elastic demand [M. De Luca and A. Maugeri) 1. Introduction The computational procedure An existence theorem An example 74 6 Smoothness of nonsmooth functions (V.F. Dem'yanov) 1. First order approximations of a function Upper and lower first order approximations Codifferentiable functions 84

3 viii Contents 4. Calculus of codifferentials Second order approximations 87 7 Exact penalty functions for nondifferentiable programming problems (G. Di Pillo and F. Facchinei) 1. Introduction Problem formulation Preliminary results Exactness of J q (x;e) Further results Conclusions Fuzzy T-operators and convolutive approximations (S. Dolecki) 1. Introduction Fuzzy T-functionals Fuzzy T-operators Semi-fuzzy Operators and convolutive representations Comparison theorems: flexibility Mixed Fuzzy T-operators Inf- and sup-convolutive approximations On cone approximations and generalized directional derivatives {K.-H. Elster and J. Thierfelder) 1. Introduction Cone approximations of sets Generalized directional derivatives Some conclusions Some techniques for flnding the search direction in nonsmooth minimization problems (M. Gaudioso and M.F. Monaco) 1. Introduction Bündle type approaches Bündle methods and quadratic approximations Conclusions Directional derivative for the value function in mathematical programming (J. Gauvin) 1. Introduction Hypotheses and preliminaries Directional derivative for the value function Examples Necessary optimality conditions via image problem (F. Giannessi, M. Pappalardo and L. Pellegrini) 1. Introduction The image problem Generalized sernidifferentiable functions Cone-approximations and image reductions Necessary conditions Concluding remarks and extensions 214

4 Contents ix 13 From convex optimization to nonconvex optimization. Necessary and sumcient conditions for global optimality (J.-B. Hiriart-Urruly) 1. Introduction Background of convex analysis and optimization Necessary conditions for local and global optimality Sufficient conditions for local and global optimality First examples of applications Nonconvex subdifferentials (.4. loffe) 1. Introduction Non-Lipschitz functions. The finite-dimensional case An example: an optimal control problem Non-Lipschitz functions. Infinite-dimensional case G-subdifferential Calculus of G-subdifferentials G-subdifferential and proximal analysis Perturbed differential inclusion problems (P.D. Loewen) 1. Introduction The value function Sensitivity analysis A formula for the generalized gradient of V Subdifferential analysis and plates subjected to unilateral constraints (A. Marino, C. Saccon) 1. Introduction A notion of subdifferential and some properties A class of functions von Karman's plate problern with obstacle Optimization problems for aircraft flight in a windshear (A. Miele, T. Wang) 1. Introduction Equations of motion System description Algorithm Take-off problem Abort landing problem Penetration landing problem Conclusions Constrained well-posed two-level optimization problems (/. Morgan) 1. Introduction Formulation of the problem Sequential stability analysis of (S): a motivation for the well-posedness notion Well-posed two-level optimization problems The reaction set is a singleton Application to the constrained linear quadratic dynamic games.. 320

5 x Contents 19 A compactness theorem for curves of maximal slope for a class of nonsmooth and nonconvex functions (Ä. Orlandoni, 0. Petrucci and M. Tosques) 1. Introduction Convergence A compactness result Basics of minimax algorithms (E. Polak) 1. Introduction Generic algorithms for simplest minimax problems Geometrie extension of steepest descent Rate of convergence of minimax algorithm General minimax problem Constrained minimax problems Conclusions Implicit funetion theorems for multi-valued mappings (B.N. Pshenichny) 1. Introduction Implicit funetion theorems for convex mappings Locally smooth maps Perturbation of generalized Kuhn-Tucker points in finitedimensional optimization (R.T. Rockafellar ) 1. Introduction Parametrization and sensitivity Nonsmooth optimization and dual bounds (N.Z. Shor) 1. Introduction Quadratic dual estimates for nonconvex polynomial problems A unified treatment of some nonstandard problems in dynamics optimization (R.B. Vinter) 1. Introduction A maximum principle Sketch of proof Local and global directional controllability: sufficient conditions and examples (J. Warga) 1. Introduction Sufficient conditions for conical controllability Examples. (Higher order conditions) Global directional controllability Examples. (Global controllability) Stability for a class of nonlinear optimization problems and applications (C. Zälinescu) 1. Introduction Asymptotic cones and asymptotically compact sets d-stability for a class of nonconvex problems Lower semicontinuity, recession functions and e-subdifferentials for some convex functions A perturbation results Conclusions 456

6 Contents xi 27 The BT-algorithm for minimizing a nonsmooth functional subject to linear constraints (J. Zowe) 1. Introduction and example Failure of smooth methods The conceptual algorithm The trajectory d(t) The BT-algorithm Convergence Numerical results 479 Addresses of Contributors 481 Index. 485