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OPTIMIZATION and VARIATIONAL INEQUALITIES Basic Statements and Constructions

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1 OPTIMIZATION and VARIATIONAL INEQUALITIES Basic statements and constructions Bernd Kummer; [email protected] ; May 2011 Abstract. This paper summarizes basic facts in both finite and infinite dimensional optimization and for variational inequalities. In addition, partially new results (concerning methods and stability) are included. They were elaborated in joint work with D. Klatte, Univ. Z¨urich and J. Heerda, HU-Berlin. Key words. Existence of solutions, (strong) duality, Karush-Kuhn-Tucker points, Kojima-function, generalized equation, (quasi -) variational inequality, multifunctions, Kakutani Theo- rem, MFCQ, subgradient, subdifferential, conjugate function, vector-optimization, Pareto- optimality, solution methods, penalization, barriers, non-smooth Newton method, per- turbed solutions, stability, Ekeland's variational principle, Lyusternik theorem, modified successive approximation, Aubin property, metric regularity, calmness, upper and lower Lipschitz, inverse and implicit functions, generalized Jacobian @cf, generalized derivatives TF , CF , D∗F , Clarkes directional derivative, limiting normals and subdifferentials, strict differentiability. Bemerkung. Das Material wird st¨andigaktualisiert. Es soll Vorlesungen zur Optimierung und zu Vari- ationsungleichungen (glatt, nichtglatt) unterst¨utzen. Es entstand aus Scripten, die teils in englisch teils in deutsch aufgeschrieben wurden. Daher gibt es noch Passagen in beiden Sprachen sowie z.T. verschiedene Schreibweisen wie etwa cT x und hc; xi f¨urdas Skalarpro- dukt und die kanonische Bilinearform. Hinweise auf Fehler sind willkommen. 2 Contents 1 Introduction 7 1.1 The intentions of this script . .7 1.2 Notations . .7 1.3 Was ist (mathematische) Optimierung ? . .8 1.4 Particular classes of problems . 10 1.5 Crucial questions . 11 2 Lineare Optimierung 13 2.1 Dualit¨atund Existenzsatz f¨urLOP . 15 2.2 Die Simplexmethode . 17 2.3 Schattenpreise . 22 2.4 Klassische Transportaufgabe . 22 2.5 Maximalstromproblem . 24 2.6 The core of a game . 25 2.7 Aquivalenz:¨ Matrixspiel und Lineare Optimierung . 26 2.7.1 Matrixspiel . 26 2.7.2 Matrixspiel als LOP; Bestimmen einer GGS . 27 2.7.3 LOP als Matrixspiel; Bestimmen einer LOP-L¨osung . 28 2.8 The Julia Robinson Algorithm . 28 3 Elements of convex analysis 31 3.1 Separation of convex sets . 31 3.2 Brouwer's and Kakutani's Fixpunktsatz . 34 3.2.1 The theorems in IRn ........................... 34 3.2.2 The theorems in normed spaces . 38 3.3 Nash Equilibria and Minimax Theorem . 38 3.3.1 The basic Existence Theorem . 39 3.3.2 Normalized Nash Equilibria . 39 3.4 Classical subdifferentials, normals, conjugation . 40 3.4.1 Definitions and existence . 41 3.4.2 Directional derivative, subdifferential, optimality condition . 43 3.4.3 Particular subdifferentials in analytical form . 46 3.5 Normal cones and Variational Inequalities . 47 3.5.1 Definitions and Basic motivations . 47 3.5.2 VI's and Fixed-points . 48 3.5.3 Monotonicity . 49 ∗ 3.5.4 Analytical formulas for CM and NM = CM and hypo-monotonicty . 50 3.6 An Application: Tschebyschev-approximation . 52 3 4 CONTENTS 4 Nonlinear Problems in IRn 53 4.1 Notwendige Optimalit¨atsbedingungen . 53 4.1.1 KKT- Bedingungen, Lagrange- Multiplikatoren . 53 4.1.2 Concrete Constraint Qualifications . 57 4.1.3 Calmness, MFCQ and Aubin-Property . 59 4.1.4 Calmness and Complementarity . 60 4.2 The standard second order condition . 61 4.3 Eine Anwendung: Brechungsgesetz . 63 5 Duality and perturbations in Banach spaces 65 5.1 Separation in a normed space . 65 5.2 Strong duality and subgradients in B-spaces . 66 5.3 Minimax and saddle points . 68 5.4 Ensuring strong duality and existence of LM's . 69 5.4.1 The convex case . 69 5.4.2 The C1- case and linear approximations . 70 5.5 Modifications for vector- optimization . 71 6 L¨osungsverfahren; NLO in endl. Dimension 73 6.1 Freie Minima, Newton Meth. und variable Metrik . 73 6.1.1 "Fixed stepsize" or Armijo- Golstein rule . 73 6.1.2 Line search . 74 6.1.3 Variable metric . 75 6.2 The nonsmooth Newton method . 75 6.3 Cyclic Projections; Feijer Methode . 78 6.4 Proximal Points, Moreau-Yosida approximation . 78 6.5 Schnittmethoden; Kelley-Schnitte . 79 6.5.1 The original version . 79 6.5.2 Minimizing a convex function by linear approximation . 80 6.6 Strafmethoden . 81 6.7 Barrieremethoden . 83 6.8 Ganzzahligkeit und "Branch and Bound" . 83 6.9 Second-order methods . 85 7 Stability: Motivations and first results 87 7.1 Two-stage optimization and the main problems . 87 7.1.1 The simplest two-stage problem . 87 7.1.2 The general case . 88 7.2 KKT under strict complementarity . 89 7.3 Basic generalized derivatives . 91 7.3.1 CF and TF ................................ 91 7.3.2 Co-derivatives and generalized Jacobians . 91 7.4 Some chain rules . 92 7.4.1 Adding a function . 93 7.4.2 Inverse mappings . 93 7.4.3 Derivatives of the inverse S = (h + F )−1 ................ 93 7.4.4 Composed mappings . 93 7.4.5 Linear transformations or diffeomorphisms of the space . 94 7.5 Loc. Lipschitzian inverse and implicit functions . 94 7.5.1 Inverse Lipschitz functions . 94 CONTENTS 5 7.5.2 Implicit Lipschitz functions . 96 8 KKT points as zeros of equations 99 8.1 (S. M. Robinson's) Generalized equations . 99 8.2 NCP- functions . 100 8.3 Kojima's function . 101 8.3.1 The product form . 101 8.3.2 Implicit KKT-points . 103 8.4 Relations to penalty-barrier functions . 104 8.4.1 Quadratic Penalties . 104 8.4.2 Quadratic and logarithmic barriers . 105 8.4.3 Modifications and estimates . 105 8.5 Regularity and Newton Meth. for KKT points and VI's . 106 9 More elements of nonsmooth analysis 109 9.1 Abstract normals, subgradients, coderivatives, tangents . 109 9.1.1 Subdifferentials and optimality conditions derived from normals . 110 9.1.2 (i) Usual localized normals . 110 9.1.3 (ii) Fr´echet-normals . 111 9.1.4 (iii) Limiting "- normals. 112 9.1.5 (iv) Limiting Fr´echet-normals. 113 9.1.6 Equivalence of the limiting concepts . 113 9.2 (v) Clarke's approach . 114 9.2.1 Basic definitions and interrelations . 114 9.2.2 Mean-value theorems and Generalized Jacobian . 118 9.3 Subdifferentials derived from optimality conditions . 120 9.3.1 The elementary principle . 120 9.3.2 The empty and limiting F-subdifferential . 121 10 Stability (regularity) of solutions 123 10.1 Definitions of Stability . 123 10.2 Interrelations and composed mappings . 124 10.3 Stability of multifunctions via Lipschitz functions . 125 10.3.1 Basic transformations . 125 10.3.2 Solutions of (exact) penalty problems . 127 10.4 Stability and algorithms . 128 10.5 The algorithmic framework . 129 10.6 Stability via approximate projections . 131 10.7 Stability of a locally Lipschitz operator . 132 10.7.1 Calmness and the relative slack for inequality systems . 133 10.7.2 The relative slack for solving C1 inequality systems . 135 10.7.3 Calmness and crucial linear inequality systems . 136 10.8 Modified successive approximation for −h(x) 2 F (x)............. 137 11 Ekelands Principle, Stability and gen. Derivatives 139 11.1 Ekeland's principle and Aubin property . 139 11.1.1 Application to the Aubin property . 140 11.1.2 Weakly stationary points . 142 11.2 Stability in terms of generalized derivatives . 143 11.2.1 Strongly Lipschitz . 143 11.2.2 Upper Lipschitz . 144 6 CONTENTS 11.2.3 Lower Lipschitz . 144 11.2.4 Aubin property . 144 11.2.5 Summary . ..
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