DOCSLIB.ORG

1.3 Hilbert Space Methods 25 1.3 Hilbert Space Methods

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1.3 Hilbert Space Methods 25 1.3 Hilbert Space Methods 1.3 Hilbert space methods 25 1.3 Hilbert space methods One of the most often used theorems of functional analysis is the Riesz rep- resentation theorem. Theorem 1.30 (Riesz representation theorem). If x∗ is a continuous linear functional on a Hilbert space H, then there is exactly one element y 2 H such that < x∗; x >= (x; y): (1.32) 1.3.1 To identify or not to identify{the Gelfand triple Riesz theorem shows that there is a canonical isometry between a Hilbert space H and its dual H∗. It is therefore natural to identify H and H∗ and is done so in most applications. There are, however, situations when it cannot be done. Assume that H is a Hilbert space equipped with a scalar product (·; ·)H and that V ⊂ H is a subspace of H which is a Hilbert space in its own right, endowed with a scalar product (·; ·)V . Assume that V is densely and continuously embedded in H that is V = H and kxkH ≤ ckxkV , x 2 V , for some constant c. There is a canonical map T : H∗ ! V ∗ which is given by restriction to V of any h∗ 2 H∗: ∗ ∗ < T h ; v >V ∗×V =< h ; v >H∗×H ; v 2 V: We easily see that ∗ ∗ kT h kV ∗ ≤ Ckh kH∗ : Indeed ∗ ∗ ∗ kT h kV ∗ = sup j < T h ; v >V ∗×V j = sup j < h ; v >H∗×H j kvkV ≤1 kvkV ≤1 ∗ ∗ ≤ kh kH∗ sup kvkH ≤ ckh kH∗ : kvkV ≤1 ∗ ∗ Further, T is injective. For, if T h1 = T h2, then ∗ ∗ ∗ ∗ 0 =< T h1 − T h2; v >V ∗×V =< h1 − h2; v >H∗×H for all v 2 V and the statement follows from density of V in H. Finally, the image of TH∗ is dense in V ∗. Indeed, let v 2 V ∗∗ be such that < v; T h∗ >= 0 for all h∗ 2 H∗. Then, by reflexivity, ∗ ∗ ∗ ∗ ∗ 0 =< v; T h >V ∗∗×V ∗ =< T h ; v >V ∗×V =< h ; v >H∗×H ; h 2 H implies v = 0. Now, if we identify H∗ with H by the Riesz theorem and using T as the canonical embedding from H∗ into V ∗, one writes 26 1 Basic Facts from Functional Analysis and Banach Lattices V ⊂ H ' H∗ ⊂ V ∗ and the injections are dense and continuous. In such a case we say that H is the pivot space. Note that the scalar product in H coincides with the duality pairing < ·; · >V ∗×V : (f; g)H =< f; g >V ∗×V ; f 2 H; g 2 V: Remembering now that V is a Hilbert space with scalar product (·; ·)V we see that identifying also V with V ∗ would lead to an absurd { we would have V = H = H∗ = V ∗. Thus, we cannot identify simultaneously both pairs. In such situations it is common to identify the pivot space H with its dual H∗ bur to leave V and V ∗ as separate spaces with duality pairing being an extension of the scalar product in H. An instructive example is H = L2([0; 1]; dx) (real) with scalar product 1 Z (u; v) = u(x)v(x)dx 0 and V = L2([0; 1]; wdx) with scalar product 1 Z (u; v) = u(x)v(x)w(x)dx; 0 where w is a nonnegative bounded measurable function. Then it is useful to ∗ −1 identify V = L2([0; 1]; w dx) and 1 1 Z Z p g(x) < f; g >V ∗×V = f(x)g(x)dx ≤ f(x) w(x) dx ≤ kfkV kgkV ∗ : pw(x) 0 0 1.3.2 The Radon-Nikodym theorem Let µ and ν be finite nonnegative measures on the same σ-algebra in Ω. We say that ν is absolutely continuous with respect to µ if every set that has µ-measure 0 also has ν measure 0. Theorem 1.31. If ν is absolutely continuous with respect to µ then there is an integrable function g such that Z ν(E) = gdµ, (1.33) E for any µ-measurable set E ⊂ Ω. 1.3 Hilbert space methods 27 Proof. Assume for simplicity that µ(Ω); ν(Ω) < 1. Let H = L2(Ω; dµ + dν) on the field of reals. Schwarz inequality shows that if f 2 H, then f 2 L1(dµ+ dν). The linear functional Z < x∗; f >= fdµ Ω To be completed 1.3.3 Projection on a convex set Corollary 1.32. Let K be a closed convex subset of a Hilbert space H. For any x 2 H there is a unique y 2 K such that kx − yk = inf kx − zk: (1.34) z2K Moreover, y 2 K is a unique solution to the variational inequality (x − y; z − y) ≤ 0 (1.35) for any z 2 K. Proof. Let d = inf kx − zk. We can assume z2 = K and so d > 0. Consider z2K f(z) = kx − zk, z 2 K and consider a minimizing sequence (zn)n2N, zn 2 K such that d ≤ f(zn) ≤ d + 1=n. By the definition of f,(zn)n2N is bounded and thus it contains a weakly convergent subsequence, say (ζn)n2N. Since K is closed and convex, by Corollary 1.24, ζn * y 2 K. Further we have 1 j(h; x−y)j = lim j(h; x−ζn)j ≤ khk lim inf kx−ζnk ≤ khk lim inf d+ = khkd n!1 n!1 n!1 n for any h 2 H and thus, taking supremum over khk ≤ 1, we get f(y) ≤ d which gives existence of a minimizer. To prove equivalence of (1.35) and (1.34) assume first that y 2 K satisfies (1.34) and let z 2 K. Then, from convexity, v = (1 − t)y + tz 2 K for t 2 [0; 1] and thus kx − yk ≤ kx − ((1 − t)y + tz)k = k(x − y) − t(z − y)k and thus kx − yk2 ≤ kx − yk2 − 2t(x − y; z − y) + t2kz − yk2: Hence tkz − yk2 ≥ 2(x − y; z − y) for any t 2 (0; 1] and thus, passing with t ! 0, (x − y; z − y) ≤ 0. Conversely, assume (1.35) is satisfied and consider 28 1 Basic Facts from Functional Analysis and Banach Lattices kx − yk2 − kx − zk2 = (x − y; x − y) − (x − z; x − z) = 2(x; z) − 2(x; y) + 2(y; y) − 2(y; z) + 2(y; z) − (y; y) = 2(x − y; z − y) − (y − z; y − z) ≤ 0 hence kx − yk ≤ kx − zk for any z 2 K. For uniqueness, let y1; y2 satisfy (x − y1; z − y1) ≤ 0; (x − y2; z − y2) ≤ 0; z 2 H: Choosing z = y2 in the first inequality and z = y1 in the second and adding 2 them, we get jy1 − y2j ≤ 0 which implies y1 = y2. We call the operator assigning to any x 2 K the element y 2 K satisfying (1.34) the projection onto K and denote it by Pk. Proposition 1.33. Let K be a nonempty closed and convex set. Then PK is non expansive mapping. Proof. Let yi = PK xi, i = 1; 2: We have (x1 − y1; z − y1) ≤ 0; (x2 − y2; z − y2) ≤ 0; z 2 H so choosing, as before, z = y2 in the first and z = y1 in the second inequality and adding them together we obtain 2 ky1 − y2k ≤ (x1 − x2; y1 − y2); hence kPK x1 − PK x2k ≤ kx1 − x2k. 1.3.4 Theorems of Stampacchia and Lax-Milgram 1.3.5 Motivation Consider the Dirichlet problem for the Laplace equation in Ω ⊂ Rn −∆u = f in Ω; (1.36) uj@Ω = 0: (1.37) Assume that there is a solution u 2 C2(Ω) \ C(Ω¯). If we multiply (1.37) by a 1 test function φ 2 C0 (Ω) and integrate by parts, then we obtain the problem Z Z ru · rφdx = fφdx. (1.38) Ω Ω 1.3 Hilbert space methods 29 Conversely, if u satisfies (1.38), then it is a distributional solution to (1.37). Moreover, if we consider the minimization problem for 1 Z Z J(u) = jruj2dx − fudx 2 Ω Ω 2 over K = fu 2 C (Ω); uj@Ω = 0g and if u is a solution to this problem then 1 for any > 0 and C0 (Ω) we have J(u + φ) ≥ J(u); then we also obtain (1.38). The question is how to obtain the solution. In a similar way, we consider the obstacle problem, to minimize J over 2 K = fu 2 C (Ω); uj@Ω = 0; u ≥ gg over some continuous function g satisfying gj@Ω < 0. Note that K is convex. Again, if u 2 K is a solution then for any > 0 and φ 2 K we obtain that u + (φ − u) = (1 − )u + φ is in K and therefore J(u + (φ − u)) ≥ J(u): Here, we obtain only Z Z ru · r(φ − u)dx ≥ f(φ − u)dx: (1.39) Ω Ω for any φ 2 K. For twice differentiable u we obtain Z Z ∆u(φ − u)dx ≥ f(φ − u)dx Ω Ω 1 and choosing φ = u + , 2 C0 (Ω) we get −∆u ≥ f almost everywhere on Ω. As u is continuous, the set N = fx 2 Ω; u(x) > 1 g(x)g is open. Thus, taking 2 C0 (N), we see that for sufficiently small > 0, u ± φ 2 K. Then, on N −∆u = f Summarizing, for regular solutions the minimizer satisfies −∆u ≥ f u ≥ g (∆u + f)(u − g) = 0 on Ω. 30 1 Basic Facts from Functional Analysis and Banach Lattices Hilbert space theory We begin with the following definition.
Load more

Recommended publications
  • FUNCTIONAL ANALYSIS 1. Banach and Hilbert Spaces in What
    FUNCTIONAL ANALYSIS PIOTR HAJLASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, k · k), where X is a linear space over K and k · k : X → [0, ∞) is a function, called a norm, such that (1) kx + yk ≤ kxk + kyk for all x, y ∈ X; (2) kαxk = |α|kxk for all x ∈ X and α ∈ K; (3) kxk = 0 if and only if x = 0. Since kx − yk ≤ kx − zk + kz − yk for all x, y, z ∈ X, d(x, y) = kx − yk defines a metric in a normed space. In what follows normed paces will always be regarded as metric spaces with respect to the metric d. A normed space is called a Banach space if it is complete with respect to the metric d. Definition. Let X be a linear space over K (=R or C). The inner product (scalar product) is a function h·, ·i : X × X → K such that (1) hx, xi ≥ 0; (2) hx, xi = 0 if and only if x = 0; (3) hαx, yi = αhx, yi; (4) hx1 + x2, yi = hx1, yi + hx2, yi; (5) hx, yi = hy, xi, for all x, x1, x2, y ∈ X and all α ∈ K. As an obvious corollary we obtain hx, y1 + y2i = hx, y1i + hx, y2i, hx, αyi = αhx, yi , Date: February 12, 2009. 1 2 PIOTR HAJLASZ for all x, y1, y2 ∈ X and α ∈ K. For a space with an inner product we define kxk = phx, xi . Lemma 1.1 (Schwarz inequality).
    [Show full text]
  • Duality, Part 1: Dual Bases and Dual Maps Notation
    Duality, part 1: Dual Bases and Dual Maps Notation F denotes either R or C. V and W denote vector spaces over F. Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Examples: Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). n n Fix (b1;:::; bn) 2 C . Define ': C ! C by '(z1;:::; zn) = b1z1 + ··· + bnzn: Then ' is a linear functional on Cn. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx. Then ' is a linear functional on P(R). Linear Functionals Definition: linear functional A linear functional on V is a linear map from V to F. In other words, a linear functional is an element of L(V; F). Examples: Define ': R3 ! R by '(x; y; z) = 4x − 5y + 2z. Then ' is a linear functional on R3. Define ': P(R) ! R by '(p) = 3p00(5) + 7p(4). Then ' is a linear functional on P(R). R 1 Define ': P(R) ! R by '(p) = 0 p(x) dx.
    [Show full text]
  • Using Functional Analysis and Sobolev Spaces to Solve Poisson’S Equation
    USING FUNCTIONAL ANALYSIS AND SOBOLEV SPACES TO SOLVE POISSON'S EQUATION YI WANG Abstract. We study Banach and Hilbert spaces with an eye to- wards defining weak solutions to elliptic PDE. Using Lax-Milgram we prove that weak solutions to Poisson's equation exist under certain conditions. Contents 1. Introduction 1 2. Banach spaces 2 3. Weak topology, weak star topology and reflexivity 6 4. Lower semicontinuity 11 5. Hilbert spaces 13 6. Sobolev spaces 19 References 21 1. Introduction We will discuss the following problem in this paper: let Ω be an open and connected subset in R and f be an L2 function on Ω, is there a solution to Poisson's equation (1) −∆u = f? From elementary partial differential equations class, we know if Ω = R, we can solve Poisson's equation using the fundamental solution to Laplace's equation. However, if we just take Ω to be an open and connected set, the above method is no longer useful. In addition, for arbitrary Ω and f, a C2 solution does not always exist. Therefore, instead of finding a strong solution, i.e., a C2 function which satisfies (1), we integrate (1) against a test function φ (a test function is a Date: September 28, 2016. 1 2 YI WANG smooth function compactly supported in Ω), integrate by parts, and arrive at the equation Z Z 1 (2) rurφ = fφ, 8φ 2 Cc (Ω): Ω Ω So intuitively we want to find a function which satisfies (2) for all test functions and this is the place where Hilbert spaces come into play.
    [Show full text]
  • Recent Developments in the Theory of Duality in Locally Convex Vector Spaces
    [ VOLUME 6 I ISSUE 2 I APRIL– JUNE 2019] E ISSN 2348 –1269, PRINT ISSN 2349-5138 RECENT DEVELOPMENTS IN THE THEORY OF DUALITY IN LOCALLY CONVEX VECTOR SPACES CHETNA KUMARI1 & RABISH KUMAR2* 1Research Scholar, University Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2*Research Scholar, University Department of Mathematics T. M. B. University, Bhagalpur Received: February 19, 2019 Accepted: April 01, 2019 ABSTRACT: : The present paper concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Key Words: LOCALLY CONVEX SPACES We shall be exclusively concerned with vector spaces over the real field: the passage to complex spaces offers no difficulty. We shall assume that the definition and properties of convex sets are known. A convex set A in a vector space E is symmetric if —A=A; then 0ЄA if A is not empty. A convex set A is absorbing if for every X≠0 in E), there exists a number α≠0 such that λxЄA for |λ| ≤ α ; this implies that A generates E. A locally convex space is a topological vector space in which there is a fundamental system of neighborhoods of 0 which are convex; these neighborhoods can always be supposed to be symmetric and absorbing. Conversely, if any filter base is given on a vector space E, and consists of convex, symmetric, and absorbing sets, then it defines one and only one topology on E for which x+y and λx are continuous functions of both their arguments.
    [Show full text]
  • Hilbert Space Methods for Partial Differential Equations
    Hilbert Space Methods for Partial Differential Equations R. E. Showalter Electronic Journal of Differential Equations Monograph 01, 1994. i Preface This book is an outgrowth of a course which we have given almost pe- riodically over the last eight years. It is addressed to beginning graduate students of mathematics, engineering, and the physical sciences. Thus, we have attempted to present it while presupposing a minimal background: the reader is assumed to have some prior acquaintance with the concepts of “lin- ear” and “continuous” and also to believe L2 is complete. An undergraduate mathematics training through Lebesgue integration is an ideal background but we dare not assume it without turning away many of our best students. The formal prerequisite consists of a good advanced calculus course and a motivation to study partial differential equations. A problem is called well-posed if for each set of data there exists exactly one solution and this dependence of the solution on the data is continuous. To make this precise we must indicate the space from which the solution is obtained, the space from which the data may come, and the correspond- ing notion of continuity. Our goal in this book is to show that various types of problems are well-posed. These include boundary value problems for (stationary) elliptic partial differential equations and initial-boundary value problems for (time-dependent) equations of parabolic, hyperbolic, and pseudo-parabolic types. Also, we consider some nonlinear elliptic boundary value problems, variational or uni-lateral problems, and some methods of numerical approximation of solutions. We briefly describe the contents of the various chapters.
    [Show full text]
  • Residues, Duality, and the Fundamental Class of a Scheme-Map
    RESIDUES, DUALITY, AND THE FUNDAMENTAL CLASS OF A SCHEME-MAP JOSEPH LIPMAN Abstract. The duality theory of coherent sheaves on algebraic vari- eties goes back to Roch's half of the Riemann-Roch theorem for Riemann surfaces (1870s). In the 1950s, it grew into Serre duality on normal projective varieties; and shortly thereafter, into Grothendieck duality for arbitrary varieties and more generally, maps of noetherian schemes. This theory has found many applications in geometry and commutative algebra. We will sketch the theory in the reasonably accessible context of a variety V over a perfect field k, emphasizing the role of differential forms, as expressed locally via residues and globally via the fundamental class of V=k. (These notions will be explained.) As time permits, we will indicate some connections with Hochschild homology, and generalizations to maps of noetherian (formal) schemes. Even 50 years after the inception of Grothendieck's theory, some of these generalizations remain to be worked out. Contents Introduction1 1. Riemann-Roch and duality on curves2 2. Regular differentials on algebraic varieties4 3. Higher-dimensional residues6 4. Residues, integrals and duality7 5. Closing remarks8 References9 Introduction This talk, aimed at graduate students, will be about the duality theory of coherent sheaves on algebraic varieties, and a bit about its massive|and still continuing|development into Grothendieck duality theory, with a few indications of applications. The prerequisite is some understanding of what is meant by cohomology, both global and local, of such sheaves. Date: May 15, 2011. 2000 Mathematics Subject Classification. Primary 14F99. Key words and phrases. Hochschild homology, Grothendieck duality, fundamental class.
    [Show full text]
  • 3. Hilbert Spaces
    3. Hilbert spaces In this section we examine a special type of Banach spaces. We start with some algebraic preliminaries. Definition. Let K be either R or C, and let Let X and Y be vector spaces over K. A map φ : X × Y → K is said to be K-sesquilinear, if • for every x ∈ X, then map φx : Y 3 y 7−→ φ(x, y) ∈ K is linear; • for every y ∈ Y, then map φy : X 3 y 7−→ φ(x, y) ∈ K is conjugate linear, i.e. the map X 3 x 7−→ φy(x) ∈ K is linear. In the case K = R the above properties are equivalent to the fact that φ is bilinear. Remark 3.1. Let X be a vector space over C, and let φ : X × X → C be a C-sesquilinear map. Then φ is completely determined by the map Qφ : X 3 x 7−→ φ(x, x) ∈ C. This can be see by computing, for k ∈ {0, 1, 2, 3} the quantity k k k k k k k Qφ(x + i y) = φ(x + i y, x + i y) = φ(x, x) + φ(x, i y) + φ(i y, x) + φ(i y, i y) = = φ(x, x) + ikφ(x, y) + i−kφ(y, x) + φ(y, y), which then gives 3 1 X (1) φ(x, y) = i−kQ (x + iky), ∀ x, y ∈ X. 4 φ k=0 The map Qφ : X → C is called the quadratic form determined by φ. The identity (1) is referred to as the Polarization Identity.
    [Show full text]
  • SERRE DUALITY and APPLICATIONS Contents 1
    SERRE DUALITY AND APPLICATIONS JUN HOU FUNG Abstract. We carefully develop the theory of Serre duality and dualizing sheaves. We differ from the approach in [12] in that the use of spectral se- quences and the Yoneda pairing are emphasized to put the proofs in a more systematic framework. As applications of the theory, we discuss the Riemann- Roch theorem for curves and Bott's theorem in representation theory (follow- ing [8]) using the algebraic-geometric machinery presented. Contents 1. Prerequisites 1 1.1. A crash course in sheaves and schemes 2 2. Serre duality theory 5 2.1. The cohomology of projective space 6 2.2. Twisted sheaves 9 2.3. The Yoneda pairing 10 2.4. Proof of theorem 2.1 12 2.5. The Grothendieck spectral sequence 13 2.6. Towards Grothendieck duality: dualizing sheaves 16 3. The Riemann-Roch theorem for curves 22 4. Bott's theorem 24 4.1. Statement and proof 24 4.2. Some facts from algebraic geometry 29 4.3. Proof of theorem 4.5 33 Acknowledgments 34 References 35 1. Prerequisites Studying algebraic geometry from the modern perspective now requires a some- what substantial background in commutative and homological algebra, and it would be impractical to go through the many definitions and theorems in a short paper. In any case, I can do no better than the usual treatments of the various subjects, for which references are provided in the bibliography. To a first approximation, this paper can be read in conjunction with chapter III of Hartshorne [12]. Also in the bibliography are a couple of texts on algebraic groups and Lie theory which may be beneficial to understanding the latter parts of this paper.
    [Show full text]
  • Homotopy Theory and Generalized Duality for Spectral Sheaves
    IMRN International Mathematics Research Notices 1996, No. 20 Homotopy Theory and Generalized Duality for Spectral Sheaves Jonathan Block and Andrey Lazarev We would like to express our sadness at the passing of Robert Thomason whose ideas have had a lasting impact on our work. The mathematical community is sorely impoverished by the loss. We dedicate this paper to his memory. 1 Introduction In this paper we announce a Verdier-type duality theorem for sheaves of spectra on a topological space X. Along the way we are led to develop the homotopy theory and stable homotopy theory of spectral sheaves. Such theories have been worked out in the past, most notably by [Br], [BG], [T], and [J]. But for our purposes these theories are inappropri- ate. They also have not been developed as fully as they are here. The previous authors did not have at their disposal the especially good categories of spectra constructed in [EKM], which allow one to do all of homological algebra in the category of spectra. Because we want to work in one of these categories of spectra, we are led to consider sheaves of spaces (as opposed to simplicial sets), and this gives rise to some additional technical difficulties. As an application we compute an example from geometric topology of stratified spaces. In future work we intend to apply our theory to equivariant K-theory. 2 Spectral sheaves There are numerous categories of spectra in existence and our theory can be developed in a number of them. Received 4 June 1996. Revision received 9 August 1996.
    [Show full text]
  • Duality for Mixed-Integer Linear Programs
    Duality for Mixed-Integer Linear Programs M. Guzelsoy∗ T. K. Ralphs† Original May, 2006 Revised April, 2007 Abstract The theory of duality for linear programs is well-developed and has been successful in advancing both the theory and practice of linear programming. In principle, much of this broad framework can be ex- tended to mixed-integer linear programs, but this has proven difficult, in part because duality theory does not integrate well with current computational practice. This paper surveys what is known about duality for integer programs and offers some minor extensions, with an eye towards developing a more practical framework. Keywords: Duality, Mixed-Integer Linear Programming, Value Function, Branch and Cut. 1 Introduction Duality has long been a central tool in the development of both optimization theory and methodology. The study of duality has led to efficient procedures for computing bounds, is central to our ability to perform post facto solution analysis, is the basis for procedures such as column generation and reduced cost fixing, and has yielded optimality conditions that can be used as the basis for “warm starting” techniques. Such procedures are useful both in cases where the input data are subject to fluctuation after the solution procedure has been initiated and in applications for which the solution of a series of closely-related instances is required. This is the case for a variety of integer optimization algorithms, including decomposition algorithms, parametric and stochastic programming algorithms, multi-objective optimization algorithms, and algorithms for analyzing infeasible mathematical models. The theory of duality for linear programs (LPs) is well-developed and has been extremely successful in contributing to both theory and practice.
    [Show full text]
  • Duality in MIP
    Duality in MIP Generating Dual Price Functions Using Branch-and-Cut Elena V. Pachkova Abstract This presentation treats duality in Mixed Integer Programming (MIP in short). A dual of a MIP problem includes a dual price function F , that plays the same role as the dual variables in Linear Programming (LP in the following). The price function is generated while solving the primal problem. However, di®erent to the LP dual variables, the characteristics of the dual price function depend on the algorithmic approach used to solve the MIP problem. Thus, the cutting plane approach provides non- decreasing and superadditive price functions while branch-and-bound algorithm generates piecewise linear, nondecreasing and convex price functions. Here a hybrid algorithm based on branch-and-cut is investigated, and a price function for that algorithm is established. This price function presents a generalization of the dual price functions obtained by either the cutting plane or the branch-and-bound method. 1 Introduction Duality in mathematical programming is used in a variety of applications. Apart from con- ceptual interest it provides interesting economic interpretations of the problem. Moreover, using dual information usually improves the performance of an algorithm. Thus, there exist many results on duality in linear programming (LP) (e.g. see Gass (1985)). Results on duality in integer programming (IP) also exist (Wolsey (1981)). While algorithms for LP produce unique dual programs (apart from degenerating programs), that are relatively easy to obtain, IP algorithms generate a dual function whose characteristics depend on the method used to solve the primal IP problem. Wolsey (1981) characterized this function for 73 the branch-and-bound and the cutting plane methods.
    [Show full text]
  • Notes for Functional Analysis
    Notes for Functional Analysis Wang Zuoqin (typed by Xiyu Zhai) November 13, 2015 1 Lecture 18 1.1 Characterizations of reflexive spaces Recall that a Banach space X is reflexive if the inclusion X ⊂ X∗∗ is a Banach space isomorphism. The following theorem of Kakatani give us a very useful creteria of reflexive spaces: Theorem 1.1. A Banach space X is reflexive iff the closed unit ball B = fx 2 X : kxk ≤ 1g is weakly compact. Proof. First assume X is reflexive. Then B is the closed unit ball in X∗∗. By the Banach- Alaoglu theorem, B is compact with respect to the weak-∗ topology of X∗∗. But by definition, in this case the weak-∗ topology on X∗∗ coincides with the weak topology on X (since both are the weakest topology on X = X∗∗ making elements in X∗ continuous). So B is compact with respect to the weak topology on X. Conversly suppose B is weakly compact. Let ι : X,! X∗∗ be the canonical inclusion map that sends x to evx. Then we've seen that ι preserves the norm, and thus is continuous (with respect to the norm topologies). By PSet 8-1 Problem 3, ∗∗ ι : X(weak) ,! X(weak) is continuous. Since the weak-∗ topology on X∗∗ is weaker than the weak topology on X∗∗ (since X∗∗∗ = (X∗)∗∗ ⊃ X∗. OH MY GOD!) we thus conclude that ∗∗ ι : X(weak) ,! X(weak-∗) is continuous. But by assumption B ⊂ X(weak) is compact, so the image ι(B) is compact in ∗∗ X(weak-∗). In particular, ι(B) is weak-∗ closed.
    [Show full text]