introduction to discrete convex analysisshioura/slide/dcaintro.pdf · applications •...
TRANSCRIPT
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Introduction toDiscrete Convex Analysis
Akiyoshi Shioura(Tohoku University)
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Discrete Convex AnalysisDiscrete Convex Analysis [Murota 1996]‐‐‐ theoretical framework for discrete optimization problems
discrete analogue of Convex Analysis
in continuous optimization
generalization of Theory of Matroid/Submodular Function
in discrete opitmization
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• key concept: two discrete convexity: L‐convexity & M‐convexity– generalization of Submodular Set Function & Matroid
• various nice properties– local optimalglobal optimal– duality theorem, separation theorem, conjugacy relation
• set/function are discrete convex problem is tractable
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Applications
• Combinatorial Optimization– matching, min‐cost flow, shortest path, min‐cost tension
• Math economics / Game theory– allocation of indivisible goods, stable marriage
• Operations research – inventory system, queueing, resource allocation
• Discrete structures – finite metric space
• Algebra– polynomial matrix, tropical geometry
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History of Discrete Convex Analysis
1935: Matroid Whitney1965: Polymatroid, Submodular Function Edmonds1983: Submodularity and Convexity
Lovász, Frank, Fujishige1992: Valuated Matroid Dress, Wenzel1996: Discrete Convex Analysis, L‐/M‐convexity Murota1996‐2000: variants of L‐/M‐convexity Fujishige, Murota, Shioura
1971: discretely convex function Miller1990: integrally convex function Favati‐Tardella
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Today’s Talk• fundamental properties of M‐convex & L‐convex functions• comparison with other discrete convexity
– convex‐extensible fn– Miller’s discretely convex fn– Favati‐Tardella’s integrally convex fn
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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Desirable Properties of Discrete Convexity
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Important Properties of Convex Fn• optimality condition by local property
local minimum in some neighborhood global minimum
• conjugacy relationship– conjugate of convex fn convex fn
• duality theorems– Fenchel duality– separation theorem
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Desirable Properties of Discrete Convex Fn
• discrete convexity = “convexity” for functions – convex extensibility
• can be extended to convex fn on – optimality condition by local property
• local minimum global minimum– local minimality depends on choice of neighborhood
– duality theorems• “discrete” Fenchel duality• “discrete” separation theorem
– conjugacy relationship• conjugate of “discrete” convex fn “discrete” convex fn
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Classes of Discrete Convex Fns• convex‐extensible fn• discretely convex fn (Miller 1971)• integrally convex fn (Favati‐Tardella 1990)• M‐convex fn, L‐convex fn (Murota 1995, 1996)
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satisfy desirable properties?
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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Convex‐Extensible Function
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Definition of Convex‐Extensible Fn• a natural candidate for “discrete convexity”• Def: is convex‐extensible , convex fn s.t.
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convex‐extensible
NOT convex‐extensible
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Definition of Convex‐Extensible Set14
convex‐extensible
NOT convex‐extensible
• Def: is convex‐extensibleindicator fn is convex‐extensible (“no‐hole” condition)
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Properties of Convex‐Extensible Fn• if n=1, satisfies various nice properties
– convex‐extensible – local min=global min, conjugacy, duality, etc. desirable concept as discrete convexity
• if n≧2,– convex‐extensible (by definition)– what else?
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Bad Results of Conv.‐Extensible Fn• any function f with is convex‐extensible no good structure
• local opt ≠ global opt: convex‐extensible fn s.t.
: local min in but NOT global min
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x=(0,0): local min in , f(0,0)>f(2,1)
Example:
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Separable‐Convex Function• Def: is separable‐convex
each is discrete convex– examples: , etc.– satisfy various nice properties
• convex‐extensible• local min w.r.t. = global min
– but, function class is too small• e.g., dom f is integer interval
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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Miller’s Discretely Convex Fn
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Definition of Discretely Convex Fn20
𝛼𝑥 1 𝛼 𝑦𝛼𝑥 1 𝛼 𝑦
• defined by discretized version of convex inequality• Def: is discretely convex (Miller 1971)
Prop:
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Definition of Discretely Convex Set21
• Def: is discretely convexindicator fn is discretely convex
discretely convex NOT discretely convex
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Property of Discretely Convex Fn22
• Thm: [local min = global min]
validity of descent alg for minimizationrepeat: (i) find with
(ii) update
※ size of neighborhood { | is ‐‐‐ exponential
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Bad Result of Discretely Convex Fn23
• Fact: discretely conv fn is NOT convex‐extensiblediscretely conv set is NOT convex‐extensible
(not satisfy “no‐hole” condition)• Example:
is discretely convex, but has a “hole”x1
x2
x3
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convex‐extensible fn
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discretely convex fn
separable‐conv. fn
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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Integrally Convex Function
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Convex Closure of Discrete Fn• Def: convex closure of ‐‐‐ point‐wise maximal convex fn satisfying
∈
|– convex closure is convex fn
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Local Convex Closure of Discrete Fn• Def: local convex closure of collection of conv. closure on each hypercube
∈
|
–– local convex closure is not convex
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Definition of Integrally Convex Fn• Def: is integrally convex (Favati‐Tardella 1990) local conv. closure is convex fn
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0 1 2
2 1 0
convex‐extensiblebut NOT integrally convex
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0 0 2
2 0 1
convex‐extensible& integrally convex
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Properties of Integrally Convex Fn• by definition, integrally convex fn is
– convex‐extensible – discretely convex
local min w.r.t. = global min
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Bad Results of Integrally Convex Fn• by definition, integrally convex fn is
– convex‐extensible – discretely convex
local min w.r.t. = global min
but, neighborhood contains vectors (exponential)
• any function f with is integrally convexno good structure
• failure of “discrete” separation theorem
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Separation Theorem for Convex Fn• Separation Theorem:
convex fn, : concave fn, affine fn s.t.
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• equivalent to Duality Theorem for nonlinear programming efficient primal‐dual‐type algorithm
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Discrete Separation Thmfor Discrete Convex Fn
• “Discrete” Separation Theorem:“discrete convex” fn, : “discrete concave” fn,
affine fn s.t.
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• equivalent to Duality Theorem for combinatorial optimization efficient primal‐dual‐type algorithm
𝑓 𝑥
𝑔 𝑥
𝑝 𝑥 𝛼
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Failure of Discrete Separation for Integrally Convex/Concave Fns
• : integrally convex, : integrally concave s.t.
but affine fn with
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‐‐‐ integrally convex,‐‐‐ integrally concave,
but no affine fn with
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convex‐extensible fn
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discretely convex fn
separable‐conv. fn
integrally convex fn
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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L‐convex Function
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Definition of L♮‐convex Fn38
• L♮ ‐‐ L‐natural, L=Lattice• Def: is L♮‐convex (Fujishige‐Murota 2000)
[discrete mid‐point convexity]
integrally convex + submodular (Favati‐Tardella 1990)
※ L♮‐convex int. convex conv.‐extensible & discr. convex
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Examples of L♮‐convex Fn• univariate convex
• separable‐convex fn• submodular set fn L♮‐conv fn with
• quadratic fn is L♮‐convex
• Range:
• min‐cost tension problem
,
: univariate discrete conv fn)
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4 13 22 3 1
1 1 5
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Optimality Condition by Local Property• Thm: [local min = global min]
※ local minimality check can be done efficiently
: submodular set fns, minimization in poly‐time
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M‐convex Function
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Characterization of Convex Function• Prop: [“equi‐distant” convexity]
is convex
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Definition of M♮‐convex Function
Def: is M♮‐convex :(i) , or
(ii)
j
ix
y
(Murota‐Shioura99)
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M=Matroid
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Examples of M♮ ‐convex Functions• Univariate convex
• Separable convex fn on polymatroid:For integral polymatroid and univariate convex
• Matroid rank function [Fujishige05]is M♮ ‐concave
• Weighted rank function [Shioura09] is M♮ ‐concave
• Gross substitutes utility in math economics/game theoryM♮ ‐concave fn on [Fujishige‐Yang03]
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Properties of M♮‐convex Fn• Thm: [local min = global min]
※ size of neighborhood =
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• M♮‐convex int. convex conv.‐extensible & discr. convex
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convex‐extensible fn
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discretely convex fn
integrally convex fn
M♮‐convex fn
L♮‐convex fn
separable‐convex fn
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Outline of Talk• Overview of Discrete Convex Analysis• Desirable Properties of Discrete Convexity• convex‐extensible fn• Miller’s discretely convex fn• Favati‐Tardella’s integrally convex fn• M‐convex & L‐convex fns• duality and conjugacy theorems for discrete convex fn
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Conjugacy and Duality
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Conjugacy for Convex Functions• Legendre transformation for :
●
convex fn is closed under Legendre transformation• Thm:
: convex ●:convex, ● ●(if is closed)
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Conjugacy for L‐/M‐Convex Functions• integer‐valued fn• discrete Legendre transformation:
●
L♮‐convex fn and M♮‐convex fn are conjugate• Thm:
(i) : M♮‐conv ●: L♮‐conv, ● ●
(ii) : L♮‐conv ●: M♮‐conv, ● ●
• generalization of relations in comb. opt.:– matroid rank fn [Whitney 35]
– polymatroid submodular fn [Edmonds 70]
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M♮‐convex fn
L♮‐convex fn
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Fenchel Duality Theorem for Convex Fn51
• Legendre transformation:●○
• Fenchel Duality Thm: : convex, : concave
∈ℝ ∈ℝ○ ●
( ●: convex, ○: concave)
• Fenchel duality thm separation theorem
![Page 52: Introduction to Discrete Convex Analysisshioura/slide/DCAintro.pdf · Applications • Combinatorial Optimization –matching, min‐cost flow, shortest path, min‐cost tension •](https://reader031.vdocument.in/reader031/viewer/2022021716/5e5d903bc8a6db3f84696e29/html5/thumbnails/52.jpg)
Fenchel‐Type Duality Theorem for L‐/M‐Convex Fn
52
• discrete Legendre transformation:●○
• Fenchel‐type Duality Thm: [Murota 96,98]
: M♮‐convex, : M♮‐concave
∈ℤ ∈ℤ○ ●
( ●: L♮‐convex, ○: L♮‐concave)
![Page 53: Introduction to Discrete Convex Analysisshioura/slide/DCAintro.pdf · Applications • Combinatorial Optimization –matching, min‐cost flow, shortest path, min‐cost tension •](https://reader031.vdocument.in/reader031/viewer/2022021716/5e5d903bc8a6db3f84696e29/html5/thumbnails/53.jpg)
Discrete Separation Thm for L♮ Convex Fn
• L♮ Separation Theorem: [Murota 96,98]
L♮‐convex fn, : L♮‐concave fn, affine fn s.t.
53
𝑓 𝑥
𝑔 𝑥
𝑥 𝑝 𝛽
![Page 54: Introduction to Discrete Convex Analysisshioura/slide/DCAintro.pdf · Applications • Combinatorial Optimization –matching, min‐cost flow, shortest path, min‐cost tension •](https://reader031.vdocument.in/reader031/viewer/2022021716/5e5d903bc8a6db3f84696e29/html5/thumbnails/54.jpg)
Discrete Separation Thm for M♮ Convex Fn
• M♮ Separation Theorem: [Murota 96,98]
M♮‐convex fn, : M♮‐concave fn, affine fn s.t.
54
𝑓 𝑥
𝑔 𝑥
𝑝 𝑥 𝛼
![Page 55: Introduction to Discrete Convex Analysisshioura/slide/DCAintro.pdf · Applications • Combinatorial Optimization –matching, min‐cost flow, shortest path, min‐cost tension •](https://reader031.vdocument.in/reader031/viewer/2022021716/5e5d903bc8a6db3f84696e29/html5/thumbnails/55.jpg)
Relation among Duality ThmsM♮ Separation Thm
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Fenchel‐type Duality Thm
○ ●
L♮ Separation Thm● ○
(poly)matroidintersection thm [Edmonds 70]
discrete separation for subm. fn [Frank 82]
Fenchel‐type duality thmfor subm. fn [Fujishige 84]
weight splitting thm for weighted matroid intersection [Iri‐Tomizawa 76, Frank 81]
weighted matroid intersection thm [Iri‐Tomizawa 76, Frank 81]