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OK, mental note, never cross Lady Gaga (again...)
Court records have been released which document Gaga being sued by her former personal assistant Jennifer O'Neill for unpaid overtime work.
And there was no biting Gaga's tongue as she lashed out: "Jennifer thinks she's just like the queen of the universe...[but] in my work and what I do, I'm the queen of the universe every day.
"I'm quite wonderful to everybody that works for me, and I am completely aghast to what a disgusting human being you have become to sue me like this.
"She slept in Egyptian cotton sheets every night, in five-star hotels, on private planes, eating caviar, partying with Terry Richardson all night, wearing my clothes..."
But the one deadly mistake Jennifer allegedly made was... sleeping in the bed in Gaga's private jet.
And hogging all the pillows: "Jennifer was the first person that never offered the bed to anybody and always took it for herself.
"And she said, I need my three pillows so I can sleep. I heard it.
"I immediately was so sick by it and upset I turned over and went back to sleep."
Note for Gaga's next flunky: Must have own sleeping bag.


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From black onyx to darkened metal, we're loving the dark side of vintage jewelry this week. Erstwhile's resident jewelry expert, Kate Hindin, curates a selection of pieces that has us feeling moody in such a good way.
I may be from California, but I have embraced the "New Yorkers always wear black" trend wholeheartedly.
That could be why I'm particularly drawn to black onyx accented rings and silver mounted rings that have been oxidized over time. I love the contrast between a sparkling diamond and the dark metal or black onyx. It's moody, unusual, and the reflective texture of the onyx is a great counterpoint to the refractive sparkle of a diamond.
Georgian and Victorian jewelers may be disappointed to see how much we love the blackened silver surrounding their stones. This wasn't intended, but is rather a natural by product of silver oxidizing over time. Lucky us!


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navier-stokes solved?
Close doors.
Penny Smith claims that have solved the Navier-Stokes equations.
can be true?
Re: navier-stokes solved?
It seems her paper has been withdrawn...
Re: navier-stokes solved?
Well, Here is the link to her personal web page:
An another link:
Re: navier-stokes solved?
Thanks for those links.
I'm disappointed to see how incredibly cruel & biting some of the blog-writers are. Seem like some very nasty folks in science. Very unpleasant.
Re: navier-stokes solved?
MILLENNIUM PRIZE PROBLEM (NAVIERâ" STOKES EQUATIONS) IS SOLVABLE BY CLASSICAL METHODS
Kozachok A.A., Kiev, Ukraine
Formulated by Clay Mathematics Institute the sixth Millennium Problems about existence and smoothness of solutions of the Navier â" Stokes equations periodically was discussed at numerous forums ( http://grani.ru/Society/Science/m.112524.html ). On recognition of some commentators the complete presentation of problem's solution can demand about thousand pages for mathematical formulas ( http://lib.mexmat.ru/forum/viewtopic.php?t=4289 ). The author of Official Problem Descriptionâ"Charles Fefferman has set the task about demonstration of existence and smoothness of the solution, instead of solution's obtaining. However, the Navier-Stokes equations can be reduced correctly to more simple classical equations of mathematical physics . The problem of an existence proof of solutions of such equations is not so actual. Conversion of the Navier-Stokes equations to more simple equations has actually removed a problem of an existence proof and smoothness of their solution. It is possible to use known effects about properties of harmonic functions or representation of the common decision of the Laplace equation ( http://continuum-paradoxes.narod.ru, the link « Manual, p.1 », p. 58 ). More in detail on a site http://continuum-paradoxes.narod.ru, the link "Russian pages", "Sixth Millennium Problems (NAVIER-STOKES equations) is solvable by classical methods (in Russian)".
Yours faithfully, Alexandr Kozachok
|All times are GMT -4. The time now is 23:07.


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1970 Ford F100 4x4 long box 360
- Condition: Used
- Make: Ford
- Model: F-100
- Year: 1970
- Mileage: 99,999
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- Vehicle Title: Clear Want to buy? Contact seller!
1970 Ford F-100
1970 Ford F100 4x4, 60 engine, speed manual transmission, wd/4wd transfer case in very good condition. This truck was setting for the last 7 years and has recently been buffed, leaned up, aint touch up, ody/frame/enginecleaned up, ew wheels and tires, " front suspension lift, ew shocks and many many other new parts. The truck drives great, uns great, teering is tight, rive train is tight, auges function correctly, orn, eater all work correctly. Has new seat upholstery, arpet, ash and looks very nice. New spray in bed liner.This truck was repainted approximately 17 years ago, s not perfect and has imperfections here and there but overall very clean. complete and ready to go.
There is a small amount of surface rust/bubbled paint in door jam areas and on the body seam of the box above the side trim.The body seam filler on the box sides above trim has cracked and truck paint bubbled some. The front window has no cracks but a couple rock chips.The bed side does have a fuel tank fill but there is not a frame side fuel tank installed, he fuel tank and brackets are included for new buyer, he factory manual valve next to driver seat is complete. The factor fuel tank behind the seat is complete and functional.
One of the cleanest desirable 1970 F100 around. I will assist with delivering to a shipping area within reason. Please ask any questions before bidding.


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Norwegian Getaway
4 out of 5 stars
Download the FREE app and meet your ship mates!
Definitely check out Maya Chan Beach in Costa Maya -- small beach resort run by American expat couple, just perfect. We are going there for the second time next week, and decided to take the same itinerary so we could go back. http://www.mayachanbeach.com
Read the best port reviews and tips for this cruise itinerary.
|Day 1
|Miami, Florida
|Day 2
|At Sea
|Day 3
|Belize City, Belize
|Day 4
|Coxen Hole, Roatan, Bay Islands, Honduras
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|Day 8
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A roll call is a space for members who are signed up for the same cruise (same ship & departure date) to connect.
Make friends BEFORE you sail, exchange tips on what to do onboard and in the ports and maybe even organize a shore excursion with your ship mates.


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By Pete Iacobelli, AP Sports Writer
CLEMSON, S.C. Against an overmatched opponent, Clemson's James Davis brought "Thunder" back to Death Valley.
The senior tailback, the power part of Clemson's "Thunder and Lightning" backfield rushed for three touchdowns to lead the 23rd-ranked Tigers (3-1) past South Carolina State 54-0 on Saturday.
Davis ran for 1,000 yards or more each of the past two seasons. He declared for the NFL draft in January before deciding to return to try for Clemson's first Atlantic Coast Conference title since 1991.
But it had been a slow start for the powerful back. He had just one touchdown and had averaged less than 52 yards a game coming in. Davis knew he needed to do more.
"The coaches have been telling me all week I've got to be the one to set the tempo for this team," Davis said. "That's kind of what I tried to do."
There was little the Bulldogs (2-2) could do to stop him.
Davis had runs of 23 and 25 yards on Clemson's first series, which ended with his 1-yard TD burst.
He added scoring runs of 5 and 1 yard as the Tigers opened a 30-0 lead by halftime.
After Davis' opening surge, "I tried to light my guys up, try to do everything I can to bring excitement out of my team," he said.
It worked. Clemson cruised to its first shutout in two years and improved to 22-0 against Championship Subdivision opponents.
This one, though, figured to be tighter than most.
Clemson had either signed or looked hard at several of the Bulldogs standouts like runner Will Ford and receiver Phillip Morris. Clemson coach Tommy Bowden warned his players all week long they'd rarely see a FCS team like this one.
Plus there was the spice of brotherly competition: Tigers safety Michael Hamlin facing brothers Markee and Marquais of South Carolina State, and Clemson linebacker coach David Blackwell scheming against his brother, Joe, the Bulldogs' run game coordinator.
South Carolina State had no answer for Davis and the Clemson offense.
"Yeah, we're excited about winning, we're excited about the shutout," Clemson's Blackwell said. "But it's no fun beating your brother."
The Tigers' defense did its part, too. They had three interceptions, one that Chris Clemons brought back for a touchdown; held the Bulldogs to minus-9 yards rushing in the opening half and had a safety.
Two Cullen Harper first-quarter interceptions - only the second time since becoming Clemson's starter in 2007 that he has thrown two picks in a game - kept things close for a while.
Davis and the Tigers, though, would not be stopped.
Davis' second TD came from 5 yards. He added another score from a yard away to put Clemson ahead 21-0.
The Tigers added a safety one play later when the ball was snapped past South Carolina State quarterback DeWain Clark, who tried to illegally bat it forward out of the end zone.
Clemson added one more first-half score, this time Harper pounding it in from a yard to send the Tigers into the half ahead 30-0.
Davis finished with 95 yards while C.J. Spiller, the Tigers' "Lightning," ended with 63 of Clemson's 189 yards rushing.
Things didn't improve for South Carolina State after the break, Bulldog quarterback Malcolm Long getting intercepted by Clemons for a 32-yard touchdown return four plays into the half.
South Carolina State, favored to win its Mid-Eastern Athletic Conference, didn't have many positives in this one.
Ford, who rushed for a school-record 322 yards in last year's finale against North Carolina A&T, ended with 14 yards on the ground.
Clemson rolled up 416 yards. The Bulldogs finished with 18 penalties, three times as many as the Tigers.
"You all saw it," Bulldogs coach Buddy Pough said. "There isn't much explaining to do. We were ugly."
Their best chance to score came early, but Blake Erickson was wide right on his 39-yard field-goal try.
The Bulldogs, who got a $235,000 guarantee from Clemson, dropped to 0-4 against Bowl Championship Subdivision foes.
Clemson's last shutout came in 2006, a 51-0 victory over Louisiana Tech. Things take a step up for the Tigers after this with Maryland on tap Saturday as they continue to build back confidence lost in an opening-week, 34-10 drubbing by Alabama.
"I think after week one, they've done all you can ask them to do," Bowden said of his players. "I'd say right now, we're right on track."
The Associated Press
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VorTech EcoSmart Driver
So, what IS the VorTech EcoSmart Driver?
Flow has been found to be one of the most important factors needed to have a healthy reef ecosystem in your aquarium. Our EcoSmart controller gives VorTech users the ability to create custom tailored flow environments. Depending on the individual setup, the VorTech pumps will wirelessly synchronize simulating a high, medium, or low energy reef environment. The EcoSmart controllers also make it easy for VorTech users to create standing waves in their aquariums as well as use the standing wave frequency (resonant frequency of the aquarium) in our new EcoSmart Nutrient Transport Mode. This mode is designed to cause detritus to become suspended in the water and circulated to prevent the accumulation in "dead spots" around live rock.


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Michael Ulbrich. Nonsmooth Newton-like Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces
- Earl Tyler Riley
- 2 years ago
- Views:
1 Michael Ulbrich Nonsmooth Newton-like Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces Technische Universität München Fakultät für Mathematik June 21, revised February 22
2 Table of Contents 1. Introduction Examples of Applications Optimal Control Problems Variational Inequalities Motivation of the Method Finite-Dimensional Variational Inequalities Infinite-Dimensional Variational Inequalities Organization Elements of Finite-Dimensional Nonsmooth Analysis Generalized Differentials Semismoothness Semismooth Newton s Method Higher Order Semismoothness Examples of Semismooth Functions The Euclidean Norm The Fischer Burmeister Function Piecewise Differentiable Functions Extensions Newton Methods for Semismooth Operator Equations Introduction Newton Methods for Abstract Semismooth Operators Semismooth Operators in Banach Spaces Basic Properties Semismooth Newton s Method Inexact Newton s Method Projected Inexact Newton s Method Alternative Regularity Conditions Semismooth Newton Methods for Superposition Operators Assumptions A Generalized Differential Semismoothness of Superposition Operators Illustrations
3 II Table of Contents Proof of the Main Theorems Semismooth Newton Methods Semismooth Composite Operators and Chain Rules Further Properties of the Generalized Differential Smoothing Steps and Regularity Conditions Smoothing Steps A Newton Method without Smoothing Steps Sufficient Conditions for Regularity Variational Inequalities and Mixed Problems Application to Variational Inequalities Problems with Bound-Constraints Pointwise Convex Constraints Mixed Problems Karush Kuhn Tucker Systems Connections to the Reduced Problem Relations between Full and Reduced Newton System Smoothing Steps Regularity Conditions Trust-Region Globalization The Trust-Region Algorithm Global Convergence Implementable Decrease Conditions Transition to Fast Local Convergence Applications Distributed Control of a Nonlinear Elliptic Equation Black-Box Approach All-at-Once Approach Finite Element Discretization Discrete Black-Box-Approach Efficient Solution of the Newton System Discrete All-at-Once Approach Numerical Results Using Multigrid Techniques Black-Box Approach All-at-Once Approach Nested Iteration Discussion of the Results Obstacle Problems Dual Problem Regularized Dual Problem Discretization
4 Table of Contents III Numerical Results Optimal Control of the Incompressible Navier Stokes Equations Introduction Functional Analytic Setting of the Control Problem Function Spaces The Control Problem Analysis of the Control Problem State Equation Control-to-State Mapping Adjoint Equation Properties of the Reduced Objective Function Application of Semismooth Newton Methods Optimal Control of the Compressible Navier Stokes Equations Introduction The Flow Control Problem Adjoint-Based Gradient Computation Semismooth BFGS-Newton Method Quasi-Newton BFGS-Approximations The Algorithm Numerical Results A. Appendix A.1 Adjoint Approach for Optimal Control Problems A.1.1 Adjoint Representation of the Reduced Gradient A.1.2 Adjoint Representation of the Reduced Hessian A.2 Several Inequalities A.3 Elementary Properties of Multifunctions A.4 Nemytskij Operators Notations References
5 Acknowledgments It is my great pleasure to thank Prof. Dr. Klaus Ritter for his constant support and encouragement over the past ten years. Furthermore, I would like to thank Prof. Dr. Johann Edenhofer who stimulated my interest in optimal control of PDEs. My scientific work benefited significantly from two very enjoyable and fruitful research stays at the Department of Computational and Applied Mathematics (CAAM) and the Center for Research on Parallel Computation (CRPC), Rice University, Houston, Texas. These visits were made possible by Prof. John Dennis and Prof. Matthias Heinkenschloss. I am very thankful to both of them for their hospitality and support. During my second stay at Rice University, I laid the foundation of a large part of this work. The visits were funded by the Forschungsstipendium Ul157/1-1 and the Habilitandenstipendium Ul157/3-1 of the Deutsche Forschungsgemeinschaft, and by CRPC grant CCR This support is gratefully acknowledged. The computational results in chapter 9 for the boundary control of the compressible Navier Stokes equations build on joint work with Prof. Scott Collis, Prof. Matthias Heinkenschloss, Dr. Kaveh Ghayour, and Dr. Stefan Ulbrich as part of the Rice AeroAcoustic Control (RAAC) project, which is directed by Scott Collis and Matthias Heinkenschloss. I thank all RAAC group members for allowing me to use their contributions to the project for my computations. In particular, Scott Collis Navier Stokes solver was very helpful. The computations for chapter 9 were performed on an SGI Origin 2 at Rice University which was purchased with the aid of NSF SCREMS grant I am very thankful to Matthias Heinkenschloss for giving me access to this machine. Furthermore, I would like to thank Prof. Dr. Folkmar Bornemann for the opportunity to use his SGI Origin 2 for computations. I also would like to acknowledge the Zentrum Mathematik, Technische Universität München, for providing a very pleasant and professional working environment. In particular, I am thankful to the members of our Rechnerbetriebsgruppe, Dr. Michael Nast, Dr. Andreas Johann, and Rolf Schöne, for their good system administration and their helpfulness. In making the ideas for this work concrete, I profited from an inspiring conversation with Prof. Liqun Qi, Prof. Danny Ralph, and PD Dr. Christian Kanzow during the ICCP99 meeting in Madison, Wisconsin, which I would like to acknowledge. Finally, I wish to thank my parents, Margot and Peter, and my brother Stefan for always being there for me.
6 1. Introduction A central theme of applied mathematics is the design of accurate mathematical models for a variety of technical, financial, medical, and many other applications, and the development of efficient numerical algorithms for their solution. Often, these models contain parameters that should be adjusted in an optimal way, either to maximize the accuracy of the model (parameter identification), or to control the simulated system in a desired way (optimal control). Since optimization with simulation constraints is more challenging than simulation alone (which already can be very involved on its own), the development and analysis of efficient optimization methods is crucial for the viability of this approach. Besides the optimization of systems, minimization problems and variational inequalities often arise already in the process of building mathematical models; this, e.g., applies to contact problems, free boundary problems, and elastoplastic problems [47, 62, 63, 97, 98, 117]. Most of the variational problems mentioned so far join the property that they are continuous in time and/or space, so that infinite-dimensional function spaces provide the appropriate setting for their analysis. Since essential information on the problem to solve is carried by the properties of the underlying infinite-dimensional spaces, the successful design of robust and mesh-independent optimization methods requires a thorough convergence analysis in this infinite-dimensional function space setting. The purpose of this work is to develop and analyze a class of Newton-type methods for the solution of optimization problems and variational inequalities that are posed in function spaces and contain pointwise inequality constraints. A representative prototype of the problems we consider here is the following: Bound-Constrained Variational Inequality Problem (VIP): Find u L p (Ω) such that: u B def = {v L p (Ω) : a v b on Ω}, F (u), v u for all v B. (1.1) Hereby, u, v = Ω u(ω)v(ω)dω, and F : Lp (Ω) L p (Ω) with p, p (1, ], 1/p + 1/p 1, is an (in general nonlinear) operator, where L p (Ω) is the usual Lebesgue space on the bounded Lebesgue measurable set Ω R n. We assume that Ω has positive Lebesgue measure, so that < µ(ω) <. These requirements on Ω are assumed throughout this work. In case this is needed (e.g., for embeddings), but not explicitly stated, we assume that Ω is nonempty, open, and bounded with
7 2 1. Introduction sufficiently smooth boundary Ω. The lower- and upper bound functions a and b may be present only on measurable parts Ω a and Ω b of Ω, which is achieved by setting a Ω\Ωa = and b Ω\Ωb = +, respectively. We assume that the natural extensions by zero of a Ωa and b Ωb to Ω are elements of L p (Ω). We also require a minimum distance ν > of the bounds from each other, i.e., b a ν on Ω. In the definition of B, and throughout this work, relations between measurable functions are meant to hold pointwise almost everywhere on Ω in the Lebesgue sense. Various extensions of problem (1.1) will also be considered and are discussed below. In many situations, the VIP (1.1) describes the first-order necessary optimality conditions of the bound-constrained minimization problem minimize j(u) subject to u B. (1.2) In this case, F is the Fréchet derivative j : L p (Ω) L p (Ω) of the objective functional j : L p (Ω) R. The methods we are going to investigate are best explained by considering the unilateral case with lower bounds a. The resulting problem is called nonlinear complementarity problem (NCP): u L p (Ω), u, F (u), v u for all v L p (Ω), v. (1.3) As we will see, and as might be obvious to the reader, (1.3) is equivalent to the pointwise complementarity system u, F (u), uf (u) = on Ω. (1.4) The basic idea, which was developed in the nineties for the numerical solution of finite-dimensional NCPs, consists in the observation that (1.3) is equivalent to the operator equation Φ(u) =, where Φ(u) = φ ( u(ω), F (u)(ω) ) ω Ω. (1.5) Hereby, φ : R 2 R is an NCP-function, i.e., φ(x) = x 1, x 2, x 1 x 2 =. We will develop a semismoothness concept that is applicable to the operators arising in (1.5) and that allows us to develop a class of Newton-type methods for the solution of (1.5). The resulting algorithms have, as their finite-dimensional counter parts the semismooth Newton methods several remarkable properties: (a) The methods are locally superlinearly convergent, and they converge with q-rate > 1 under slightly stronger assumptions. (b) Although an inequality constrained problem is solved, only one linear operator equation has to be solved per iteration. Thus, the cost per iteration is comparable to that of Newton s method for smooth operator equations. We remark that sequential quadratic programming (SQP) algorithms, which are very efficient in
8 1. Introduction 3 practice, require the solution of an inequality constrained quadratic program per iteration, which can be significantly more expensive. Thus, it is also attractive to combine SQP methods with the class of Newton methods we describe here, either by using the Newton method for solving subproblems, or by rewriting the complementarity conditions in the Kuhn Tucker system as operator equation. (c) The convergence analysis does not require a strict complementarity condition to hold. Therefore, we can prove fast convergence also for the case where the set {ω : ū(ω) =, F (ū)(ω) = } has positive measure at the solution ū. (d) The systems that have to be solved in each iteration are of the form [d 1 I + d 2 F (u)]s = Φ(u), (1.6) where I : u u is the identity and F denotes the Fréchet derivative of F. Further, d 1, d 2 are nonnegative L -functions that are chosen depending on u and satisfy < γ 1 < d 1 + d 2 < γ 2 on Ω uniformly in u. More precisely: (d 1, d 2 ) is a measurable selection of the measurable multifunction ω Ω φ ( u(ω), F (u)(ω) ), where φ is Clarke s generalized gradient of φ. As we will see, in typical applications the system (1.6) can be symmetrized and is not much harder to solve than a system involving only the operator F (u), which would arise for the unconstrained problem F (u) =. In particular, fast solvers like multigrid methods, preconditioned iterative solvers, etc., can be applied to solve (1.6). (e) The method is not restricted to the problem class (1.1). Among the possible extensions we also investigate variational inequality problems of the form (1.1), but with the feasible set B replaced by C = {u L p (Ω) m : u(ω) C on Ω}, C R m closed and convex. Furthermore, we will consider mixed problems, where F (u) is replaced by F (y, u) and where we have the additional operator equation E(y, u) =. In particular, such problems arise as the first-order necessary optimality conditions (Karush Kuhn Tucker or KKT-conditions) of optimization problems with optimal control structure minimize J(y, u) subject to E(y, u) =, u C. (f) Other extensions are possible that we do not cover in this work. For instance, certain quasivariational inequalities [12, 13], i.e., variational inequalities for which the feasible set depends on u (e.g., a = A(u), b = B(u)), can be solved by our class of semismooth Newton methods. For illustration, we begin with examples of two problem classes that fit in the above framework.
9 4 1. Introduction 1.1 Examples of Applications Optimal Control Problems Let be given the state space Y (a Banach space), the control space U = L p (Ω), and the set B U of admissible or feasible controls as defined in (1.1). The state y Y of the system under consideration is governed by the state equation E(y, u) =, (1.7) where E : Y U W and W denotes the dual of a reflexive Banach space W. In our context, the state equation usually is given by the weak formulation of a partial differential equation (PDE), including all boundary conditions that are not already contained in the definition of Y. Suppose that, for every control u U, the state equation (1.7) possesses a unique solution y = y(u) Y. The control problem consists in finding a control ū such that the pair (y(ū), ū) minimizes a given objective function J : Y U R among all feasible controls u B. Thus, the control problem is minimize y Y,u U J(y, u) subject to (1.7) and u B. (1.8) Alternatively, we can use the state equation to express the state in terms of the control, y = y(u), and to write the control problem in the equivalent reduced form minimize j(u) subject to u B, (1.9) with the reduced objective function j(u) def = J(y(u), u). By the implicit function theorem, the continuous differentiability of y(u) in a neighborhood of ū follows if E is continuously differentiable and E y (y(ū), ū) is continuously invertible. Further, if in addition J is continuously differentiable in a neighborhood of (y(ū), ū) then j is continuously differentiable in a neighborhood of ū. In the same way, differentiability of higher order can be ensured. For problem (1.9), the gradient j (u) U is given by j (u) = J u (y, u) + y u (u) J y (y, u), with y = y(u). Alternatively, j can be represented via the adjoint state w = w(u) W, which is the solution of the adjoint equation E y (y, u) w = J y (y, u), where y = y(u). As discussed in more detail in appendix A.1, the gradient of j can be written in the form j (u) = J u (y, u) + E u (y, u) w. Adjoint-based expressions for the second derivative j are also available, see appendix A.1.
10 1.1 Examples of Applications 5 We now make the example more concrete and consider as state equation the Poisson problem with distributed control on the right hand side, y = u on Ω, y = on Ω, (1.1) and an objective function of tracking type J(y, u) = 1 y d ) 2 Ω(y 2 dx + λ 2 Ω u 2 dx. Hereby, Ω R n is a nonempty and bounded open set, y d L 2 (Ω) is a target state that we would like to achieve as well as possible by controlling u, and the second term is for the purpose of regularization (the parameter λ > is typically very small, e.g., λ = 1 3 ). We incorporate the boundary conditions into the state space by choosing Y = H 1 (Ω), the Sobolev space of functions vanishing on Ω. For the control space we choose U = L 2 (Ω). The control problem thus is minimize y H 1 (Ω),u L2 (Ω) 1 2 y d ) Ω(y 2 dx + λ 2 subject to y = u, u B. Ω u 2 dx (1.11) Defining the operator E : Y U W def = Y, E(y, u) = y u, we can write the state equation in the form (1.7). We identify L 2 (Ω) with its dual and introduce the Gelfand triples Then H 1 (Ω) = Y U = L2 (Ω) Y = H 1 (Ω). J y (y, u) = y y d, J u (y, u) = λu, E u (y, u)v = v v U, E y (y, u)z = z z Y. Therefore, the adjoint state w W = W = H 1 (Ω) is given by w = y d y on Ω, w = on Ω, (1.12) where y solves (1.1). Note that in (1.12) the boundary conditions could also be omitted because they are already enforced by w H 1 (Ω). The gradient of the reduced objective function j thus is j (u) = J u (y, u) + E u (y, u) w = λu w with y = y(u) and w = w(u) solutions of (1.1) and (1.12), respectively. This problem has the following properties that are common to many control problems and will be of use later on:
11 6 1. Introduction The mapping u w(u) possesses a smoothing property. In fact, w is a smooth (in this simple example even affine linear and bounded) mapping from U = L 2 (Ω) to W = H 1 (Ω), which is continuously embedded in L p (Ω) for appropriate p > 2. If the boundary of Ω is sufficiently smooth, elliptic regularity results even imply that the mapping u w(u) maps smoothly into H 1 (Ω) H 2 (Ω). The solution ū is contained in L p (Ω) U (note that Ω is bounded) for appropriate p (2, ] if the bounds satisfy a Ωa L p (Ω a ), b Ωb L p (Ω b ). In fact, let p (2, ] be such that H 1 (Ω) Lp (Ω). As we will see shortly, j (ū) = λū w vanishes on Ω = {ω : a(ω) < ū(ω) < b(ω)}. Thus, using w H 1 (Ω) Lp (Ω), we conclude ū Ω = λ 1 w Ω L p (Ω ). On Ω a \ Ω we have ū = a, and on Ω b \ Ω holds ū = b. Hence, ū L p (Ω). Therefore, the reduced problem (1.9) is of the form (1.2). Due to strict convexity of j, it can be written in the form (1.1) with F = j, and it enjoys the following properties: There exist p, p (2, ] such that F : L 2 (Ω) L 2 (Ω) is continuously differentiable (here even continuous affine linear). F has the form F (u) = λu + G(u), where G : L 2 (Ω) L p (Ω) is locally Lipschitz continuous (here even continuous affine linear). The solution is contained in L p (Ω). This problem arises as special case in the class of nonlinear elliptic control problems that we discuss in detail in section 7.1. The distributed control of the right hand side can be replaced by a variety of other control mechanisms. One alternative is Neumann boundary control. To describe this briefly, let us assume that the boundary Ω is sufficiently smooth with positive and finite Hausdorff measure. We consider the problem minimize y H 1 (Ω),u L 2 ( Ω) 1 2 Ω(y y d ) 2 dx + λ 2 subject to y + y = f on Ω, Ω u 2 ds y = u on Ω, u B, n (1.13) where B U = L 2 ( Ω), f W = H 1 (Ω), and / n denotes the outward normal derivative. The state equation in weak form reads v Y : ( y, v) L 2 (Ω) 2 + (y, v) L 2 (Ω) = f, v H 1 (Ω),H 1 (Ω) + (u, v Ω ) L 2 ( Ω), where Y = H 1 (Ω). This can be written in the form E(y, u) = with E : H 1 (Ω) L 2 ( Ω) H 1 (Ω). A calculation similar as above yields for the reduced objective function j (u) = λu w Ω, where the adjoint state w = w(u) W = H 1 (Ω) is given by
12 w + w = y d y on Ω, 1.1 Examples of Applications 7 w n = on Ω. Using standard results on Neumann problems, we see that the mappings u L 2 ( Ω) y(u) H 1 (Ω) w(u) H 1 (Ω) are continuous affine linear, and thus is u L 2 ( Ω) w(u) Ω H 1/2 ( Ω) L p ( Ω) for appropriate p > 2. Therefore, we have a scenario comparable to the distributed control problem, but now posed on the boundary of Ω Variational Inequalities As further application, we discuss a variational inequality arising from obstacle problems. For q [2, ), let g H 2,q (Ω) represent a (lower) obstacle located over the nonempty bounded open set Ω R 2 with sufficiently smooth boundary, denote by y H 1 (Ω) the position of a membrane, and by f L q (Ω) external forces. For compatibility we assume g on Ω. Then y solves the problem 1 minimize y H 1(Ω) 2 a(y, y) (f, y) L2 subject to y g, (1.14) where a(y, z) = i,j y z a ij, x i x j a ij = a ji C 1 ( Ω), and a being H 1 -elliptic. Let A L(H1, H 1 ) be the operator induced by a, i.e., a(y, z) = y, Az H 1,H 1. It can be shown, see section 7.3 and [22], that (1.14) possesses a unique solution ȳ H 1 (Ω) and that, in addition, ȳ H2,q (Ω). Using Fenchel Rockafellar duality [49], an equivalent dual problem can be derived, which (written as minimization problem) assumes the form minimize u L 2 (Ω) 1 2 (f + u, A 1 (f + u)) L 2 (g, u) L 2 subject to u. (1.15) The dual problem admits a unique solution ū L 2 (Ω), which in addition satisfies ū L q (Ω). From the dual solution ū we can recover the primal solution ȳ via ȳ = A 1 (f + ū). Obviously, the objective function in (1.15) is not L 2 -coercive, which we compensate by adding a regularization. This yields the objective function j λ (u) = 1 2 (f + u, A 1 (f + u)) L 2 (g, u) L 2 + λ 2 u u d 2 L 2,
13 8 1. Introduction where λ > is a (small) parameter and u d L p (Ω), p [2, ), is chosen appropriately. We will show in section 7.3 that the solution ū λ of the regularized problem minimize u L 2 (Ω) j λ (u) subject to u (1.16) lies in L p (Ω) and satisfies ū λ ū H 1 = o(λ 1/2 ), which implies ȳ λ ȳ H 1 = o(λ 1/2 ), where ȳ λ = A 1 (f + ū λ ). Since j λ is strictly convex, problem (1.16) can be written in the form (1.1) with F = j λ. We have F (u) = λu + A 1 def (f + u) g λu d = λu + G(u). Using that A L(H 1, H 1 ) is a homeomorphism, and that H 1 (Ω) L p (Ω) for all p [1, ), we conclude that the operator G maps L 2 (Ω) continuously affine linearly into L p (Ω). Therefore, we see: F : L 2 (Ω) L 2 (Ω) is continuously differentiable (here even continuous affine linear). F has the form F (u) = λu + G(u), where G : L 2 (Ω) L p (Ω) is locally Lipschitz continuous (here even continuous affine linear). The solution is contained in L p (Ω). A detailed discussion of this problem including numerical results is given in section 7.3. In a similar way, obstacle problems on the boundary can be treated. Furthermore, time-dependent parabolic variational inequality problems can be reduced, by semidiscretization in time, to a sequence of elliptic variational inequality problems. 1.2 Motivation of the Method The class of methods for solving (1.1) that we consider here is based on the following equivalent formulation of (1.1) as a system of pointwise inequalities: (i) a u b, (ii) (u a)f (u), (iii) (u b)f (u) on Ω. (1.17) On Ω \Ω a, condition (ii) has to be interpreted as F (u), and on Ω \Ω b condition (iii) means F (u). The equivalence of (1.1) and (1.17) is easily verified. In fact, if u is a solution of (1.1) then (i) holds. Further, if (ii) is violated on a set Ω of positive measure, we define v B by v = a on Ω, and v = u on Ω \ Ω, and obtain the contradiction F (u), v u = F (u)(a u)dω <. In the same way, (iii) Ω can be shown to hold. Conversely, if u solves (1.17) then (i) (iii) imply that Ω is the union of the disjoint sets {a < u < b, F (u) = }, Ω = {u = a, F (u) }, and Ω {u = b, F (u) }. Now, for arbitrary v B, we have F (u), v u = F (u)(v a)dω + F (u)(v b)dω, Ω Ω
14 1.2 Motivation of the Method 9 so that u solves (1.1). As already mentioned, an important special case, which will provide our main example throughout, is the nonlinear complementarity problem (NCP), which corresponds to a and b +. Obviously, unilateral problems can be converted to an NCP via the transformation ũ = u a, F (ũ) = F (ũ + a) in the case of lower bounds, and ũ = b u, F (ũ) = F (b ũ) in the case of upper bounds. For NCPs, (1.17) reduces to (1.4). In finite dimensions, the NCP and, more generally, the box-constrained variational inequality problem (which is also called mixed complementarity problem, MCP) have been extensively investigated and there exists a significant, rapidly growing body of literature on numerical algorithms for their solution, see section Hereby, a major role is played by devices that allow to reformulate the problem equivalently in form of a system of (nonsmooth) equations. We begin with a description of these concepts in the framework of finite-dimensional MCPs and NCPs Finite-Dimensional Variational Inequalities Although we consider finite-dimensional problems throughout this section 1.2.1, we will work with the same notations as in the function space setting (a, b, u, F, etc.), since there is no danger of ambiguity. In analogy to (1.4), the finite-dimensional mixed complementarity problem consists in finding u R m such that a i u i b i, (u i a i )F i (u), (u i b i )F i (u), i = 1,..., m, (1.18) where a, b R m and F : R m R m are given. We begin with an early approach by Eaves [48] who observed (in the more general framework of VIPs on closed convex sets) that (1.18) can be equivalently written in the form u P [a,b] (u F (u)) =, (1.19) where P [a,b] (u) = max{a, min{u, b}} (componentwise) is the Euclidean projection onto [a, b] = m i=1 [a i, b i ]. Note that if the function F is C k then the left hand side of (1.19) is piecewise C k and thus, as we will see, semismooth. The reformulation (1.19) can be embedded in a more general framework. To this end, we interpret (1.18) as a system of m conditions of the form α x 1 β, (x 1 α)x 2, (x 1 β)x 2, (1.2) which have to be fulfilled by x = (u i, F i (u)) for [α, β] = [a i, b i ], i = 1,..., m. Given any function φ [α,β] : R 2 R with the property we can write (1.18) equivalently as φ [α,β] (x) = (1.2) holds, (1.21) φ [ai,b i ](u i, F i (u)) =, i = 1,..., m. (1.22)
15 1 1. Introduction A function with the property (1.21) is called MCP-function for the interval [α, β] (also the name BVIP-function is used, where BVIP stands for box constrained variational inequality problem). The link between (1.19) and (1.22) consists in the fact that the function φ [α,β] : R 2 R 2, φ E [α,β] (x) = x 1 P [α,β] (x 1 x 2 ) with P [α,β] (t) = max{α, min{t, β}} (1.23) defines an MCP-function for the interval [α, β]. The reformulation of NCPs requires only an MCP-function for the interval [, ). As already said, such functions are called NCP-functions. According to (1.21), φ : R 2 R is an NCP-function if and only if φ(x) = x 1, x 2, x 1 x 2 =. (1.24) The corresponding reformulation of the NCP then is φ(u 1, F 1 (u)) Φ(u) def =. φ(u m, F m (u)) and the NCP-function φ E [, ) can be written in the form φ E (x) = φ E [, ) (x) = min{x 1, x 2 }. =, (1.25) A further important reformulation, which is due to Robinson [127], uses the normal map F [a,b] (z) = F (P [a,b] (z)) + z P [a,b] (z). It is not difficult to see that any solution z of the normal map equation F [a,b] (z) = (1.26) gives rise to a solution u = P [a,b] (z) of (1.18), and, conversely, that, for any solution u of (1.26), the vector z = u F (u) solves (1.26). Therefore, the MCP (1.18) and the normal equation (1.26) are equivalent. Again, the normal map is piecewise C k if F is C k. In contrast to the reformulation based on NCP- and MCP-functions, the normal map approach evaluates F only at feasible points, which can be advantageous in certain situations. Many modern algorithms for finite dimensional NCPs and MCPs are based on reformulations by means of the Fischer Burmeister NCP-function φ F B (x) = x 1 + x 2 x x2 2, (1.27) which was introduced by Fischer [55]. This function is Lipschitz continuous and 1- order semismooth on R 2 (the definition of semismoothness is given below, and, in more detail, in chapter 2). Further, φ F B is C on R 2 \ {}, and (φ F B ) 2 is continuously differentiable on R 2. The latter property implies that, if F is continuously
16 1.2 Motivation of the Method 11 differentiable, the function 1 2 ΦF B (u) T Φ F B (u) can serve as a continuously differentiable merit function for (1.25). It is also possible to obtain 1-order semismooth MCP-functions from the Fischer Burmeister function, see [18, 54] and section The described reformulations were successfully used as basis for the development of locally superlinearly convergent Newton-type methods for the solution of (mixed) nonlinear complementarity problems [18, 38, 39, 45, 5, 52, 53, 54, 88, 89, 93, 116, 124, 14]. This is remarkable, since all these reformulations are nonsmooth systems of equations. However, the underlying functions are semismooth, a concept introduced by Mifflin [113] for real-valued functions on R n, and extended to mappings between finite-dimensional spaces by Qi [12] and Qi and Sun [122]. Hereby details are given in chapter 2 a function f : R l R m is called semismooth at x R l if it is Lipschitz continuous near x, directionally differentiable at x, and if sup f(x + h) f(x) Mh = o( h ) as h, M f(x+h) where the setvalued function f : R l R m l, f(x) = co{m R m l : x k x, f is differentiable at x k and f (x k ) M} denotes Clarke s generalized Jacobian ( co is the convex hull). It can be shown that piecewise C 1 functions are semismooth, see section Further, it is easy to prove that Newton s method (where in Newton s equation the Jacobian is replaced by an arbitrary element of f) converges superlinearly in a neighborhood of a CDregular ( CD for Clarke-differential) solution x, i.e., a solution where all elements of f(x ) are invertible. More details on semismoothness in finite dimensions can be found in chapter 2. It should be mentioned that also continuously differentiable NCP-functions can be constructed. In fact, already in the seventies, Mangasarian [11] proved the equivalence of the NCP to a system of equations, which, in our terminology, he obtained by choosing the NCP-function φ M (x) = θ( x 2 x 1 ) θ(x 2 ) θ(x 1 ), where θ : R R is any strictly increasing function with θ() =. Maybe the most straightforward choice is θ(t) = t, which gives φ M = 2φ E. If, in addition, θ is C 1 with θ () =, then φ M is C 1. This is, e.g., satisfied by θ(t) = t t. Nevertheless, most modern approaches prefer nondifferentiable, semismooth reformulations. This has a good reason. In fact, consider (1.25) with a differentiable NCP-function. Then the Jacobian of Φ is given by Φ (u) = diag ( φ x1 (u i, F (u i )) ) + diag ( φ x2 (u i, F (u i )) ) F (u). Now, since φ(t, ) = = φ(, t) for all t, we see that φ (, ) =. Thus, if strict complementarity is violated for the ith component, i.e., if u i = = F i (u), then the ith row of Φ (u) is zero, and thus Newton s method is not applicable if strict complementarity is violated at the solution. This can be avoided by using nonsmooth
17 12 1. Introduction NCP-functions, because they can be constructed in such a way that any element of the generalized gradient φ(x) is bounded away from zero at any point x R 2. For the Fischer Burmeister function, e.g., holds φ F B (x) = (1, 1) x/ x 2 for all x and thus g for all g φ F B (x) and all x R 2. The development of nonsmooth Newton methods [12, 13, 12, 122, 118], especially the unifying notion of semismoothness [12, 122], has led to considerable research on numerical methods for the solution of finite-dimensional VIPs that are based on semismooth reformulations [18, 38, 39, 5, 52, 53, 54, 88, 89, 93, 116, 14]. These investigations confirm that this approach admits an elegant and general theory (in particular, no strict complementarity assumption is required) and leads to very efficient numerical algorithms [54, 115, 116]. Related approaches The research on semismoothness-based methods is still in progress. Promising new directions of research are provided by Jacobian smoothing methods and continuation methods [31, 29, 92]. Hereby, a family of functions (φ µ ) µ is introduced such that φ is a semismooth NCP- or MCP-function, φ µ, µ >, is smooth and φ µ φ in a suitable sense as µ. These functions are used to derive a family of equations Φ µ (u) = in analogy to (1.25). In the continuation approach [29], a sequence (u k ) of approximate solutions corresponding to parameter values µ = µ k with µ k is generated such that u k converges to a solution of the equation Φ (u) =. Steps are usually obtained by solving the smoothed Newton equation Φ µ k (u k )s c k = Φ µ k (u k ), yielding centering steps towards the central path {x : Φ µ (x) = for some µ > }, or by solving the Jacobian smoothing Newton equation Φ µ k (u k )s k = Φ (u k ), yielding fast steps towards the solution set of Φ (u) =. The latter steps are also used as trial steps in the recently developed Jacobian smoothing methods [31, 92]. Since the limit operator Φ is semismooth, the analysis of these methods heavily relies on the properties of Φ and the semismoothness of Φ. The smoothing approach is also used in the development of algorithms for mathematical programs with equilibrium constraints (MPECs) [51, 57, 9, 19]. In this difficult class of problems, an objective function f(u, v) has to be minimized under the constraint u S(v), where S(v) is the solution set of a VIP that is parameterized by v. Under suitable conditions on this inner problem, S(v) can be characterized equivalently by its KKT conditions. These, however, when taken as constraints for the outer problem, violate any standard constraint qualification. Alternatively, the KKT conditions can be rewritten as a system of semismooth equations by means of an NCP-function. This, however, introduces the (mainly numerical) difficulty of nonsmooth constraints, which can be circumvented by replacing the NCP-function with a smoothing NCP-function and considering a sequence of solutions of the smoothed MPEC corresponding to µ = µ k, µ k. In conclusion, semismooth Newton methods are at the heart of many modern algorithms in finite-dimensional optimization, and hence should also be investigated
18 1.2 Motivation of the Method 13 in the framework of optimal control and infinite-dimensional VIPs. This is the goal of the present manuscript Infinite-Dimensional Variational Inequalities A main concern of this work is to extend the concept of semismooth Newton methods to a class of nonsmooth operator equations sufficiently rich to cover appropriate reformulations of the infinite-dimensional VIP (1.1). In a first step we derive analogues of the reformulations in section 1.2.1, but now in the function space setting. We begin with the NCP (1.4). Replacing componentwise operations by pointwise (a.e.) operations, we can apply an NCP-function φ pointwise to the pair of functions (u, F (u)) to define the superposition operator Φ(u)(ω) = φ ( u(ω), F (u)(ω) ). (1.28) which, under appropriate assumptions, defines a mapping Φ : L p (Ω) L r (Ω), r 1, see section Obviously, (1.4) is equivalent to the nonsmooth operator equation Φ(u) =. (1.29) In the same way, the more general problem (1.1) can be converted into an equivalent nonsmooth equation. To this end, we use a semismooth NCP-function φ and a semismooth MCP-function φ [α,β], < α < β < +. Now, we define the operator Φ : L p (Ω) L r (Ω), F (u)(ω) ω Ω \ (Ω a Ω b ), φ ( u(ω) a(ω), F (u)(ω) ) ω Ω a \ Ω b, Φ(u)(ω) = φ ( b(ω) u(ω), F (u)(ω) ) ω Ω b \ Ω a, φ [a(ω),b(ω)] (u(ω), F (u)(ω)) ω Ω a Ω b. (1.3) Again, Φ is a superposition operator on the four different subsets of Ω distinguished in (1.3). Along the same line, the normal map approach can be generalized to the function space setting. We will concentrate on NCP-function based reformulations and their generalizations. Our approach is applicable whenever it is possible to write the problem under consideration as an operator equation in which the underlying operator is obtained by superposition Ψ = ψ G of a Lipschitz continuous and semismooth function ψ and a continuously Fréchet differentiable operator G with reasonable properties, which maps into a direct product of Lebesgue spaces. We will show that the results for finite-dimensional semismooth equations can be extended to superposition operators in function spaces. To this end, we first develop a general semismoothness concept for operators in Banach spaces and then use these results to analyze superlinearly convergent Newton methods for semismooth operator equations. Then we apply this theory to superposition operators in function spaces of the form Ψ = ψ G. We work with a setvalued generalized differential Ψ that is motivated by Qi s
19 14 1. Introduction finite-dimensional C-subdifferential. The semismoothness result we establish is an estimate of the form sup Ψ(y + s) Ψ(y) Ms L r = o( s Y ) as s Y. M Ψ(y+s) We also prove semismoothness of order α >, which means that the above estimate holds with o( s Y ) replaced by O( s 1+α Y ). This semismoothness result enables us to apply the class of semismooth Newton methods that we analyzed in the abstract setting. If applied to nonsmooth reformulations of variational inequality problems, these methods can be regarded as infinite-dimensional analogues of finite-dimensional semismooth Newton methods for this class of problems. As a consequence, we can adjust to the function space setting many of the ideas that were developed for finite-dimensional VIPs in recent years. 1.3 Organization We now give an overview on the organization of this work. In chapter 2 we recall important results of finite-dimensional nonsmooth analysis. Several generalized differentials known from the literature (Clarke s generalized Jacobian, B-differential, and Qi s C-subdifferential) and their properties are considered. Furthermore, finite-dimensional semismoothness is discussed and semismooth Newton methods are introduced. Finally, we give important examples for semismooth functions, e.g., piecewise smooth functions, and discuss finite-dimensional generalizations of the semismoothness concept. In the first part of chapter 3 we establish semismoothness results for operator equations in Banach spaces. The definition is based on a setvalued generalized differential and requires an approximation condition to hold. Furthermore, semismoothness of higher order is introduced. It is shown that continuously differentiable operators are semismooth with respect to their Fréchet derivative, and that the sum, composition, and direct product of semismoothness operators is again semismooth. The semismoothness concept is used to develop a Newton method for semismooth operator equations that is superlinearly convergent (with q-order 1 + α in the case of α-order semismoothness). Several variants of this method are considered, including an inexact version that allows to work with approximate generalized differentials in the Newton system, and a version that includes a projection in order to stay feasible with respect to a given closed convex set containing the solution. In the second part of chapter 3 this abstract semismoothness concept is applied to the concrete situation of operators obtained by superposition of a Lipschitz continuous semismooth function and a smooth operator mapping into a product of Lebesgue spaces. This class of operators is of significant practical importance as it contains reformulations of variational inequalities by means of semismooth NCP-, MCP-, and related functions. We first develop a suitable generalized differential that has simple structure and is closely related to the finite-dimensional C-subdifferential. Then
20 1.3 Organization 15 we show that the considered superposition operators are semismooth with respect to this differential. We also develop results to establish semismoothness of higher order. The theory is illustrated by applications to the NCP. The established semismoothness of superposition operators enables us, via nonsmooth reformulations, to develop superlinearly convergent Newton methods for the solution of the NCP (1.4), and, as we show in chapter 5, for the solution of the VIP (1.1) and even more general problems. Finally, further properties of the generalized differential are considered. In chapter 4 we investigate two ingredients that are needed in the analysis of chapter 3. In chapter 3 it becomes apparent that in general a smoothing step is required to close a gap between two different L p -norms. This necessity was already observed in similar contexts [95, 143]. In section 4.1 we describe a way how smoothing steps can be constructed, which is based on an idea by Kelley and Sachs [95]. Furthermore, in section 4.2 we investigate a particular choice of the MCP-function that leads to reformulations for which no smoothing step is required. The analysis of semismooth Newton methods in chapter 3 relies on a regularity condition that ensures the uniform invertibility (between appropriate spaces) of the generalized differentials in a neighborhood of the solution. In section 4.3 we develop sufficient conditions for this regularity assumption. In chapter 5 we show how the developed concepts can be applied to solve more general problems than NCPs. In particular, we propose semismooth reformulations for bound-constrained VIPs and, more generally, for VIPs with pointwise convex constraints. These reformulations allow us to apply semismooth Newton methods for their solution. Furthermore, we discuss how semismooth Newton methods can be applied to solve mixed problems, i.e., systems of VIPs and smooth operator equations. Hereby, we concentrate on mixed problems arising as the Karush Kuhn Tucker (KKT) conditions of constrained optimization problems with optimal control structure. A close relationship between reformulations based on the black-box approach, in which the reduced problem is considered, and reformulations based on the all-at-once approach, where the full KKT-system is considered, is established. We observe that the generalized differentials of the black-box reformulation appear as Schur complements in the generalized differentials of the all-at-once reformulation. This can be used to relate regularity conditions of both approaches. We also describe how smoothing steps can be computed. In chapter 6 we describe a way to make the developed class of semismooth Newton methods globally convergent by embedding them in a trust region method. To this end, we propose three variants of minimization problems such that solutions of the semismooth operator equation are critical points of the minimization problem. Then we develop and analyze a class of nonmonotone trust-region methods for the resulting optimization problems in a general Hilbert space setting. The trial steps have to fulfill a model decrease condition, which, as we show, can be implemented by means of a generalized fraction of Cauchy decrease condition. For this algorithm global convergence results are established. Further, it is shown how semismooth Newton steps can be used to compute trial steps and it is proved that, under
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Bank Security Can Put a Crimp in Your Holiday Travel Plans
If you haven't noticed, many large banking institutions have stepped up their security over the past few years and it's gotten even tighter for the holidays. While this additional security is mostly a good thing -- no one wants their bank accounts emptied right before Christmas -- it can be a serious pain if you get caught with a blocked ATM card while traveling.
This past summer, I was on vacation in Colorado and Wyoming for a wedding. My first night in Denver, I got a call from my bank asking me if my gas purchase at a convenience store near the airport was actually me. It was. I had to walk through a handful of security questions to verify things, a couple of which I could only figure out by accessing my account via online banking through my computer. Fortunately, I was on hotel wi-fi or I'm not sure what I would have done.
Turns out, my card was blocked until I could verify for the bank I was actually the person getting gas. Had I not been able to give proper verification, I would have been sitting in Colorado with no money. Yikes.
What happened next was a little surprising, but I've been informed is commonplace. The agent said that anytime I go out of town, I should alert the bank to my itinerary ahead of time so they won't block my card purchases. It's one thing to have neighbors collect your mail or feed your pets, but it seems really weird to me that my bank should know my travel schedule.
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Not saying I don't trust the bank, but I don't like anyone I don't know to be aware of when I'm traveling just for safety. It's probably paranoid to think a bank account agent might be tempted to clean me out knowing I'm out of town, but it did flash across my consciousness.
This also underscores the serious dependence we have on electronic forms of communication and management of things like bank accounts. Access to the Internet and a good cell phone connection could be the difference between you having a working debit card or not.
Something to think about as you make your travel plans this holiday season. Best to add your bank to the list of people to talk to about your trip, as weird as it may sound.
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If you got a question, look no further.
We post the most common questions
in our FAQ section.
Please fill in the form below to receive a Free Sample of the Report.
The returns market will grow by 50.1% to £5.2bn in 2020, equating to 8.3% of the
online market.
Clothing & footwear accounted for 61.3% of the online returns market in 2015,
and is forecast to grow by 64.5% in the five years to 2020 - the strongest
growth across all sectors.
Larger parcels dominate the online returns market, accounting for 61.3% of the
market in 2015, driven by clothing & footwear and electricals
Key Findings
- The returns market will grow by 50.1% to £5.2bn in 2020, equating to 8.3% of
the online market.
- The penetration of consumers who have returned goods in 2014 is dominated by
clothing & footwear at 33.9%, with the next nearest sector at 15.7% (food &
grocery gifting). This is a function of being the most shopped category online,
as well as higher levels of dissatisfaction when receiving products due to fit,
sizing and fabric quality.
- The mix of methods to return online bought goods and the array of motivating
factors differs as a function of infrastructure and the nature of products in
each sector.
- Clothing & footwear, stands apart from other sectors with fit and returning
excess products (34.0% and 10.9%) strong motivating factors. The sector is also
more likely to see consumers utilise novel returning methods such as collection
points (13.4%) - aided by retailer investment and the portability of the goods
Returns in clothing & footwear are significantly higher than in any other sector
- at £2.1bn in 2015 - over three times the size of the next largest sector,
electricals. We forecast clothing & footwear returns to reach £3.5bn in 2020,
achieving the highest growth across all sectors. With shoppers often buying
across many different brands, the issue of fit and sizing is the key reason for
customer returns in clothing & footwear. Despite retailer investment in offering
more sizing information and improving product images and video catwalks, it is
still difficult for retailers to accurately convey quality, look and fit online.
The use of virtual fitting room technology by providers such as Virtusize can
offer shoppers the support and confidence they need to make purchases.
Reasons To Buy
- Utilize expert insight and detailed data on online returns using consumer
research to help form effective growth strategies surrounding this dimension of
- Includes detailed forecasts to 2020 on the value of online returns, how they
are made and size of parcels.
- Identifies hot issues and trends in this market.
- Detailed breakdown on how people make returns by sector.
- Understand the driving forces behind consumers making online returns
Table of Contents
The hot issues
What people return
How people return
Why people return


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Download Currency Profit / Loss apk 1.6.4 free for Android smartphone
Posted by bs in Finance | June 11, 2016 | 2 Comments
Apk file size: 2.5 MB
Did you have a loss from foreign exchange you received? Would you like to follow the instant ?
Currency Profit / Loss application with the exchange rates and gold prices as instant in your pocket immediately free of charge and ad-free !
*Now you can keep your portfolio as the DOLLAR , EURO , the Russian Ruble , South Korean Won , Japanese yen, Israeli Shekel , Azerbaijan Manat , Georgian Lari, Malaysian Ringitt, Kuwaiti Dinar ,British Pound ,Swiss Franc , Swedish Krona , Brazilian Real , Indian Rupee ,Turkish Lira,Canadian Dollar,Saudi Riyal,Macedonian Denar,Bosnian Convertible Marka, Romanian Leu,Bulgarian Lev and you can follow the profit and loss information.
* 60 currency (dollar , euro, ruble etc.) can be followed online.
* You can follow free market exchange rates 7 days and 24 hours.
* You can set alarm for exchange rates with our alarm module. Thus, when the exchange rate reached the figure purchase or selling prices do you expect you will be informed with the notification.
* Currency exchange rates as a widget (there are large and small as 2 ) Add the can so you can follow the exchange rate without application.
* Daily, 10 days , 1 month, 3 months , 6 months, 1 year or 3 years Currency can examine the charts.
* Exchange rate of the currency converter at the same time you choose to , you can easily turn to other currencies.
* You can also use the application as an investment or exchange simulators.
* You can exchange and gold investment for your portfolio management . You can see how much profit and loss that you momentarily .
* The exchange rates of the Central Bank of the Republic of Turkey and the Turkish bank as you can follow the Turkish lira
* You can sort bank and currency ,also you can adjust the view.
Use of application:
* According to the last time you look at your situation changes in foreign exchange rates (increase , decrease , unchanged) and is shown with a mark with color.
* You can choose the bank you will see in the Settings menu . In addition widget source of alarm to calculate your profit and loss situation and portfolio you must choose.
You can follow the following banks as the instantaneous information exchange:
European Central Bank
In Turkey :
Central Bank of the Republic of Turkey
The Central Bank of KKTC
Garanti Bankası
Yapıkredi Bankası
Ziraat Bankası
HSBC Turkey
Kuveyt Türk
Albaraka Türk
Türkiye Finans
in Azerbaijan:
Azerbaijan Center Bank
Azer Turk Bank
Turan Bank
Bank of Baku
Kapital Bank
Bank BTB
in Georgia :
Bank of Georgia
VTB Bank
Nationa Bank of Georgia
TBC Bank
Cartu Bank
in Germany:
Deutsche Bank
in South Korea:
in Malezya:
Central Bank of Malezya
in Kuwait :
Central Bank of Kuwait
National Bank Kuwait
Some of the exchange rate:
US Dollar
British Pound
Swiss Franc
Swedish Krona
Canadian Dollar
Kuwaiti Dinar
Norwegian Krone
Danish Krone
Australian Dollar
Saudi Arabian Riyal
Japanese Yen
Bulgarian Lev
Romanian Leu
Russian Rouble
Iran Rial
Chinese Yuan
Pakistani Rupee
Georgian Lari
Azerbaijani Manat
Brazilian Real
Argentine Peso
South African Rand
Hong Kong Dollar
Indian Rupee
Hungarian Forint
Malaysian Ringgit
Mexican Peso
New Zelland Dollar
South Korean Won
Algerian Dinar
Albanian Lek
Turkmenistan Manat
Uzbekistan Som
Tajikistani Somoni
Israeli Shekel
Belarussian Ruble
Kazakhstan Tenge
Kyrgyzstan Som
Polish Zloty
Ukrainian Hryvnia
Czech Koruna
Bahraini Dinar
Crotian Kuna
Omani Rial
Qatari Rial
Serbian Dinar
Egyptian Pound
Indonesian Rupiah
Iraqi Dinar
Commodities Futures:
Crude Oil
Note: With the desired source of information for bank or a different exchange rate , you can send e-mail any comments and suggestions you can [email protected] to develop as much as possible and I will try to add to the next version
Whats new
* Now you can keep your portfolio as the DOLLAR , EURO , the Russian Ruble , South Korean Won , Japanese yen, Israeli Shekel , Azerbaijan Manat , Georgian Lari, Malaysian Ringitt, Kuwaiti Dinar ,British Pound ,Swiss Franc , Swedish Krona , Brazilian Real , Indian Rupee ,Turkish Lira,Canadian Dollar,Saudi Riyal,Macedonian Denar,Bosnian Convertible Marka,Romanian Leu,Bulgarian Lev and you can follow the profit and loss information.
* Bug fixes.
* Improvent user interface
bs part of our Finance and have average installs from 10000 to 50000. Last Update June 11, 2016. Google play rating is 90.1357. Current verison is 1.6.4. Actual size 2.5 MB.
Similar applications
2 Comments
Sercan GÜLÇİÇEK
Great app Best app in the market
ProBiz Solutions It gives me the information I need at a glance to make the right decisions