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Minimization Algorithms forDiscrete Convex Functions
Akiyoshi Shioura(Tohoku University)
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Minimization of L‐/M‐convex Functions
• fundamental problems in discrete convex analysis• many examples & applications• various algorithmic approaches
– Greedy, Scaling, Continuous Relaxation, etc.
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Outline of Talk• Overview of Discrete Convex Analysis• Definitions of L‐/M‐convex Functions• Algorithms for Unconstrained Minimization
– Greedy– Scaling– Continuous Relaxation
• Algorithms for More Difficult Problems
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Outline of Talk• Overview of Discrete Convex Analysis• Definitions of L‐/M‐convex Functions• Algorithms for Unconstrained Minimization
– Greedy– Scaling– Continuous Relaxation
• Algorithms for More Difficult Problems
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Overview of 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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History of Discrete Convex Analysis
1935: Matroid Whitney1965: Polymatroid, Submodular Function Edmonds1983: relation between 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
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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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Outline of Talk• Overview of Discrete Convex Analysis• Definitions of L‐/M‐convex Functions• Algorithms for Unconstrained Minimization
– Greedy– Scaling– Continuous Relaxation
• Algorithms for More Difficult Problems
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Definition of L♮‐convex Fn9
• L♮ ‐‐ L‐natural, L=Lattice• Def: is L♮‐convex (Fujishige‐Murota 2000)
[discrete mid‐point convexity]
integrally convex + submodular (Favati‐Tardella 1990)
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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 conv fn)
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4 13 22 3 1
1 1 5
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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♮‐convex fn: a variant of M‐convex fn
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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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Relationship of L‐/M‐convex Fns
M♮‐conv fn
L♮‐conv fn
convex‐extensible fns
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Outline of Talk• Overview of Discrete Convex Analysis• Definitions of L‐/M‐convex Functions• Algorithms for Unconstrained Minimization
– Greedy– Scaling– Continuous Relaxation
• Algorithms for More Difficult Problems
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Our Problems• Minimization of L♮ ‐convex function
• Minimization of M♮ ‐convex function– special case:
is unique minimal vector in dom =
( dom is integral polymatroid)
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Optimality Criterionfor Minimization Problems
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Optimality Criterion: General CaseDesirable property of “discrete convex” fn:
: global opt : local opt w.r.t. some neighborhood univariate convex fn
Prop: ∗: global opt local opt w.r.t ∗ ∗
‐variate “discrete convex” fn• local opt global opt?
– NOT for convex‐extensible fn• which neighborhood?
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∗
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Optimality Criterion: L♮ ‐convex Function
Thm:
∗ global opt local opt in ∗
Local optimality check:• need to check vectors? ‐‐‐ No!
• can be reduced to submoduar set fn min ‐‐‐ poly time ∗
is submodular set fn ∗ is local opt takes min at
(Murota98, 03)
∗
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Optimality Criterion: M♮ ‐convex Function
Thm:
∗ global opt local opt in ∗ , ∗
∗ ∗
Local optimality check:• vectors poly time
(Murota96)
∗
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Greedy Algorithm
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Greedy Algorithm: General Case
• Greedy Algorithm≒Steepest Descent Local Search• “Global opt=Local opt” Greedy works
Repeat: • find local min • set Stop if: is local opt
• Greedy terminates in finite # of iters. (can be exponential)• (pseudo)‐poly. iteration?
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Greedy Algorithm: L♮ ‐convex FunctionL♮ ‐convex fn: global opt local opt w.r.t.
Greedy works with
Key Lemma: in each iteration,“positive gap” ∗ ∗ decreases, or“negative gap” ∗ ∗ increases
∗
(Kolmogorov‐Shioura09)
Thm: : initial sol., ∗ “nearest” global opt# of iter ∗
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positive gap
negative gap
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Greedy Algorithm: M♮ ‐convex Function
Greedy works with
Thm: : initial sol., ∗ “nearest” global opt # of iter ∗
M♮ ‐convex fn:
global opt local opt w.r.t.
Minimizer Cut Thm:
: local opt ∗: global opt s.t.
∗ ∗
(Shioura98)
(Murota03)
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Greedy Algorithm for Special CaseSpecial Case: is unique minimal in dom
Initial vector: Repeat:• find • set Stop if:
Minimizer Cut Thm 2:(i) minimizes ∗: opt. s.t. ∗
(ii) ∗: opt. s.t. ∗
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Scaling and Proximity
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Scaling and Proximity: General CaseScaling of
( scaling parameter)= restriction of to
‐4 ‐2 0 2 4
“Proximity Thm”:global minimizer ● exists
in a neighborhood of scaled (local) minimizer ●
univariate convex fn Prop: |●ー●|≦α‐1
efficient algorithm
‐variate “discrete convex” fn• ||●ー●|| is bounded? How large?
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Scaling and Proximity: L♮ ‐convex Function
Thm: scaled local minimizer, ∗: global minimizers.t. ∗
Prop: is L♮ ‐convex fn
scaled (local) minimizer can be computed efficientlyefficient scaling algorithm
(Iwata‐Shigeno03)
Step 0: sufficiently large integerStep 1: find minimizer of in a neighborhood of Step 2: if , then stop ( is global opt)Step 3: set ; go to Step 1
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Scaling and Proximity: M♮ ‐convex Function
Thm: scaled local minimizer, ∗: global minimizers.t. ∗
But: is NOT M♮ ‐convex difficult to compute a scaled local minimizersimple scaling algo does not workapply scaling approach in a different way
(Moriguchi‐Murota‐Shioura02)
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Scaling Algorithm for Special CaseSpecial Case: is unique minimal vector in dom
Update of using step size : if , set otherwise, set with
maximum under
apply scaling technique to Greedy Algo
∗Prop: output of scaled greedy algo,
∗: global minimizers.t. ∗
efficient algorithm※ can be extended to general M♮ ‐convex fn
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Continuous Relaxation and Proximity
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Continuous Relaxation and Proximity: General Case
Assumption: convex fn with is given
“Proximity Thm”:int. minimizer ● exists in a neighborhood of real minimizer ●
univariate convex fn
Prop: |●ー●|< 1
efficient algorithm
‐variate “discrete convex” fn• ||●ー●|| is bounded? How large?
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Continuous L♮‐convex Function
Prop:• restriction of cont. L♮‐conv. fn on discrete L♮‐conv.• ∀discrete L♮‐conv. fn , ∃cont. L♮‐conv. fn
s.t.
(Murota‐Shioura00,04)
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Def: convex fn is continuous L♮‐convex is submodular
Assumption: continuous L♮‐convex fnwith is given
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Continuous Relaxation and Proximity:L♮‐convex Function
Assumption: continuous L♮‐convex fnwith is given
Thm: ℝ real minimizer, ∗: integral minimizers.t. ∗ ℝ
(Moriguchi‐Tsuchimura09)
if ℝ can be computed efficiently (e.g., quadratic ) efficient algorithm for int. minimizer
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Continuous M♮‐convex Function
Prop:∀discrete M♮‐conv. fn , ∃cont. M♮‐conv. fns.t.
(Murota‐Shioura00,04)
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※restriction of cont. M♮‐conv. fn on is NOT discrete M♮‐conv.
Def: convex fn is continuous M♮‐convex
(i) , or(ii) s.t.
Assumption: continuous M♮‐convex fnwith is given
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Continuous Relaxation and Proximity:M♮‐convex Function
Assumption: continuous M♮‐convex fnwith is given
Thm: ℝ real minimizer, ∗: integral minimizers.t. ∗ ℝ
(Moriguchi‐Shioura‐Tsuchimura11)
if ℝ can be computed efficiently (e.g., quadratic ) efficient algorithm for int. minimizer
Special case: separable convex fn on polymatroid:
Thm: ℝ real minimizer, ∗: integral minimizers.t. ∗ ℝ
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∗ ℝ
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Outline of Talk• Overview of Discrete Convex Analysis• Definitions of L‐/M‐convex Functions• Algorithms for Unconstrained Minimization
– Greedy– Scaling– Continuous Relaxation
• Algorithms for More Difficult Problems
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Minimization of Sum of Two M♮ ‐convex Fns
• Minimization of Sum of two M♮ ‐convex fns– sum of two M♮ ‐convex fns is NOT M♮ ‐convex– contains Polymatroid constrained problem:
– generalization of polymatroid intersection problem
Minimize sub. to
Minimize
where (if (otherwise)
• poly.‐time solvable– polymatroid intersection algorithms can be extended– use new techniques & analysis
(Murota96,99,Iwata‐Shigeno03,Iwata‐Moriguchi‐Murota05)
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Minimization of Sum of Many M♮ ‐convex Fns
• Minimization of Sum of more than two M♮ ‐convex fns
– contains Polymatroid constrained problem:
– generalization of three polymatroid intersection problem• NP‐hard
Minimize sub. to
• (1‐1/e)‐approximation (for maximization version) for monotone & polymatroid const. (Shioura09)– continuous relaxation + pipage rounding (Calinescu et al. 07)
– Key Property: convex closure of M♮ ‐convex fn can be computed in poly‐time cont. relaxation in poly‐time
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Convex Closure of M♮ ‐convex Fnconvex closure of ‐‐‐ point‐wise maximal convex fn satisfying
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Define
Prop: (i) restriction of on is L♮ ‐concave(ii) if f is integer‐valued has integral opt reduced to L♮ ‐concave fn maximization
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M♮ ‐concave Function Maximization with Knapsack Constraints
• Maximization of M♮ ‐concave fn
under knapsack constraints – NP‐hard
• polynomial‐time approximation scheme (Shioura11)
– continuous relaxation + simple rounding– near integrality of continuous opt. solution– Key Property: convex closure of M♮ ‐convex fn can be computed in poly‐timecont. relaxation in poly‐time
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