DOCSLIB.ORG

An Optimization Primer

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Optimization Primer Johannes O. Royset Roger J-B Wets An Optimization Primer April 16, 2021 Springer Nature to our families Contents Preface ................................................... ............. vi Acknowledgements ................................................... .. xi 1 PRELUDE ................................................... ...... 1 1.A The Mathematical Curtain Rise .................................... 1 1.B Data Smoothing ................................................ 5 1.C Optimization under Uncertainty ................................... 7 1.D Convex Analysis ................................................ 10 1.E Estimation and Classification ..................................... 25 1.F Gradient Descent Method ........................................ 28 1.G Newton’s Method ............................................... 34 1.H Acceleration and Regularization ................................... 39 1.I Quasi-Newton Methods .......................................... 45 1.J Coordinate Descent Algorithms ................................... 48 2 CONVEX OPTIMIZATION ......................................... 52 2.A Formulations ................................................... 52 2.B Subderivatives and Subgradients ................................... 60 2.C Subgradient Calculus ............................................ 68 2.D Proximal Gradient Methods ....................................... 75 2.E Linear Constraints ............................................... 81 2.F Karush-Kuhn-Tucker Condition ................................... 86 2.G Interior-Point Method ............................................ 89 2.H Support Vector Machines ......................................... 92 2.I Subgradient Method ............................................. 96 2.J Conic Constraints ............................................... 101 2.K Polyhedral Analysis ............................................. 106 3 OPTIMIZATION UNDER UNCERTAINTY .......................... 116 3.A Product Mix Optimization ........................................ 116 3.B Expectation Functions ........................................... 120 ii Contents iii 3.C Risk Modeling .................................................. 124 3.D Models of Uncertainty ........................................... 133 3.E Risk-Adaptive Design ............................................ 143 3.F Optimality in Stochastic Optimization .............................. 147 3.G Stochastic Gradient Descent ...................................... 153 3.H Simple Recourse Problems ....................................... 160 3.I Control of Water Pollution ........................................ 168 3.J Linear Recourse Problems ........................................ 170 3.K Network Capacity Expansion ..................................... 177 4 MINIMIZATION PROBLEMS ...................................... 179 4.A Formulations ................................................... 179 4.B Network Design and Operation .................................... 189 4.C Epigraphical Approximation Algorithm ............................. 192 4.D Constraint Softening ............................................. 205 4.E Set Analysis ................................................... 207 4.F Robotic Path Planning ........................................... 213 4.G Tangent and Normal Cones I ...................................... 216 4.H Tangent and Normal Cones II ..................................... 229 4.I Subdifferentiability .............................................. 235 4.J Optimality Conditions ........................................... 249 4.K SQP and Interior-Point Methods ................................... 257 5 PERTURBATION AND DUALITY .................................. 261 5.A Rockafellians ................................................... 261 5.B Quantitative Stability ............................................ 272 5.C Lagrangians and Dual Problems ................................... 280 5.D Lagrangian Relaxation ........................................... 291 5.E Saddle Points ................................................... 295 5.F Strong Duality .................................................. 303 5.G Reformulations ................................................. 311 5.H L-Shaped Method ............................................... 316 5.I Monitoring Functions ............................................ 323 5.J Lagrangian Finite-Generation Method .............................. 328 6 WITHOUT CONVEXITY OR SMOOTHNESS ....................... 333 6.A Second-Order Analysis ........................................... 333 6.B Augmented Lagrangians ......................................... 339 6.C Epigraphical Nesting ............................................ 351 6.D Optimality Conditions ........................................... 359 6.E Sup-Projections ................................................. 365 6.F Proximal Composite Method ...................................... 374 6.G Design of Multi-Component Systems ............................... 383 6.H Difference-of-Convex Functions ................................... 389 6.I DC in Regression and Classification ................................ 395 iv Contents 6.J Approximation Errors............................................ 399 7 GENERALIZED EQUATIONS ...................................... 415 7.A Formulations ................................................... 415 7.B Equilibrium in Energy Markets .................................... 423 7.C Traffic Equilibrium .............................................. 429 7.D Reformulation as Minimization Problems ........................... 433 7.E Projection Methods .............................................. 441 7.F Nonsmooth Newton-Raphson Algorithm ............................ 443 7.G Continuity of Set-Valued Mappings ................................ 448 7.H Graphical Approximation Algorithm ............................... 453 7.I Consistent Approximations ....................................... 465 7.J Approximation Errors............................................ 475 8 RISK MODELING AND SAMPLE AVERAGES ....................... 486 8.A Estimation of Optimality Gaps .................................... 486 8.B Risk and Regret ................................................. 491 8.C Risk-Adaptive Data Analytics ..................................... 505 8.D Duality ................................................... ..... 513 8.E Subgradients of Functionals ...................................... 520 8.F Residual Risk and Surrogates ..................................... 528 8.G Sample Average Approximations .................................. 536 8.H Concentration Inequalities ........................................ 549 8.I Diametrical Stochastic Optimization ............................... 556 9 GAMES AND MINSUP PROBLEMS................................. 564 9.A Generalized Nash Games ......................................... 564 9.B Formulation as Minsup Problems .................................. 568 9.C Bifunctions and Solutions ........................................ 572 9.D Lopsided Approximation Algorithm ................................ 580 9.E Lop-Convergence I .............................................. 584 9.F Lop-Convergence II ............................................. 590 9.G Approximation of Games ......................................... 594 9.H Walras Barter Model............................................. 599 10 DECOMPOSITION ................................................ 606 10.AProximal Alternating Gradient Method ............................. 606 10.BLinkage Constraints ............................................. 612 10.CProgressive Decoupling Algorithm ................................. 617 10.DLocal Elicitation ................................................ 631 10.E Decoupling in Stochastic Optimization ............................. 637 10.F Strong Monotonicity ............................................. 647 10.GVariational Convexity and Elicitation ............................... 651 10.HNonlinear Linkage .............................................. 660 References ................................................... .......... 665 Contents v Index ................................................... .............. 669 Preface The concerns with finding minima and maxima were for a long time mainly restricted to mathematical expressions stemming from physical phenomena. The preeminence of being able to do this was underlined by Leonard Euler when he wrote in 1744: “Nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.” The mathematical tools required were supplied by differential calculus as developed by Newton, Leibnitz and many others, not forgetting the Lagrange multiplier strategy for finding the local minima and maxima constrained by smooth equalities. This resulted in a fertile interaction between physics and mathematics, which was instrumental to their shared progress. By the mid-twentieth century, “optimizers” were confronted by a radically new variety of problems coming from organizational and technological issues. Companies wanted to forecast the demand for their products and line up their supply chain accordingly. Airlines had to match crews as well as aircraft-type to specific routes. Engineers needed to plan transportation networks that best serve urban as well as rural areas. Governments aimed to allocate resources efficiently to sectors of the economy. While the decision criteria
Load more

Recommended publications
  • Chapter 9. Coping with NP-Completeness
    Chapter 9 Coping with NP-completeness You are the junior member of a seasoned project team. Your current task is to write code for solving a simple-looking problem involving graphs and numbers. What are you supposed to do? If you are very lucky, your problem will be among the half-dozen problems concerning graphs with weights (shortest path, minimum spanning tree, maximum flow, etc.), that we have solved in this book. Even if this is the case, recognizing such a problem in its natural habitat—grungy and obscured by reality and context—requires practice and skill. It is more likely that you will need to reduce your problem to one of these lucky ones—or to solve it using dynamic programming or linear programming. But chances are that nothing like this will happen. The world of search problems is a bleak landscape. There are a few spots of light—brilliant algorithmic ideas—each illuminating a small area around it (the problems that reduce to it; two of these areas, linear and dynamic programming, are in fact decently large). But the remaining vast expanse is pitch dark: NP- complete. What are you to do? You can start by proving that your problem is actually NP-complete. Often a proof by generalization (recall the discussion on page 270 and Exercise 8.10) is all that you need; and sometimes a simple reduction from 3SAT or ZOE is not too difficult to find. This sounds like a theoretical exercise, but, if carried out successfully, it does bring some tangible rewards: now your status in the team has been elevated, you are no longer the kid who can't do, and you have become the noble knight with the impossible quest.
    [Show full text]
  • CS 561, Lecture 24 Outline All-Pairs Shortest Paths Example
    Outline CS 561, Lecture 24 • All Pairs Shortest Paths Jared Saia • TSP Approximation Algorithm University of New Mexico 1 All-Pairs Shortest Paths Example • For the single-source shortest paths problem, we wanted to find the shortest path from a source vertex s to all the other vertices in the graph • We will now generalize this problem further to that of finding • For any vertex v, we have dist(v, v) = 0 and pred(v, v) = the shortest path from every possible source to every possible NULL destination • If the shortest path from u to v is only one edge long, then • In particular, for every pair of vertices u and v, we need to dist(u, v) = w(u → v) and pred(u, v) = u compute the following information: • If there’s no shortest path from u to v, then dist(u, v) = ∞ – dist(u, v) is the length of the shortest path (if any) from and pred(u, v) = NULL u to v – pred(u, v) is the second-to-last vertex (if any) on the short- est path (if any) from u to v 2 3 APSP Lots of Single Sources • The output of our shortest path algorithm will be a pair of |V | × |V | arrays encoding all |V |2 distances and predecessors. • Many maps contain such a distance matric - to find the • Most obvious solution to APSP is to just run SSSP algorithm distance from (say) Albuquerque to (say) Ruidoso, you look |V | times, once for every possible source vertex in the row labeled “Albuquerque” and the column labeled • Specifically, to fill in the subarray dist(s, ∗), we invoke either “Ruidoso” Dijkstra’s or Bellman-Ford starting at the source vertex s • In this class, we’ll focus
    [Show full text]
  • Approximability Preserving Reduction Giorgio Ausiello, Vangelis Paschos
    Approximability preserving reduction Giorgio Ausiello, Vangelis Paschos To cite this version: Giorgio Ausiello, Vangelis Paschos. Approximability preserving reduction. 2005. hal-00958028 HAL Id: hal-00958028 https://hal.archives-ouvertes.fr/hal-00958028 Preprint submitted on 11 Mar 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 227 Septembre 2005 Approximability preserving reductions Giorgio AUSIELLO, Vangelis Th. PASCHOS Approximability preserving reductions∗ Giorgio Ausiello1 Vangelis Th. Paschos2 1 Dipartimento di Informatica e Sistemistica Università degli Studi di Roma “La Sapienza” [email protected] 2 LAMSADE, CNRS UMR 7024 and Université Paris-Dauphine [email protected] September 15, 2005 Abstract We present in this paper a “tour d’horizon” of the most important approximation-pre- serving reductions that have strongly influenced research about structure in approximability classes. 1 Introduction The technique of transforming a problem into another in such a way that the solution of the latter entails, somehow, the solution of the former, is a classical mathematical technique that has found wide application in computer science since the seminal works of Cook [10] and Karp [19] who introduced particular kinds of transformations (called reductions) with the aim of study- ing the computational complexity of combinatorial decision problems.
    [Show full text]
  • Admm for Multiaffine Constrained Optimization Wenbo Gao†, Donald
    ADMM FOR MULTIAFFINE CONSTRAINED OPTIMIZATION WENBO GAOy, DONALD GOLDFARBy, AND FRANK E. CURTISz Abstract. We propose an expansion of the scope of the alternating direction method of multipliers (ADMM). Specifically, we show that ADMM, when employed to solve problems with multiaffine constraints that satisfy certain easily verifiable assumptions, converges to the set of constrained stationary points if the penalty parameter in the augmented Lagrangian is sufficiently large. When the Kurdyka-Lojasiewicz (K-L)property holds, this is strengthened to convergence to a single con- strained stationary point. Our analysis applies under assumptions that we have endeavored to make as weak as possible. It applies to problems that involve nonconvex and/or nonsmooth objective terms, in addition to the multiaffine constraints that can involve multiple (three or more) blocks of variables. To illustrate the applicability of our results, we describe examples including nonnega- tive matrix factorization, sparse learning, risk parity portfolio selection, nonconvex formulations of convex problems, and neural network training. In each case, our ADMM approach encounters only subproblems that have closed-form solutions. 1. Introduction The alternating direction method of multipliers (ADMM) is an iterative method that was initially proposed for solving linearly-constrained separable optimization problems having the form: ( inf f(x) + g(y) (P 0) x;y Ax + By − b = 0: The augmented Lagrangian L of the problem (P 0), for some penalty parameter ρ > 0, is defined to be ρ L(x; y; w) = f(x) + g(y) + hw; Ax + By − bi + kAx + By − bk2: 2 In iteration k, with the iterate (xk; yk; wk), ADMM takes the following steps: (1) Minimize L(x; yk; wk) with respect to x to obtain xk+1.
    [Show full text]
  • Approximation Algorithms
    Lecture 21 Approximation Algorithms 21.1 Overview Suppose we are given an NP-complete problem to solve. Even though (assuming P = NP) we 6 can’t hope for a polynomial-time algorithm that always gets the best solution, can we develop polynomial-time algorithms that always produce a “pretty good” solution? In this lecture we consider such approximation algorithms, for several important problems. Specific topics in this lecture include: 2-approximation for vertex cover via greedy matchings. • 2-approximation for vertex cover via LP rounding. • Greedy O(log n) approximation for set-cover. • Approximation algorithms for MAX-SAT. • 21.2 Introduction Suppose we are given a problem for which (perhaps because it is NP-complete) we can’t hope for a fast algorithm that always gets the best solution. Can we hope for a fast algorithm that guarantees to get at least a “pretty good” solution? E.g., can we guarantee to find a solution that’s within 10% of optimal? If not that, then how about within a factor of 2 of optimal? Or, anything non-trivial? As seen in the last two lectures, the class of NP-complete problems are all equivalent in the sense that a polynomial-time algorithm to solve any one of them would imply a polynomial-time algorithm to solve all of them (and, moreover, to solve any problem in NP). However, the difficulty of getting a good approximation to these problems varies quite a bit. In this lecture we will examine several important NP-complete problems and look at to what extent we can guarantee to get approximately optimal solutions, and by what algorithms.
    [Show full text]
  • ABOUT STATIONARITY and REGULARITY in VARIATIONAL ANALYSIS 1. Introduction the Paper Investigates Extremality, Stationarity and R
    1 ABOUT STATIONARITY AND REGULARITY IN VARIATIONAL ANALYSIS 2 ALEXANDER Y. KRUGER To Boris Mordukhovich on his 60th birthday Abstract. Stationarity and regularity concepts for the three typical for variational analysis classes of objects { real-valued functions, collections of sets, and multifunctions { are investi- gated. An attempt is maid to present a classification scheme for such concepts and to show that properties introduced for objects from different classes can be treated in a similar way. Furthermore, in many cases the corresponding properties appear to be in a sense equivalent. The properties are defined in terms of certain constants which in the case of regularity proper- ties provide also some quantitative characterizations of these properties. The relations between different constants and properties are discussed. An important feature of the new variational techniques is that they can handle nonsmooth 3 functions, sets and multifunctions equally well Borwein and Zhu [8] 4 1. Introduction 5 The paper investigates extremality, stationarity and regularity properties of real-valued func- 6 tions, collections of sets, and multifunctions attempting at developing a unifying scheme for defining 7 and using such properties. 8 Under different names this type of properties have been explored for centuries. A classical 9 example of a stationarity condition is given by the Fermat theorem on local minima and max- 10 ima of differentiable functions. In a sense, any necessary optimality (extremality) conditions de- 11 fine/characterize certain stationarity (singularity/irregularity) properties. The separation theorem 12 also characterizes a kind of extremal (stationary) behavior of convex sets. 13 Surjectivity of a linear continuous mapping in the Banach open mapping theorem (and its 14 extension to nonlinear mappings known as Lyusternik-Graves theorem) is an example of a regularity 15 condition.
    [Show full text]
  • Branch and Price for Chance Constrained Bin Packing
    Branch and Price for Chance Constrained Bin Packing Zheng Zhang Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai 200240, China, [email protected] Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, [email protected] Brian Denton Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, [email protected] Xiaolan Xie Department of Industrial Engineering and Management, Shanghai Jiao Tong University, Shanghai 200240, China, [email protected] Center for Biomedical and Healthcare Engineering, Ecole Nationale Superi´euredes Mines, Saint Etienne 42023, France, [email protected] This article considers two versions of the stochastic bin packing problem with chance constraints. In the first version, we formulate the problem as a two-stage stochastic integer program that considers item-to- bin allocation decisions in the context of chance constraints on total item size within the bins. Next, we describe a distributionally robust formulation of the problem that assumes the item sizes are ambiguous. We present a branch-and-price approach based on a column generation reformulation that is tailored to these two model formulations. We further enhance this approach using antisymmetry branching and subproblem reformulations of the stochastic integer programming model based on conditional value at risk (CVaR) and probabilistic covers. For the distributionally robust formulation we derive a closed form expression for the chance constraints; furthermore, we show that under mild assumptions the robust model can be reformulated as a mixed integer program with significantly fewer integer variables compared to the stochastic integer program. We implement a series of numerical experiments based on real data in the context of an application to surgery scheduling in hospitals to demonstrate the performance of the methods for practical applications.
    [Show full text]
  • Distributed Optimization Algorithms for Networked Systems
    Distributed Optimization Algorithms for Networked Systems by Nikolaos Chatzipanagiotis Department of Mechanical Engineering & Materials Science Duke University Date: Approved: Michael M. Zavlanos, Supervisor Wilkins Aquino Henri Gavin Alejandro Ribeiro Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mechanical Engineering & Materials Science in the Graduate School of Duke University 2015 Abstract Distributed Optimization Algorithms for Networked Systems by Nikolaos Chatzipanagiotis Department of Mechanical Engineering & Materials Science Duke University Date: Approved: Michael M. Zavlanos, Supervisor Wilkins Aquino Henri Gavin Alejandro Ribeiro An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mechanical Engineering & Materials Science in the Graduate School of Duke University 2015 Copyright c 2015 by Nikolaos Chatzipanagiotis All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract Distributed optimization methods allow us to decompose certain optimization prob- lems into smaller, more manageable subproblems that are solved iteratively and in parallel. For this reason, they are widely used to solve large-scale problems arising in areas as diverse as wireless communications, optimal control, machine learning, artificial intelligence, computational biology, finance and statistics, to name a few. Moreover, distributed algorithms avoid the cost and fragility associated with cen- tralized coordination, and provide better privacy for the autonomous decision mak- ers. These are desirable properties, especially in applications involving networked robotics, communication or sensor networks, and power distribution systems. In this thesis we propose the Accelerated Distributed Augmented Lagrangians (ADAL) algorithm, a novel decomposition method for convex optimization prob- lems.
    [Show full text]
  • Approximation Algorithms for Optimization of Combinatorial Dynamical Systems
    1 Approximation Algorithms for Optimization of Combinatorial Dynamical Systems Insoon Yang, Samuel A. Burden, Ram Rajagopal, S. Shankar Sastry, and Claire J. Tomlin Abstract This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0–1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems. I. INTRODUCTION The dynamics of critical infrastructures and their system elements—for instance, electric grid infrastructure and their electric load elements—are interdependent, meaning that the state of each infrastructure or its system elements influences and is influenced by the state of the others [1].
    [Show full text]
  • Scribe Notes 2 from 2018
    Advanced Algorithms 25 September 2018 Lecture 2: Approximation Algorithms II Lecturer: Mohsen Ghaffari Scribe: Davin Choo 1 Approximation schemes Previously, we described simple greedy algorithms that approximate the optimum for minimum set cover, maximal matching and vertex cover. We now formalize the notion of efficient (1 + )-approximation algorithms for minimization problems, a la [Vaz13]. Let I be an instance from the problem class of interest (e.g. minimum set cover). Denote jIj as the size of the problem (in bits), and jIuj as the size of the problem (in unary). For example, if the input is n just a number x (of at most n bits), then jIj = log2(x) = O(n) while jIuj = O(2 ). This distinction of \size of input" will be important later when we discuss the knapsack problem. Definition 1 (Polynomial time approximation algorithm (PTAS)). For cost metric c, an algorithm A is a PTAS if for each fixed > 0, c(A(I)) ≤ (1 + ) · c(OP T (I)) and A runs in poly(jIj). By definition, the runtime for PTAS may depend arbitrarily on . A stricter related definition is that of fully polynomial time approximation algorithms (FPTAS). Assuming P 6= NP, FPTAS is the best one can hope for on NP-hard optimization problems. Definition 2 (Fully polynomial time approximation algorithm (FPTAS)). For cost metric c, an algo- 1 rithm A is a FPTAS if for each fixed > 0, c(A(I)) ≤ (1 + ) · c(OP T (I)) and A runs in poly(jIj; ). As before, (1−)-approximation, PTAS and FPTAS for maximization problems are defined similarly.
    [Show full text]
  • Monotonic Transformations: Cardinal Versus Ordinal Utility
    Natalia Lazzati Mathematics for Economics (Part I) Note 10: Quasiconcave and Pseudoconcave Functions Note 10 is based on Madden (1986, Ch. 13, 14) and Simon and Blume (1994, Ch. 21). Monotonic transformations: Cardinal Versus Ordinal Utility A utility function could be said to measure the level of satisfaction associated to each commodity bundle. Nowadays, no economist really believes that a real number can be assigned to each commodity bundle which expresses (in utils?) the consumer’slevel of satisfaction with this bundle. Economists believe that consumers have well-behaved preferences over bundles and that, given any two bundles, a consumer can indicate a preference of one over the other or the indi¤erence between the two. Although economists work with utility functions, they are concerned with the level sets of such functions, not with the number that the utility function assigns to any given level set. In consumer theory these level sets are called indi¤erence curves. A property of utility functions is called ordinal if it depends only on the shape and location of a consumer’sindi¤erence curves. It is alternatively called cardinal if it also depends on the actual amount of utility the utility function assigns to each indi¤erence set. In the modern approach, we say that two utility functions are equivalent if they have the same indi¤erence sets, although they may assign di¤erent numbers to each level set. For instance, let 2 2 u : R+ R where u (x) = x1x2 be a utility function, and let v : R+ R be the utility function ! ! v (x) = u (x) + 1: These two utility functions represent the same preferences and are therefore equivalent.
    [Show full text]
  • The Ekeland Variational Principle, the Bishop-Phelps Theorem, and The
    The Ekeland Variational Principle, the Bishop-Phelps Theorem, and the Brøndsted-Rockafellar Theorem Our aim is to prove the Ekeland Variational Principle which is an abstract result that found numerous applications in various fields of Mathematics. As its application to Convex Analysis, we provide a proof of the famous Bishop- Phelps Theorem and some related results. Let us recall that the epigraph and the hypograph of a function f : X ! [−∞; +1] (where X is a set) are the following subsets of X × R: epi f = f(x; t) 2 X × R : t ≥ f(x)g ; hyp f = f(x; t) 2 X × R : t ≤ f(x)g : Notice that, if f > −∞ on X then epi f 6= ; if and only if f is proper, that is, f is finite at at least one point. 0.1. The Ekeland Variational Principle. In what follows, (X; d) is a metric space. Given λ > 0 and (x; t) 2 X × R, we define the set Kλ(x; t) = f(y; s): s ≤ t − λd(x; y)g = f(y; s): λd(x; y) ≤ t − sg ⊂ X × R: Notice that Kλ(x; t) = hyp[t − λ(x; ·)]. Let us state some useful properties of these sets. Lemma 0.1. (a) Kλ(x; t) is closed and contains (x; t). (b) If (¯x; t¯) 2 Kλ(x; t) then Kλ(¯x; t¯) ⊂ Kλ(x; t). (c) If (xn; tn) ! (x; t) and (xn+1; tn+1) 2 Kλ(xn; tn) (n 2 N), then \ Kλ(x; t) = Kλ(xn; tn) : n2N Proof. (a) is quite easy.
    [Show full text]