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The Quadratic Sub-Lagrangian of Prox-Regular Funct Ions

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The Quadratic Sub-Lagrangian of Prox-Regular Funct Ions University of Alberta The Quadratic Sub-Lagrangian of Prox-Regular Funct ions Wzren L. Hare O A thesis submitted to the Faculty of Graduate Studies and Research in partial ikiEUment of the requirements for the degree of Mâster of Science in Mathernat ics Department of Mathematics Edmonton, Alberta Spring 2000 National Library Bibliothèque nationale 1*1 ofCanada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395. rue Wellington Ottawa ON KIA ON4 Ottawa ON KIA ON4 Canada racla The author has granted a non- L'auteur a accordé une licence non exclusive licence allowing the exclusive permettant à la National Lhrary of Canada to Bibliothèque nationale du Canada de reproduce, loan, distri%uteor seIl reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fonne de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantid extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othenvise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation, Abstract Given a finite valured convex function, f, the Cr-Lagrangian, as defined by Lemaréchal, Oustr-jr, and Sagastiz&bal, provides an envelcpe of f that maintains these propemties. By selecting the appropriate UV decomposition on Rnone can also ensure that the U-Lagrangian is not only continuous at the origin, but differentiab-le there. This thesis extends t hese properties to pros- regular functions by creating a new envelope which we cal1 the Quadrat ic Sub- Lagrangian. In furthen exploration of this envelope it will be demonstrated how to use a quadratic expansion of the Quadratic Sub-Lagrangian to create a quadratic expansiom for the original function. Finally some properties that guarantee such am expansion for the Quadratic Sub-Lagrangian will be developed. Acknowledgements and Dedicat ion To Dr. H. Bell, for making me think it would be possible; To Dr. R. Kerman, for making me think it would be interesting; and, To Dr. R. Poliquin, for making both these things true- 1 would also like to express my deep gratitude to the Naturai Sciences and Engineering Research Council of Canada and the University of Alberta for the financial support they have given me throughout my studies. Finally f would like to thank Dr. Tuszynski and Dr. Van Roessel, who dedicated much of their time and e:!ergy to the irnprovement of this thesis. Contents 1 Introduction and Background 1 1.1 Introduction ............................ 1 1.2 Basic Definitions ..........................3 1.2.1 Lower Semi.Continuity, and Proper Functions ..... 9, Lower Semi-continuou: Upper Semi-continuous functions . 2 Proper functions ........................ 3 1.2.2 The Geometry of Functions ............... 3 Relative Interior of a set ................... 3 Outer Semi.continuous, Inner Semi-continuous functions . 4 Normal, Tangent Cones .................... 4 Epi-Limits and Epi-Derivatives ................ 5 1.2.3 Subgradients, Subderivatives. and Strict Differentiability 6 Subgradients and Subdifferentiability ............ 6 f-attentiveconvergence .................... 6 Subdifferentially Regular functions .............. 7 -? Strict Differentiability ..................... i 1.3 Moreau Envelopes and U-Lagrangians .............. 7 1.3.1 Moreau Eavelopes .................... 8 The argrnin function ...................... 8 Threshold of Prox-boundedness ................ 8 1.3.2 Lagrangians, and the U-Lagrangian ........... 9 1.4 Prox-Regularity .......................... 11 f-attentive e-localization .................... Il StrongIy arnenable and Lower-C2 functions ......... 12 2 The Quadratic Sub-Lagrangian and its Basic Properties 13 2.1 The Quadratic Sub-Lagrangian ................. 13 Definition of QR. WR.and hR ................. 13 2.2 Basic Properties of the Quadratic Sub-Lagrangian ....... 14 aR compared to e~ ...................... 15 R @R+fi/~R-+~...................... 15 Properness of Qrz ....................... 20 Loaer Semi-continuity of QR ................. 34 2.3 Subgradieùt Properties ...................... 24 3 Results for CcGood"UV Decompositions 27 3.1 UV Decompositions and Smoothness of the Quadratic Sub- Lagrangian ............................ 28 Continuity and Prox-regularity of qR ............ 28 3.2 Effects of UV Decomposition on f ................ 31 Lipschitz Continuity ...................... 33 3.3 UV Decomposition and First Order Behaviour of The Quad- ratic Sub-Lagrangian ....................... 33 Differentiability of QR ..................... 34 4 Second Order Properties of the Quadratic Sub-Lagrangian 37 4.1 Definition of fi .......................... 38 4.2 Existence of Quadratic Expansions of QR ............ 39 Twice Epi-differentiability of aR ............... 40 5 Conclusion. and F'urther Areas of Exploration 43 5.1 Summary of Results ....................... 43 5.2 Future Areas of Study ...................... 46 5.3 Conclusion ............................. 48 Chapter 1 Introduction and Background Si quid calurnnietur levinus esse quam decet theologzlrn, aut mordacius quam deceat Christianum - non Ego, sed Dionysos dixit. Laurence Sterne 1.1 Introduction The theory of Optimization may be said to have begun in late 1620's, when Fermat solved the problem of hding the maximum of x y given the constraint condition that x + y = 10. Amazingly he accomplished this twenty years before either Newton or Leibniz were born (1642 and 1646 respectively), and sixty years before Calculus was first presented to the public view. In fact Fermat died in 1665, almost twenty years before Leibniz hst published his new methods for solving maxima and minima problems in 1684 ([9], pp. 396, 430, 461, 473, 477). Unlike in Fermat's time, the study of optimization today has a fuüy de- veloped field of Calculus to support it. Therefore the question of optimizing a differentiable function is mote; so, we turn ouattention to non-differentiable functions. Since the area of differentiable functions is so well developed one of the primary goals in non-differentiable optimization is to determine ways of estima ting non-Merentiable functions via differentiable ones. We call 'If anyone faIsely accuses that this is more light-hearted than becornes a theologian, or more biting than becomes a Christian - not 1, but Dionysm wrote it. such estimations envelope functions and in this thesis we will focus on the developrnent of a new one. Until recently, much of the research in optimization was directed towards convex fuictions. The reasons for this are plentifd, but largely based on the fact that convex functions usually obtain their minimum. The drawback of convex functions is that they are not dense in the space of measurable functions. This prompted the development of a new set of functions, known as prox-regular (see Section 1.4 below), which are dense in the measurable function space. In this thesis we shall generalize some the results of envelopes of convex functions to the broader set of functions known as prox-regular. 1.2 Basic Definitions Before approaching the subject of this thesis it is prudent to ensure that some basic background material is covered. 1.2.1 Lower Semi-Continuity, and Proper Functions We assume that the reader is familiar with the greatest lower bound axiom of the extended real numbers: and the definition of inf f (x). Rom here we x€x begin by defining the liminf and lirnsup of a function f at the point 3 as: lim inf f (x):= lim [ inf f (x)] x+Z .r/O xéB,(~) , limsup f(x) := lim[ sup f(x)]. X+E 710 zESs(2) Although we shall deal mostly with liminf it will be important in several circumstances to know limsup as well. Most importantly we note that liminf f = - limsup(- f), so most stazements on liminf can be easily in- verted to statement on limsup. It is clear that the liminf of f at 5 is always less than or equal to f (3) since ii e B, (z)for al1 T > O. Thus we are lead to our fkst important definition, lower semi-continuity (or lsc). We call a function lower semi-continuous at Z if lirninf f (x) = f 2+x (z), and Say the function is lower semi-continuous if this holds for all 5 E Zn. Conversely upper semi-continuity (usc) corresponds to limsup f (x) = f (3). x+z It can be shown that a function is continuous if and only if it is both upper and lower semi-continuous ([8] Example, 1.12). Since the study of optirnization generally focuses on the achievement of minimums, lower serni-continuity will play a much larger role than that of upper serni-continuity- Next we call a function, f, proper if f is not constantly infinity and f (x) # -m for all x E Rn. That is inff is a reai nurnber when examined over any compact set. The study of Optimization often focuses on proper lower semi-continuous functions because of the remarkable fact that these functions always obtain their irifimum when examined mer any compact set which intersects their domain (181, Corollary 1.10). Justigng the examination of proper fuhctions is not dSicult, as if the function is not proper one Ends inf f (x) is not a real nurnber whenever I~'P p becomes sufficently large. To justify the focus on lower serni-continuous functions, the concept of lower semi-continuity shall be examined a little more closely. For any function, f , the epi-graph of f is the set of all points lying on or above the graph of f. More rigorously we define the epigraph of f by epi(f) := {(x,a) €Rn x 'il?: : 3 f (x))- The importance of this set is made clear by the fact that f is lower semi- continuous if and only if the epigraph of f is a closed subset of 8" x R ([8], Theorem 1.6). Thus given any proper function one can create it's lower semi-continuous regularization (or closure) by defining f to be the function associated with the closure of the epigraph of f.
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