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Discrete Convex Analysis I Hausdorff School: Economics and Tropical Geometry Bonn, May 9-13, 2016 Discrete Convex Analysis I: Concepts of Discrete Convex Functions Kazuo Murota (Tokyo Metropolitan University) 160509BonnHCMecon1 1 Convex Function f(x) f(x) convex nonconvex f : Rn ! R is convex () λf(x) + (1 − λ)f(y) ≥ f(λx + (1 − λ)y) (0 < 8 λ < 1) f(x) 6 x z y - k R = R [ f+1g λx + (1 − λ)y 2 Features of Convex Functions • Occurrence in many models motivations, applications • Operations and transformations • Sufficient structure for a theory mathematically beautiful, practically useful • Minimization algorithms 3 Features of Convex Functions = Issues in discrete convex analysis • Occurrence in many models ? motivations, applications • Operations and transformations ? • Sufficient structure for a theory ? mathematically beautiful, practically useful • Minimization algorithms ? 4 Contents of Part I Concepts of Discrete Convex Functions C1. Univariate Discrete Convex Functions C2. Classes of Discrete Convex Functions C3. L-convex Functions C4. M-convex Functions C5. Remarks on Submodular Set Functions Part II: Properties, Part III: Algorithms 5 C1. Univariate Discrete Convex Functions Ingredients of convex analysis 6 Definition of Convex Function f : Z ! R R = R [ f+1g f(x − 1) + f(x + 1) ≥ 2f(x) () f(x) + f(y) ≥ f(x + 1) + f(y − 1) (x < y) () f is convex-extensible, i.e., 9 convex f : R ! R s.t. f(x) = f(x) (8x 2 Z) convex non-convex 7 Local vs Global Optimality f : Z ! R Theorem: x∗: global opt (min) () x∗: local opt (min) f(x∗) ≤ minff(x∗ − 1); f(x∗ + 1)g convex non-convex 8 Intuition of Legendre Transformation y 6 y 6 f(x) f(x) - - x x 9 Intuition of Legendre Transformation y 6 y 6 f(x) - - x x 10 Intuition of Legendre Transformation y 6 y 6 f(x) - - x x 6 6 y y - - x x 11 Legendre Transformation f : Z ! Z (integer-valued) Define discrete Legendre transform of f by f •(p) = supfpx − f(x) j x 2 Zg (p 2 Z) 6 6 y f(x) y −f •(p) slope p - - x x Theorem: (1) f • is Z-valued convex function, f • : Z ! Z (2) (f •)• = f (biconjugacy) 12 Separation Theorem f : Z ! R f(x) f(x) convex p∗ h : Z ! R concave h(x) h(x) Theorem (Discrete Separation Theorem) (1) f(x) ≥ h(x)(8x 2 Z) ) 9 α∗ 2 R, 9 p∗ 2 R: f(x) ≥ α∗ + p∗ x ≥ h(x)(8x 2 Z) (2) f, h: integer-valued ) α∗ 2 Z, p∗ 2 Z 13 Separation Theorem f(x) p∗ f : Z ! R convex h : Z ! R concave h(x) Theorem (Discrete Separation Theorem) (1) f(x) ≥ h(x)(8x 2 Z) ) 9 α∗ 2 R, 9 p∗ 2 R: f(x) ≥ α∗ + p∗ x ≥ h(x)(8x 2 Z) (2) f, h: integer-valued ) α∗ 2 Z, p∗ 2 Z 14 Fenchel Duality (Min-Max) f : Z ! Z: convex, h : Z ! Z: concave Legendre transforms: f•(p) = supfpx − f(x) j x 2 Zg h◦(p) = inffpx − h(x) j x 2 Zg Theorem: inf ff(x) − h(x)g = sup fh◦(p) − f•(p)g x2Z p2Z 15 Five Properties of \Convex" Functions 1. convex extension 2. local opt = global opt 3. Legendre transform (biconjugacy) 4. separation theorem 5. Fenchel duality hold for univariate discrete convex functions 16 C2. Classes of Discrete Convex Functions 17 Classes of Discrete Convex Functions 1. Submodular set fn (on f0; 1gn) 1. Separable-convex fn on Zn 1. Integrally-convex fn on Zn 2. L-convex (L\-convex) fn on Zn 2. M-convex (M\-convex) fn on Zn 3. M-convex fn on jump systems 3. L-convex fn on graphs 18 Submodular Function R = R [ f+1g Set function ρ : 2V ! R is submodular () X [ Y ρ(X) + ρ(Y ) ≥ ρ(X [ Y ) + ρ(X \ Y ) XYX \ Y cf. jXj + jY j = jX [ Y j + jX \ Y j Set function () Function on f0; 1gn 19 Separable-convex Function f : Zn ! R is separable-convex () f(x) = '1(x1) + '2(x2) + ··· + 'n(xn) 'i: univariate convex 'i 6 - 20 Five Properties of \Convex" Functions 1. convex extension 2. local opt = global opt 3. Legendre transform (biconjugacy) 4. separation theorem 5. Fenchel duality hold for separable discrete convex functions 21 Some History 1935 Matroid Whitney, Nakasawa 1965 Submodular function Edmonds 1969 Convex network flow (electr.circuit) Iri 1982 Submodularity and convexity Frank, Fujishige, Lov´asz 1990 Valuated matroid Dress{Wenzel Integrally convex fn Favati{Tardella 1996 Discrete convex analysis Murota 2000 Submodular minimization algorithm Iwata{Fleischer{Fujishige, Schrijver 2006 M-convex fn on jump system Murota 2012 L-convex fn on graph Hirai, Kolmogorov 22 Motivations/Applications/Connections 1. submodular MANY problems graph cut, convex game 1. separable-conv MANY problems min-cost flow, resource allocation 1. integrally-conv [mathematical aesthetics] 2. L-conv (Zn) network tension, image processing OR (inventory, scheduling) 2. M-conv (Zn) network flow, conjestion game economics (game, auction) mixed polynomial matrix 3. M-conv (jump) deg sequence, (2-)matching polynomial (half-plane property) 3. L-conv (graph) multiflow, multifacility location 23 Books (discrete convex analysis) 2000: Murota, Matrices and Matroids for Systems Analysis, Springer 2003: Murota, Discrete Convex Analysis, SIAM 2005: Fujishige, Submodular Functions and Optimization, 2nd ed., Elsevier 2014: Simchi-Levi, Chen, Bramel, The Logic of Logistics, 3rd ed., Springer 24 Convex Extension • f : Zn ! R is convex-extensible , 9 convex f : Rn ! R: f(x) = f(x) (8x 2 Zn) • f is a convex extension of f • convex closure = (pointwise) max convex extension convex-extensible NOT convex-extensible 25 Integrally Convex Function (Favati-Tardella 1990) n n N(x) = fy 2 Z j kx − yk1 < 1g (x 2 R ) x x Local convex extension: f~(x) = supfhp; xi + α j hp; yi + α ≤ f(y)(8y 2 N(x))g p,α Def: f is integrally convex () f~ is convex Ex: f(x1; x2) = jx1 − 2x2j is NOT integrally convex f = 1 f = 0 f~(x) = 1 x = (1; 1=2) f(x) = 0 f = 0 f = 1 26 Integrally Convex Set YES NO 27 Five Properties of \Convex" Functions 1. convex extension 2. local opt = global opt hold for integrally convex functions 3. Legendre transform (biconjugacy) 4. separation theorem 5. Fenchel duality fail for integrally convex functions 28 Definitions 1. submodular (set fn) ρ(X) + ρ(Y ) ≥ ρ(X [ Y ) + ρ(X \ Y ) 1. separable f(x) = '1(x1) + '2(x2) + ··· + 'n(xn) -conv 'i(t − 1) + 'i(t + 1) ≥ 2'i(t)(8t 2 Z) 1. integrally -conv Local convex ext f~(x) is convex 2. L-conv(Zn) 2. M-conv(Zn) 3. M-conv(jump) 3. L-conv(graph) 29 Classes of Discrete Convex Functions f : Zn ! R convex-extensible integrally convex submod M\-convex set fn separable convex L\-convex L-convex M-convex on graph on jump 30 Bivariate L\- and M\-convex Functions f(x) g(p) x2 p2 p1 x1 L\-convex fn M\-convex fn 31 C3. L-convex Functions 32 L-convex Function (L = Lattice) (Murota 98) g : Zn ! R [ f+1g q p _ q p _ q compnt-max p ^ q compnt-min © p p ^ q Def: g is L-convex () • Submodular: g(p) + g(q) ≥ g(p _ q) + g(p ^ q) • Translation: 9r; 8p: g(p + 1) = g(p) + r 1 = (1; 1;:::; 1) 6 L - L\ 33 L\-convexity from Submodularity (Murota 98, Fujishige{Murota 2000) g : Zn ! R L\-convex () g~(p0; p) = g(p − p01) is submodular in (p0; p) g~ : Zn+1 ! R, 1 = (1; 1;:::; 1) q~ p~ _ q~ g~(~p) +g ~(~q) ≥ g~ (~p _ q~) +g ~ (~p ^ q~) p~ p~ ^ q~ \ Ln+1 ' Ln ) Ln 34 L\-convexity from Mid-pt-convexity (Favati-Tardella 1990, Fujishige{Murota 2000) 6 q q =) p+q p - 2 p q p+q p 2 Mid-point convex (g: Rn ! R): p+q g(p) + g(q) ≥ 2g 2 n =) Discrete mid-pointl convexm (jg : Z !kR) p+q p+q g(p) + g(q) ≥ g 2 + g 2 L\-convex function (L = Lattice) 35 Mid-pt Convexity for 01-Vectors l m 6 p+q q = p _ q For p; q 2 f0; 1gn 2 p j k - p+q 2 = p ^ q Discrete mid-pt convexity:l m j k p+q p+q g(p) + g(q) ≥ g 2 + g 2 () Submodularity: g(p) + g(q) ≥ g (p _ q) + g (p ^ q) 36 Translation Submodularity (L\) g(p) + g(q) ≥ g((p − α1) _ q) + g(p ^ (q + α1)) (α ≥ 0) p q q + α1 α = 2 © p − α1 p q discrete mid-pt convex g~(p0; p) = g(p − p01) is submodular in (p0; p) (Fujishige-Murota 00) , translation submodular (Fujishige-Murota 00) , discrete mid-pt convex (Favati-Tardella 90) , submod. integ. convex 37 Rem: L\-convex vs Submodular n = 1 Fact 1: Every g : Z ! R is submodular Fact 2: Function g : Z ! R is L\-convex () g(p − 1) + g(p + 1) ≥ 2g(p) for all p 2 Z 38 L\-convex Function: Examples X X \ Quadratic: g(p) = aijpipj is L -convex i j X , aij ≤ 0 (i 6= j); aij ≥ 0 (8i) j Energy function: For univariate convex i and ij X X g(p) = i(pi) + ij(pi − pj) i i6=j Range: g(p) = maxfp1; p2; : : : ; png − minfp1; p2; : : : ; png Submodular set function: ρ : 2V ! R \ , ρ(X) = g(χX) for some L -convex g Multimodular: h : Zn ! R is multimodular , \ h(p) = g(p1; p1 + p2; : : : ; p1 + ··· + pn) for L -convex g 39 Five Properties of \Convex" Functions 1.
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