text: − 1 p T , H = 1 + p G (R2), H = 1 + p G (R2), a 1+T 2l a 2 b 2u a 2 c H = 1 + p G (T). 1 3 a 3 Proof See Appendix . 0.9 Expression in (6) Simulation Applying Proposition 2 in asymptotic regions leads to the 0.8 following proposition. Proposition 3: (Asymptotic SIR coverage probability) As 0.7 → ∞ λ =20 λ , SIR coverage probability P (λ , λ , T) converges 0.6 u tob a ﬁnite value as follows. c b u 0.5 λu=200 λu=2000 0.4 − li→m∞ Pc(λb, λu, T) = e λuπcT(α,αc,Rc,Rb) (13) 0.3 λb 0.2 Full load 0.1 Proof From (11) and (12) in Proposition 1, we have − 0 lsiimncλeb→al∞l tPhecLBoth=er tleimrmλsb→te∞ndPtcUoB0 =as λH11→e λ∞uπ.cAT(cαc,oαrcd,Rincg,Rtbo) 100 1B0S2 density λ (BSs/km120)4 106 b − b tlih−me λSbq→u∞eezHe1the=ore1m,sliinmceλb→pa∞ P→c =0.H1T1 ehusλulπimm.λMb→e∞anPwchil=e, Fduiga.l-3sl:oEpeffemcot doefl.λu on coverage probability under bounded e λuπcT(α,αc,Rc,Rb). Proposition 3 emphasizes the importance of considering UE density in a UDN. The converged value is a decreasing → ∞ function of λu and it tends to 0 when λu , i.e., in ΓUB = λ p PUB log (1 + T). (16) a fully loaded network. Thus, deploying inﬁnite number of b a c 2 BSs will not bring the UE performance to a unprecedented level. In contrast, extreme densiﬁcation will put the coverage Proof See Appendix . probability into the danger of decreasing to 0, as shown in Fig. 3 in the next section. The converged result also depends on Similar with its asymptotic behavior, the ASE will increase environmental parameters (α, αc, Rc, Rb). It will increase as with−BS density when λb is small and ﬁnally converge to both path loss exponents grow since the coverage probability λue λuπcT log2(1 + T) as shown in Proposition 4. is a increasing function of path loss exponents [9]. A larger IV. NUMERICAL RESULTS R or R will decline the performance since it either reduce b c the signal power or amplify the interference in the near-ﬁeld. In this section, we present the numerical results to study We now turn to the network perspective and study the the performance of network densiﬁcation and validate our asymptotic behavior of the ASE. Combining the deﬁnition in theoretical analysis. We assume R = 1m, R = 70m, b c (5) with Proposition 3, we can easily obtain the following α = 2.5, α = 4 and set the SIR threshold T = 10dB in all c proposition. of our results. To calculate or simulate ‘fully loaded network’, conPvreorpgoessittioona 4ﬁ:ni(tAe svyamlupetoatsicfoAllSowEs). As λb → ∞, ASE Γ wsoethseatt λpu ≈= 12.× 108/km2 which is a sufﬁciently large value a A. Effect of UE density − li→m∞ Γ (λb, λu, T) = λue λuπcT(α,αc,Rc,Rb) log2(1 + T) (14) Fig. 3 shows the effect of UE density on coverage prob- λb ability. An exact match between simulation and analysis is observed. Meanwhile, we ﬁnd that coverage probabilities show Proposition 4 shows that the asymptotic ASE will increase → ∞ completely different trends among different UE densities. with λ when λ is small and tends to 0 as λ . u u u In full load model, the diminishing of coverage probability Unlike the coverage probability, the asymptotic ASE will be starts when interfering BSs fall into the near-ﬁeld of the beneﬁcial from ultra-desiﬁcation to some extent and but still typical UE and keeps decreasing since interference will con- highly depends on the UE density. Returning from the asymptotic regions, we demonstrate how tinue increasing. When λu is ﬁnite, the interferer coordinates converge to UE coordinates in a UDN regime [11]. Thus the the ASE scales with BS density in the next proposition. distance from the typical UE to its closest interferer can be Proposition 5: (ASE scaling) The ASE scales with − approximated as the distance to the its closest neighbor UE λbpae λbpaπcT(α,αc,Rc,Rb) and is bounded by , which has an expected value of √1 . When UE density is low (e.g. λ = 20), the expected2 vλaulue is larger than the u ΓLB = λ p PLB log (1 + T) (15) critical distance, which means the probability of no interferer b a c 2 inside the near-ﬁeld of the typical UE is very high. 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text: 1010 Bounded dual−slope (BD) Bounded dual−slope (BD) Lower bound of BD Lower bound of BD 0.25 Upper bound of BD Upper bound of BD Unbounded single−slope (US) 105 Unbounded single−slope (US) Unbounded dual−slope (UD) Unbounded dual−slope (UD) 0.2 0.15 100 0.1 10−5 0.05 0 10−10 100 102 104 106 BS density λb (BSs/km2) 100 1B0S2 density λ (BSs/km120)4 106 b (a) Coverage Probability when network is fully loaded (a) ASE when network is fully loaded 1 103 Bounded dual−slope (BD) 0.9 Lower bound of BD 0.8 Upper bound of BD Unbounded single−slope (US) 0.7 Unbounded dual−slope (UD) 102 0.6 0.5 101 0.4 0.3 Bounded dual−slope (BD) 0.2 100 Lower bound of BD Upper bound of BD 0.1 Unbounded single−slope (US) Unbounded dual−slope (UD) 010 0 102 104 106 10−1 BS density λb (BSs/km2) 100 1B0S2 density λ (BSs/km120)4 106 b (b) Coverage Probability when λ = 200 u (b) ASE when λ = 200 u Fig. 4: Coverage probability bounds and comparison with Fig. 5: ASE bounds and comparison with previous models. previous models − in a single-slope model. In contrast, higher UE density (e.g. e λuπcT(α,αc,Rc,Rb) which is smaller than 1 (assume λu > 0) when applying a bounded model. Consistent with the result in λ = 200 or 2000) leads to more potential interferers within u Proposition 3, the converged value will decrease as UE density critical distance. Thus coverage probability will decrease for grows and ﬁnally falls to zero in the full load case as shown the same reason as in fully loaded network. Nevertheless, when in Fig. 3. This is because the signal is limited by the bound all the UEs in the near-ﬁeld get service, coverage probability effect and the overall performance will be dominated by the will start increasing again since the interference are saturated interference which depends on UE density. and no longer increase. Therefore, it is important to estimate C. ASE analysis the active UE density for efﬁcient network deployment or operation in order to avoid the decreasing region of coverage Figure 5 depicts the scaling of ASE with regard to BS probability. density. Aligning with proposition 5, the ASE ﬁrst increase B. Coverage probability analysis with BS density and then converge to a constant. The constant In Fig. 4, we compare the coverage probability under our is larger than 0 in partially loaded network and decreases to → ∞ model with the previous models. We observe that the perfor- 0 when the network is full load (λ ) as proved in u mance is overestimated with unbounded or single-slope mod- Proposition 4. els in highly densiﬁed regions. The reason is that those models By comparing Fig. 5 with Fig. 4, we can observe a trade- either exaggerate the received power inside the bounded region off between UE and network performance during BS densiﬁ- or underestimate the interference in the near-ﬁeld. cation. In full load case, there exists an BS density threshold The inaccuracy of path loss models may mislead the pre- around λ = 104. Before the threshold, although individual b diction of asymptotic behavior. For instance, the coverage performance gets worse, densiﬁcation is still beneﬁcial from probability will converge to 1 with unbounded models but to the network perspective. 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text: (cid:88) trade-off appears approximately between λ = 101.5 to λ = b b 103. The phenomenon further demonstrates the necessity of L E − E − applying a dual-slope model because the coverage probability I(s) = I[e sI(cid:89)] = Φ∗b,gi[exp( s( ∈ ∗ gi(cid:96)(di)))] is a non-decreasing function of BS density in a single-slope   x Φ b model. intOegurralboexupnrdesssinionPraonpdossithioonwn2 ianndFi5g.ar4e acnodmp5a. rTedhewﬁitghutrhees (=a) EΦ∗b(cid:18) x∈Φ∗ 1(cid:90)+ s1(cid:96)((cid:18)di) (cid:19) (cid:19) (18) → ∞ b ∞ verify the asymptotic tightness of the bounds for λ . The (b) − ∗ − 1 = exp 2πλ 1 dv upper bound is tighter in small UE density scenarios while the b 1 + s(cid:96)(v) r lower bound ﬁts better for large UE densities. This is because (cid:16) (cid:17) ∼ the upper bound is close to single-slope model which is similar where (a) is because g exp(1) and (b) follows the with small UE density scenario. The bounds can be used as probability generating functional (PGFL) of the PPP. Plugg√ing approximations in large BS density regions. in s = T and employing a change of variables v = tr (cid:96)(r) V. CONCLUSION results in     (cid:18) (cid:19) (cid:90)     In this paper, we investigate the asymptotic behavior of ∞ ultra-densiﬁcation of base stations. To our best knowledge, L T − T = exp 2πλ P dt . this is the ﬁrst work incorporating two key aspects of UDN I (cid:96)(r) b a 1 T + (cid:96)√(r) (cid:96)( tr) modeling: a partially loaded network due to a ﬁnite active UE (19) density and a dual-slope path loss model with a bounded loss within a unit distance. With such models, we ﬁnd that the → Plugging (19) into (17) with z r2 gives the coverage asymptotic behavior of ultra-dense base station deployment is probability under a general path loss fucntion in (20) as:    different from what was known with simpler assumptions, e.g. (cid:90) (cid:90) unit or zero convergence of coverage probability. Depending    ∞ ∞ oconnvtheergUe Eto deeinthseitry,zebroothorUaEcoconvstearnatgevapluroeb. aEbvileintybaenfodreAtShEe Pcl(λb,λu,T) = λbπ 0 exp −λbπz 1 + pa 1 1 + 1(cid:96)(√√z) dt dz asymptotic regions, our results suggest that the densiﬁcation T(cid:96)( tz) (20) cannot always improve the individual UE performance or boost the network throughput as well. The increment are Based on (20), we can substitute our bounded dual-slope prevented by introducing extra interference in the near-ﬁeld model (2) into it and get the expression in (6). until all the UEs in the near-ﬁeld are served. Our work APPENDIX B provides insights into the scaling of the network densiﬁcation, PROOF OF PROPOSITION 2 and thus gives a guideline for the network deployment. According to the expression of (7) and (9), rG (r, T) and APPENDIX A rG (T) are linear functions of r. Thus we can rewr1ite the ﬁrst 3 PROOF OF PROPOSITION 1 and third integral in (6) as follows: (cid:90) (cid:16) (cid:17) We start from the coverage probability expression under Rb2 e−λbπr(1+paG1(r))dr = 1 e−λbpaπcT − e−λbπ(Rb2H1+pacT) general path loss model and then plug in with our bounded H dual-slope model. According to the deﬁnition, the coverage 0 1 (21) probability can be expressed as: (cid:90) (cid:16) (cid:17) (cid:20) (cid:21) ∞ P(cid:90) (cid:20) P |h|2(cid:96)(cid:21)(r) Rc2 e−λbπr(1+paG3(r))dr = H13 e−λbπRc2H3 . (22) Pl = [SIR > T] = > T c I (cid:48) In the second integral, from G (r) < 0 we can get (=a) (cid:90) P (cid:18)|h|2 >(cid:19)TI |r f (r)dr (17) G2(Rb2) ≥ G(r) ≥ G2(Rc2). With2 the inequality, we can (cid:96)(r) r provide bounds for the second integral as: r>0 (cid:90) (cid:90) (b) L T = f (r)dr r>0 I (cid:96)(r) r Rc2 e−λbπr(1+paG2(r))dr ≤ Rc2 e−λbπr(1+paG2(Rb2))dr Rb2 Rb2 where (a) follows from BS distribution and (b) is due to = 1 (e−λbπH2uRb2 − e−λbπH2uRc2) the fact that |h|2 ∼ exp(1), L (s) is the Laplace transform of H2u (23) I interference which can be derived as ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: trade-off

text: appears

text: approximately

text: between

text: λ

text: to

text: λ

text: b

text: b

text: The

text: phenomenon

text: further

text: demonstrates

text: the

text: necessity

text: of

text: L

text: E

text: E

text: applying

text: a

text: dual-slope

text: model

text: because

text: the

text: coverage

text: probability

text: I(s)

text: I[e

text: Φ∗b,gi[exp(

text: s(

text: is

text: a

text: non-decreasing

text: function

text: of

text: BS

text: density

text: in

text: a

text: single-slope

text: x

text: Φ

text: b

text: model.

text: rTedhewﬁitghutrhees

text: (=a)

text: x∈Φ∗

text: b

text: verify

text: the

text: asymptotic

text: tightness

text: of

text: the

text: bounds

text: for

text: λ

text: The

text: (b)

text: exp

text: 2πλ

text: dv

text: upper

text: bound

text: is

text: tighter

text: in

text: small

text: UE

text: density

text: scenarios

text: while

text: the

text: b

text: r

text: lower

text: bound

text: ﬁts

text: better

text: for

text: large

text: UE

text: densities.

text: This

text: is

text: because

text: the

text: upper

text: bound

text: is

text: close

text: to

text: single-slope

text: model

text: which

text: is

text: similar

text: where

text: (a)

text: is

text: because

text: g

text: and

text: (b)

text: follows

text: the

text: with

text: small

text: UE

text: density

text: scenario.

text: The

text: bounds

text: can

text: be

text: used

text: as

text: probability

text: generating

text: functional

text: (PGFL)

text: of

text: the

text: PPP.

text: Plugg√ing

text: approximations

text: in

text: large

text: BS

text: density

text: regions.

text: in

text: s

text: T

text: and

text: employing

text: a

text: change

text: of

text: variables

text: v

text: tr

text: V.

text: CONCLUSION

text: results

text: in

text: In

text: this

text: paper,

text: we

text: investigate

text: the

text: asymptotic

text: behavior

text: of

text: ultra-densiﬁcation

text: of

text: base

text: stations.

text: To

text: our

text: best

text: knowledge,

text: L

text: T

text: T

text: exp

text: 2πλ

text: P

text: dt

text: this

text: is

text: the

text: ﬁrst

text: work

text: incorporating

text: two

text: key

text: aspects

text: of

text: UDN

text: I

text: b

text: a

text: T

text: tr)

text: modeling:

text: a

text: partially

text: loaded

text: network

text: due

text: to

text: a

text: ﬁnite

text: active

text: UE

text: density

text: and

text: a

text: dual-slope

text: path

text: loss

text: model

text: with

text: a

text: bounded

text: loss

text: within

text: a

text: unit

text: distance.

text: With

text: such

text: models,

text: we

text: ﬁnd

text: that

text: the

text: Plugging

text: into

text: with

text: z

text: gives

text: the

text: coverage

text: asymptotic

text: behavior

text: of

text: ultra-dense

text: base

text: station

text: deployment

text: is

text: probability

text: under

text: a

text: general

text: path

text: loss

text: fucntion

text: in

text: as:

text: different

text: from

text: what

text: was

text: known

text: with

text: simpler

text: assumptions,

text: e.g.

text: unit

text: or

text: zero

text: convergence

text: of

text: coverage

text: probability.

text: Depending

text: oconnvtheergUe

text: Eto

text: aEbvileintybaenfodreAtShEe

text: Pcl(λb,λu,T)

text: λbπ

text: exp

text: −λbπz

text: pa

text: dt

text: dz

text: asymptotic

text: regions,

text: our

text: results

text: suggest

text: that

text: the

text: densiﬁcation

text: tz)

text: cannot

text: always

text: improve

text: the

text: individual

text: UE

text: performance

text: or

text: boost

text: the

text: network

text: throughput

text: as

text: well.

text: The

text: increment

text: are

text: Based

text: on

text: we

text: can

text: substitute

text: our

text: bounded

text: dual-slope

text: prevented

text: by

text: introducing

text: extra

text: interference

text: in

text: the

text: near-ﬁeld

text: model

text: into

text: it

text: and

text: get

text: the

text: expression

text: in

text: until

text: all

text: the

text: UEs

text: in

text: the

text: near-ﬁeld

text: are

text: served.

text: Our

text: work

text: APPENDIX

text: B

text: provides

text: insights

text: into

text: the

text: scaling

text: of

text: the

text: network

text: densiﬁcation,

text: PROOF

text: OF

text: PROPOSITION

text: and

text: thus

text: gives

text: a

text: guideline

text: for

text: the

text: network

text: deployment.

text: According

text: to

text: the

text: expression

text: of

text: and

text: rG

text: (r,

text: T)

text: and

text: APPENDIX

text: A

text: rG

text: (T)

text: are

text: linear

text: functions

text: of

text: r.

text: Thus

text: we

text: can

text: the

text: ﬁrst

text: PROOF

text: OF

text: PROPOSITION

text: and

text: third

text: integral

text: in

text: as

text: follows:

text: We

text: start

text: from

text: the

text: coverage

text: probability

text: expression

text: under

text: e−λbpaπcT

text: general

text: path

text: loss

text: model

text: and

text: then

text: plug

text: in

text: with

text: our

text: bounded

text: H

text: dual-slope

text: model.

text: According

text: to

text: the

text: deﬁnition,

text: the

text: coverage

text: probability

text: can

text: be

text: expressed

text: as:

text: P

text: Pl

text: [SIR

text: T]

text: T

text: c

text: I

text: In

text: the

text: second

text: integral,

text: from

text: G

text: (r)

text: we

text: can

text: get

text: (=a)

text: P

text: |r

text: f

text: (r)dr

text: G(r)

text: the

text: inequality,

text: we

text: can

text: r

text: provide

text: bounds

text: for

text: the

text: second

text: integral

text: as:

text: (b)

text: L

text: T

text: f

text: (r)dr

text: I

text: r

text: where

text: (a)

text: follows

text: from

text: BS

text: distribution

text: and

text: (b)

text: is

text: due

text: to

text: the

text: fact

text: that

text: L

text: (s)

text: is

text: the

text: Laplace

text: transform

text: of

text: I

text: interference

text: which

text: can

text: be

text: derived

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 (a) (b) | | FIG. 2: A component of the spatial acceleration, ar = x¨ r , plotted against the proper time, 0 0 − s/r for the smeared Dirac-delta regularization, r /(cid:96) = 3, (a): m/m = 1.95 (dashed line), 0 0 B − − m/m = 1.98 (solid line) and m/m = 2 (dotted line) and (b): m/m = 3.8 (dashed line), B B B m/m = 3.91 (solid line) and m/m = 4.1 (dotted line). B B motion, found by the help of the smeared Dirac-delta, (14), has inﬁnitely long memory and (cid:90) (cid:32) (cid:112) (cid:33) contains the velocities only in the right hand side, − (cid:48) √ r0 m 0 − (x x )2 − (x−x(cid:48))2 { − (cid:48) (cid:48) − − (cid:48) (cid:48) } x¨ = du 1 e (x x )(x˙x˙ ) [x˙(x x )]x˙ ] . (29) (cid:96) 3(cid:96)4 m −∞ 2(cid:96) B We impose the initial condition that the charge is at rest, x(t) = (t, 0), for t < t and the 0 charge follows a prescribed trajectory, x (t) for t < t < t + t , with x (t ) = 0. A certain i 0 0 i i 0 external source, k (s), is supposed to generate this motion which is turned oﬀ after this i initial phase and the invariant length, s, of the world line is measured from the time t + t . 0 i The numerical solution of the equation of motion becomes straightforward with such initial conditions: One introduces a small but ﬁnite ∆t step size and writes eqs. (28) or (29) as diﬀerential equation and ﬁnds the retarded time, u, or calculates the integral of the memory → term numerically, respectively at each step, t t + ∆t. The equation of motion has two free, adjustable parameters, the cutoﬀ, Λ, and the bare mass, m . Hence we need a renormalization condition to ﬁx the theory, it is chosen to be B 2 χr = 1 + ir ω, (30) ω 3 0 cf. eq. (26). The numerical solution of the equation of motion indicates stable dynamics for suﬃciently | | weak force, ie. small m/m . The acceleration changes in a monotonous, exponential B ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

label: caption

text: FIG.

text: A

text: component

text: of

text: the

text: spatial

text: acceleration,

text: ar

text: x¨

text: r

text: plotted

text: against

text: the

text: proper

text: time,

text: s/r

text: for

text: the

text: smeared

text: Dirac-delta

text: regularization,

text: r

text: (a):

text: m/m

text: (dashed

text: line),

text: B

text: m/m

text: (solid

text: line)

text: and

text: m/m

text: (dotted

text: line)

text: and

text: (b):

text: m/m

text: (dashed

text: line),

text: B

text: B

text: B

text: m/m

text: (solid

text: line)

text: and

text: m/m

text: (dotted

text: line).

text: B

text: B

text: motion,

text: found

text: by

text: the

text: help

text: of

text: the

text: smeared

text: Dirac-delta,

text: has

text: inﬁnitely

text: long

text: memory

text: and

text: contains

text: the

text: velocities

text: only

text: in

text: the

text: right

text: hand

text: side,

text: m

text: (x

text: x

text: x¨

text: du

text: e

text: (x

text: x

text: )(x˙x˙

text: [x˙(x

text: x

text: )]x˙

text: m

text: B

text: We

text: impose

text: the

text: initial

text: condition

text: that

text: the

text: charge

text: is

text: at

text: rest,

text: x(t)

text: (t,

text: for

text: t

text: t

text: and

text: the

text: charge

text: follows

text: a

text: prescribed

text: trajectory,

text: x

text: (t)

text: for

text: t

text: t

text: t

text: t

text: with

text: x

text: (t

text: A

text: certain

text: i

text: i

text: i

text: external

text: source,

text: k

text: (s),

text: is

text: supposed

text: to

text: generate

text: this

text: motion

text: which

text: is

text: turned

text: oﬀ

text: after

text: this

text: i

text: initial

text: phase

text: and

text: the

text: invariant

text: length,

text: s,

text: of

text: the

text: world

text: line

text: is

text: measured

text: from

text: the

text: time

text: t

text: t

text: i

text: The

text: numerical

text: solution

text: of

text: the

text: equation

text: of

text: motion

text: becomes

text: straightforward

text: with

text: such

text: initial

text: conditions:

text: One

text: introduces

text: a

text: small

text: but

text: ﬁnite

text: ∆t

text: step

text: size

text: and

text: writes

text: eqs.

text: or

text: as

text: diﬀerential

text: equation

text: and

text: ﬁnds

text: the

text: retarded

text: time,

text: u,

text: or

text: calculates

text: the

text: integral

text: of

text: the

text: memory

text: term

text: numerically,

text: respectively

text: at

text: each

text: step,

text: t

text: t

text: ∆t.

text: The

text: equation

text: of

text: motion

text: has

text: two

text: free,

text: adjustable

text: parameters,

text: the

text: cutoﬀ,

text: Λ,

text: and

text: the

text: bare

text: mass,

text: m

text: Hence

text: we

text: need

text: a

text: renormalization

text: condition

text: to

text: ﬁx

text: the

text: theory,

text: it

text: is

text: chosen

text: to

text: be

text: B

text: χr

text: ir

text: ω,

text: ω

text: cf.

text: eq.

text: The

text: numerical

text: solution

text: of

text: the

text: equation

text: of

text: motion

text: indicates

text: stable

text: dynamics

text: for

text: suﬃciently

text: weak

text: force,

text: ie.

text: small

text: m/m

text: The

text: acceleration

text: changes

text: in

text: a

text: monotonous,

text: exponential

text: B

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 25 (a) (b) (c) (d) (e) FIG. 6: The Λ˜-dependence of the three zeros of (B2) in the smeared Dirac-delta scheme, (a): Reω˜ , + (b): Reω˜−, (c): Imω˜ , (d): Imω˜ and (e): Imω˜ . + + 0 The Gaussian smearing, (14), leads to a particularly simple form of the susceptibility, Λ˜ω˜ − − χr = 1 + (4Λ˜2 3iΛ˜ω˜ ω˜2) (B2) G ω˜ 6(ω˜ + iΛ˜)3 with three zeros which can be traced analytically. Their dependence on the cutoﬀ, shown in Fig. 6, is qualitatively similar to those, found for the previous regulator. In fact, the zeros ˜ have negative imaginary part for Λ < 4 and they enter into the upper half plane by the further increase of the cutoﬀ. The imaginary root converges to the Abraham-Lorentz value ≥ ˜ and the zeros with real components diverge with the cutoﬀ. The part Λ 6 of Fig. 6 (e) ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: (a)

text: (b)

text: (c)

text: (d)

text: (e)

text: FIG.

text: The

text: Λ˜-dependence

text: of

text: the

text: three

text: zeros

text: of

text: in

text: the

text: smeared

text: Dirac-delta

text: scheme,

text: (a):

text: Reω˜

text: (b):

text: Reω˜−,

text: (c):

text: Imω˜

text: (d):

text: Imω˜

text: and

text: (e):

text: Imω˜

text: The

text: Gaussian

text: smearing,

text: leads

text: to

text: a

text: particularly

text: simple

text: form

text: of

text: the

text: susceptibility,

text: Λ˜ω˜

text: χr

text: 3iΛ˜ω˜

text: ω˜2)

text: G

text: ω˜

text: 6(ω˜

text: with

text: three

text: zeros

text: which

text: can

text: be

text: traced

text: analytically.

text: Their

text: dependence

text: on

text: the

text: cutoﬀ,

text: shown

text: in

text: Fig.

text: is

text: qualitatively

text: similar

text: to

text: those,

text: found

text: for

text: the

text: previous

text: regulator.

text: In

text: fact,

text: the

text: zeros

text: have

text: negative

text: imaginary

text: part

text: for

text: Λ

text: and

text: they

text: enter

text: into

text: the

text: upper

text: half

text: plane

text: by

text: the

text: further

text: increase

text: of

text: the

text: cutoﬀ.

text: The

text: imaginary

text: root

text: converges

text: to

text: the

text: Abraham-Lorentz

text: value

text: and

text: the

text: zeros

text: with

text: real

text: components

text: diverge

text: with

text: the

text: cutoﬀ.

text: The

text: part

text: Λ

text: of

text: Fig.

text: (e)

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: 4 change of the eﬀective equation of motion, based on physical intuition [16], on magnetic moment charge [17], or on quantum eﬀects [18]. So far no generally accepted solution has been found to cure the instability. The third step is to embed the radiation reaction problem into QED by considering it as (cid:126) → saddle point physics. The Abraham-Lorentz force can be identiﬁed in the 0 limit of the shift of the position of a wave packet [19] and the Landau-Lifsitz form of the equation [2] follows from considering the shift of the momentum of the particle [20]. The more direct derivation which goes parallel with the classical arguments is based on the retarded Green’s function. It has already been considered in the presence of a cutoﬀ [21] but the non-locality prevented a systematic, analytical exploitation of the equation of motion. We are accustomed to UV ﬁnite classical theories because the iterative solution of the O (cid:126) classical equations of motion can be presented in the form of series, involving ( 0), tree- level graphs. But this holds for elementary, closed theories only. If there are unobserved degrees of freedom, an environment, and we are looking for a closed system of equations for the observed quantities, then the elimination of the unobserved coordinates generates loop diagrams. The iterative solution of the classical equation of motion of a charge, moving on the background of a ﬁxed electromagnetic ﬁeld, can be represented by an inﬁnite series of tree-level graphs where the electromagnetic ﬁeld is attached to the world line as an external leg. However the elimination of the electromagnetic ﬁeld by the help of the Maxwell equations, using the world line as a source, couples the pairs of the external legs and forms loops. The loop integrals of classical eﬀective theories have already been spotted as the on- shell contributions to the loop integrals of the full quantum theory [22]. The self-interaction O − of a point charge with its own ﬁeld is UV singular due to the (r 1) near ﬁeld and the usual regularization and renormalization procedure of quantum ﬁled theories must be employed in this saddle point problem of QED. The UV divergences are treated by a regulator which modiﬁes the dynamics at distance or energy scales beyond the tested laws of physics. However the regulator should not violate known, general properties of physics which are expected to reign even within the not yet explored scale regimes. The properties we are mainly concerned with here are the Lorentz symmetry and the stability of the motion. The physics of the classical electron radius, despite being “simple classical physics” turns out to be surprisingly problematic because there is apparently no relativistic regulator which automatically (ie. without a detailed study of

text: change

text: of

text: the

text: eﬀective

text: equation

text: of

text: motion,

text: based

text: on

text: physical

text: intuition

text: on

text: magnetic

text: moment

text: charge

text: or

text: on

text: quantum

text: eﬀects

text: So

text: far

text: no

text: generally

text: accepted

text: solution

text: has

text: been

text: found

text: to

text: cure

text: the

text: instability.

text: The

text: third

text: step

text: is

text: to

text: embed

text: the

text: radiation

text: reaction

text: problem

text: into

text: QED

text: by

text: considering

text: it

text: as

text: saddle

text: point

text: physics.

text: The

text: Abraham-Lorentz

text: force

text: can

text: be

text: identiﬁed

text: in

text: the

text: limit

text: of

text: the

text: shift

text: of

text: the

text: position

text: of

text: a

text: wave

text: packet

text: and

text: the

text: Landau-Lifsitz

text: form

text: of

text: the

text: equation

text: follows

text: from

text: considering

text: the

text: shift

text: of

text: the

text: momentum

text: of

text: the

text: particle

text: The

text: more

text: direct

text: derivation

text: which

text: goes

text: parallel

text: with

text: the

text: classical

text: arguments

text: is

text: based

text: on

text: the

text: retarded

text: Green’s

text: function.

text: It

text: has

text: already

text: been

text: considered

text: in

text: the

text: presence

text: of

text: a

text: cutoﬀ

text: but

text: the

text: non-locality

text: prevented

text: a

text: systematic,

text: analytical

text: exploitation

text: of

text: the

text: equation

text: of

text: motion.

text: We

text: are

text: accustomed

text: to

text: UV

text: ﬁnite

text: classical

text: theories

text: because

text: the

text: iterative

text: solution

text: of

text: the

text: O

text: classical

text: equations

text: of

text: motion

text: can

text: be

text: presented

text: in

text: the

text: form

text: of

text: series,

text: involving

text: tree-

text: level

text: graphs.

text: But

text: this

text: holds

text: for

text: elementary,

text: closed

text: theories

text: only.

text: If

text: there

text: are

text: unobserved

text: degrees

text: of

text: freedom,

text: an

text: environment,

text: and

text: we

text: are

text: looking

text: for

text: a

text: closed

text: system

text: of

text: equations

text: for

text: the

text: observed

text: quantities,

text: then

text: the

text: elimination

text: of

text: the

text: unobserved

text: coordinates

text: generates

text: loop

text: diagrams.

text: The

text: iterative

text: solution

text: of

text: the

text: classical

text: equation

text: of

text: motion

text: of

text: a

text: charge,

text: moving

text: on

text: the

text: background

text: of

text: a

text: ﬁxed

text: electromagnetic

text: ﬁeld,

text: can

text: be

text: represented

text: by

text: an

text: inﬁnite

text: series

text: of

text: tree-level

text: graphs

text: where

text: the

text: electromagnetic

text: ﬁeld

text: is

text: attached

text: to

text: the

text: world

text: line

text: as

text: an

text: external

text: leg.

text: However

text: the

text: elimination

text: of

text: the

text: electromagnetic

text: ﬁeld

text: by

text: the

text: help

text: of

text: the

text: Maxwell

text: equations,

text: using

text: the

text: world

text: line

text: as

text: a

text: source,

text: couples

text: the

text: pairs

text: of

text: the

text: external

text: legs

text: and

text: forms

text: loops.

text: The

text: loop

text: integrals

text: of

text: classical

text: eﬀective

text: theories

text: have

text: already

text: been

text: spotted

text: as

text: the

text: on-

text: shell

text: contributions

text: to

text: the

text: loop

text: integrals

text: of

text: the

text: full

text: quantum

text: theory

text: The

text: self-interaction

text: O

text: of

text: a

text: point

text: charge

text: with

text: its

text: own

text: ﬁeld

text: is

text: UV

text: singular

text: due

text: to

text: the

text: (r

text: near

text: ﬁeld

text: and

text: the

text: usual

text: regularization

text: and

text: renormalization

text: procedure

text: of

text: quantum

text: ﬁled

text: theories

text: must

text: be

text: employed

text: in

text: this

text: saddle

text: point

text: problem

text: of

text: QED.

text: The

text: UV

text: divergences

text: are

text: treated

text: by

text: a

text: regulator

text: which

text: modiﬁes

text: the

text: dynamics

text: at

text: distance

text: or

text: energy

text: scales

text: beyond

text: the

text: tested

text: laws

text: of

text: physics.

text: However

text: the

text: regulator

text: should

text: not

text: violate

text: known,

text: general

text: properties

text: of

text: physics

text: which

text: are

text: expected

text: to

text: reign

text: even

text: within

text: the

text: not

text: yet

text: explored

text: scale

text: regimes.

text: The

text: properties

text: we

text: are

text: mainly

text: concerned

text: with

text: here

text: are

text: the

text: Lorentz

text: symmetry

text: and

text: the

text: stability

text: of

text: the

text: motion.

text: The

text: physics

text: of

text: the

text: classical

text: electron

text: radius,

text: despite

text: being

text: “simple

text: classical

text: physics”

text: turns

text: out

text: to

text: be

text: surprisingly

text: problematic

text: because

text: there

text: is

text: apparently

text: no

text: relativistic

text: regulator

text: which

text: automatically

text: (ie.

text: without

text: a

text: detailed

text: study

text: of


text: 24 Napat Rujeerapaiboon, Kilian Schindler, Daniel Kuhn, Wolfram Wiesemann For an empirical distribution Pˆn ∈ PE(Rd, n), the algorithm constructs a sQe- quence of reduced sets R containing the atoms of the reduced distribution . Iζn(cid:48) ∈eaRch, titheartatmioanx,imStaelply2resdelueccetss tthhee WexachssaenrgsteeiRn ∪di{sζta}n\ce{ζD(cid:48)}l,(Pˆζn,∈Rs)u. pFpo(rPˆnp)erafonrd- mance reasons, this ‘best ﬁt’ strategy can also be replaced with a ‘ﬁrst ﬁt’ strategy ∪ { } \ { (cid:48)} which conducts the ﬁrst exchange R ζ ζ found that leads to a reduction of Dl(Pˆn, R). After termination, the reduced distribution Q can be recovered from the reduced set R in the same way as in Section . It follows from Arya et al (2004) that the above algorithm (with either ‘best ﬁt’ or ‘ﬁrst ﬁt’) has an approximation ratio of 5 for the discrete scenario reduction problem for all d. We now show that the algorithm also provides solutions to the continuous scenario reduction problem with an approximation ratio of at most 10. Corollary 1 The problems Dl(Pˆn, m) and Cl(Pˆn, m) are related as follows. 1. rAantiyoaαppgriovxeismraitsieontoaalgnoraipt√hpmroxfoimr aDti2o(nPˆna,lgmor)ituhnmdefrotrhCe 22(-Pˆnnor,mm)wuitnhdaeprptrhoexi2m-naotiromn with approximation ratio 2α. 2. rAantiyoaαppgriovxeismraitsieontoalagonriatphpmrofxoirmDat1io(Pˆnna,lmgo)riuthnmderfoarnyC1n(oPˆrnm, mw)ithunadpeprrotxhiemasatimone norm with approximation ratio 2α. (cid:117)(cid:116) Proof The two statements follow directly from Theorems 5 and 6, respectively. As presented, the local search algorithm is not guaranteed to terminate in polynomial time. This can be remedied by a variant of the algorithm that only accepts exchange−s R ∪ {ζ} \ {ζ(cid:48)} that reduce the Wasserstein distance Dl(Pˆn, R) by at least (cid:15)/((n m)m) for some constant (cid:15) > 0. It follows from Arya et al (2004) that for any (cid:15), this variant terminates in polynomial time and provides a (5 + (cid:15))- approximation for the discrete scenario reduction problem. The algorithm can also be extended to accommodate multiple swaps in every iteration, which lowers the approximation ratio to 3 + (cid:15) at the expense of additional computations. We remark that there is a wealth of algorithms for the k-median problem that can be adapted to the discrete scenario reduction problem. For example, Charikar and Li (2012) present a rounding scheme for the k-median problem that gives rise to a polynomial-time algorithm for D1(Pˆn, R) with an approximation ratio of 3.25. Likewise, Kanungo et al (2004) propose a local search algorithm for the k-median problem that gives rise to a polynomial-time algorithm for D2(Pˆn, R) under the 2-norm with an approximation ratio of 9+(cid:15). In both cases, Corollary 1 allows us to extend these guarantees to the corresponding versions of the continuous scenario reduction problem. 4.3 Mixed-Integer Reformulations of the Discrete and Continuous Scenario Reduction Problems We ﬁrst review a well-known mixed-integer linear programming (MILP) refor- mulation of the discrete scenario reduction problem Dl(Pˆn, m): Theorem 9 The discrete scenario reduction problem can be formulated as the MILP ##LTLine## ##LTLine##

text: Napat

label: author

text: Rujeerapaiboon,

text: Kilian

text: Schindler,

text: Daniel

text: Kuhn,

text: Wolfram

text: Wiesemann

text: For

text: an

text: empirical

text: distribution

text: Pˆn

text: PE(Rd,

text: n),

text: the

text: algorithm

text: constructs

text: a

text: sQe-

text: quence

text: of

text: reduced

text: sets

text: R

text: containing

text: the

text: atoms

text: of

text: the

text: reduced

text: distribution

text: ∈eaRch,

text: tthhee

text: WexachssaenrgsteeiRn

text: pFpo(rPˆnp)erafonrd-

text: mance

text: reasons,

text: this

text: ‘best

text: ﬁt’

text: strategy

text: can

text: also

text: be

text: replaced

text: with

text: a

text: ‘ﬁrst

text: ﬁt’

text: strategy

text: which

text: conducts

text: the

text: ﬁrst

text: exchange

text: R

text: ζ

text: ζ

text: found

text: that

text: leads

text: to

text: a

text: reduction

text: of

text: Dl(Pˆn,

text: R).

text: After

text: termination,

text: the

text: reduced

text: distribution

text: Q

text: can

text: be

text: recovered

text: from

text: the

text: reduced

text: set

text: R

text: in

text: the

text: same

text: way

text: as

text: in

text: Section

text: It

text: follows

text: from

text: Arya

text: et

text: al

text: that

text: the

text: above

text: algorithm

text: (with

text: either

text: ‘best

text: ﬁt’

text: or

text: ‘ﬁrst

text: ﬁt’)

text: has

text: an

text: approximation

text: ratio

text: of

text: for

text: the

text: discrete

text: scenario

text: reduction

text: problem

text: for

text: all

text: d.

text: We

text: now

text: show

text: that

text: the

text: algorithm

text: also

text: provides

text: solutions

text: to

text: the

text: continuous

text: scenario

text: reduction

text: problem

text: with

text: an

text: approximation

text: ratio

text: of

text: at

text: most

text: Corollary

text: The

text: problems

text: Dl(Pˆn,

text: m)

text: and

text: Cl(Pˆn,

text: m)

text: are

text: related

text: as

text: follows.

text: with

text: approximation

text: ratio

text: mw)ithunadpeprrotxhiemasatimone

text: norm

text: with

text: approximation

text: ratio

text: Proof

text: The

text: two

text: statements

text: follow

text: directly

text: from

text: Theorems

text: and

text: respectively.

text: As

text: presented,

text: the

text: local

text: search

text: algorithm

text: is

text: not

text: guaranteed

text: to

text: terminate

text: in

text: polynomial

text: time.

text: This

text: can

text: be

text: remedied

text: by

text: a

text: variant

text: of

text: the

text: algorithm

text: that

text: only

text: accepts

text: exchange−s

text: R

text: {ζ}

text: that

text: reduce

text: the

text: Wasserstein

text: distance

text: Dl(Pˆn,

text: R)

text: by

text: at

text: least

text: m)m)

text: for

text: some

text: constant

text: It

text: follows

text: from

text: Arya

text: et

text: al

text: that

text: for

text: any

text: this

text: variant

text: terminates

text: in

text: polynomial

text: time

text: and

text: provides

text: a

text: approximation

text: for

text: the

text: discrete

text: scenario

text: reduction

text: problem.

text: The

text: algorithm

text: can

text: also

text: be

text: extended

text: to

text: accommodate

text: multiple

text: swaps

text: in

text: every

text: iteration,

text: which

text: lowers

text: the

text: approximation

text: ratio

text: to

text: at

text: the

text: expense

text: of

text: additional

text: computations.

text: We

text: remark

text: that

text: there

text: is

text: a

text: wealth

text: of

text: algorithms

text: for

text: the

text: k-median

text: problem

text: that

text: can

text: be

text: adapted

text: to

text: the

text: discrete

text: scenario

text: reduction

text: problem.

text: For

text: example,

text: Charikar

text: and

text: Li

text: present

text: a

text: rounding

text: scheme

text: for

text: the

text: k-median

text: problem

text: that

text: gives

text: rise

text: to

text: a

text: polynomial-time

text: algorithm

text: for

text: R)

text: with

text: an

text: approximation

text: ratio

text: of

text: Likewise,

text: Kanungo

text: et

text: al

text: propose

text: a

text: local

text: search

text: algorithm

text: for

text: the

text: k-median

text: problem

text: that

text: gives

text: rise

text: to

text: a

text: polynomial-time

text: algorithm

text: for

text: R)

text: under

text: the

text: with

text: an

text: approximation

text: ratio

text: of

text: In

text: both

text: cases,

text: Corollary

text: allows

text: us

text: to

text: extend

text: these

text: guarantees

text: to

text: the

text: corresponding

text: versions

text: of

text: the

text: continuous

text: scenario

text: reduction

text: problem.

text: Mixed-Integer

text: Reformulations

text: of

text: the

text: Discrete

text: and

text: Continuous

text: Scenario

text: Reduction

text: Problems

text: We

text: ﬁrst

text: review

text: a

text: well-known

text: mixed-integer

text: linear

text: programming

text: (MILP)

text: refor-

text: mulation

text: of

text: the

text: discrete

text: scenario

text: reduction

text: problem

text: Dl(Pˆn,

text: m):

text: Theorem

text: The

text: discrete

text: scenario

text: reduction

text: problem

text: can

text: be

text: formulated

text: as

text: the

text: MILP

text: ##LTLine##

text: ##LTLine##


text: Scenario Reduction Revisited: Fundamental Limits and Guarantees 25 Dll(Pˆn, m) = mΠ,iλn n1 (cid:104)Π, D(cid:105) ≤ (cid:62) (cid:62) (21) s.t. Πe = e, Π eλ , λ e = m Π ∈ Rn×n, λ ∈ {0, 1}n, + ∈ S (cid:107) − (cid:107) with D n and d = ξ ξ l encoding the distances among the atoms ij i j in supp(Pˆn). (cid:117)(cid:116) Proof See e.g. Heitsch and R¨omisch (2003). In problem (21), the decision variable π determines how much of the proba- ij bility mass of atom ξi in Pˆn is shifted to the atom ζj in the reduced distribution Q, whereas the decision variabQle λj determi(cid:80)nes whether the atom ξj ∈ supp(Pˆn) is contained in the support of . A solution (Π(cid:63), λ(cid:63)) to problem (21) allows us to recover the reduced distribution via Q = 1 n e(cid:62)Π(cid:63)e · δ . Problem (21) has n j=1 j ξj n binary and n2 continuous variables as well as n2 + n + 1 constraints. We now consider the continuous scenario reduction problem. Due to its bilinear objective function, which involves products of transportation weights π and the (cid:107) − (cid:107) ij distances ξ ζ l containing the continuous decision variables ζ , this problem i j j may not appear to be amenable to a reformulation as a mixed-integer convex optimization problem. We now show that such a reformulation indeed exists. Theorem 10 The continuous scenario reduction problem can be formulated as the mixed-integer convex optimization problem Cll(Pˆn, m) = Π,mc,i{nζj} n1 e(cid:62)c s.t. Πe = e (22) (cid:107) − (cid:107) ≤ − ∀ ∈ ∀ ∈ ξ ζ l c + M(1 π ) i I, j J Πi∈ {0j, 1}n+×mi , c ∈ Rn+, iζj1, . . . , ζm ∈ Rd, where M = maxi,j∈I (cid:107)ξi − ξj(cid:107)l denotes the diameter of the support of Pˆn. In problem (22), the decision variable π determines whether or not the prob- ij ability mass of atom ξi in tQhe empirical distribution Pˆn is shifted to the atom ζj in the reduced distribution , whereas the decision variable c records the cost of i moving the atom ξ under the transportation plan Π. i Proof of Theorem 10 We prove the statement by showing that optimal solutions to problem (22) correspond to feasible solutions in problem (2) with the same objective function value in their respective problems and vice versa. Fix a minimizer (Π(cid:63), c(cid:63), ζ(cid:63), . . . , ζ(cid:63) ) to problem (22), which corresponds to a { } 1 m { ∈ } feasible solution ( I , ζ(cid:63), . . . , ζ(cid:63) ) in problem (2) if we set I = i I : π(cid:63) = 1 ∈ j 1 (cid:107) m− (cid:107) ∈ ∈ j ij for all j J. Note that c(cid:63) = ξ ζ(cid:63) l for j J and i I . Thus, both solutions i i j j adopt the same objective value. { } Conversely, ﬁx a minimizer ( I(cid:63) , ζ(cid:63), . . . , ζ(cid:63) ) to problem (2). This solution j 1 m corresponds to a feasible solution (Π, c, ζ(cid:63), . . . , ζ(cid:63) ) to problem (22) if we set π = 1 ∈ ∈1 m (cid:107) − (cid:107) ij ∈ if i I(cid:63) and π = 0 otherwise for all j J, as well as c = ξ ζ(cid:63) l for all i I ∈j ij i i j (cid:117)(cid:116)j and j J. By construction, both solutions adopt the same objective value. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Scenario

label: title

text: Reduction

text: Revisited:

text: Fundamental

text: Limits

text: and

text: Guarantees

text: Dll(Pˆn,

text: m)

text: mΠ,iλn

text: s.t.

text: Πe

text: e,

text: Π

text: eλ

text: λ

text: e

text: m

text: Π

text: Rn×n,

text: λ

text: S

text: with

text: D

text: n

text: and

text: d

text: ξ

text: ξ

text: l

text: encoding

text: the

text: distances

text: among

text: the

text: atoms

text: ij

text: i

text: j

text: in

text: supp(Pˆn).

text: Proof

text: See

text: e.g.

text: Heitsch

text: and

text: R¨omisch

text: In

text: problem

text: the

text: decision

text: variable

text: π

text: determines

text: how

text: much

text: of

text: the

text: proba-

text: ij

text: bility

text: mass

text: of

text: atom

text: ξi

text: in

text: Pˆn

text: is

text: shifted

text: to

text: the

text: atom

text: ζj

text: in

text: the

text: reduced

text: distribution

text: Q,

text: whereas

text: the

text: decision

text: variabQle

text: λj

text: whether

text: the

text: atom

text: ξj

text: supp(Pˆn)

text: is

text: contained

text: in

text: the

text: support

text: of

text: A

text: solution

text: to

text: problem

text: allows

text: us

text: to

text: recover

text: the

text: reduced

text: distribution

text: via

text: Q

text: n

text: δ

text: Problem

text: has

text: n

text: j

text: ξj

text: n

text: binary

text: and

text: continuous

text: variables

text: as

text: well

text: as

text: n

text: constraints.

text: We

text: now

text: consider

text: the

text: continuous

text: scenario

text: reduction

text: problem.

text: Due

text: to

text: its

text: bilinear

text: objective

text: function,

text: which

text: involves

text: products

text: of

text: transportation

text: weights

text: π

text: and

text: the

text: ij

text: distances

text: ξ

text: ζ

text: l

text: containing

text: the

text: continuous

text: decision

text: variables

text: ζ

text: this

text: problem

text: i

text: j

text: j

text: may

text: not

text: appear

text: to

text: be

text: amenable

text: to

text: a

text: reformulation

text: as

text: a

text: mixed-integer

text: convex

text: optimization

text: problem.

text: We

text: now

text: show

text: that

text: such

text: a

text: reformulation

text: indeed

text: exists.

text: Theorem

text: The

text: continuous

text: scenario

text: reduction

text: problem

text: can

text: be

text: formulated

text: as

text: the

text: mixed-integer

text: convex

text: optimization

text: problem

text: Cll(Pˆn,

text: m)

text: Π,mc,i{nζj}

text: s.t.

text: Πe

text: e

text: ξ

text: ζ

text: l

text: c

text: π

text: i

text: I,

text: j

text: J

text: Πi∈

text: c

text: Rn+,

text: ζm

text: Rd,

text: where

text: M

text: maxi,j∈I

text: denotes

text: the

text: diameter

text: of

text: the

text: support

text: of

text: Pˆn.

text: In

text: problem

text: the

text: decision

text: variable

text: π

text: determines

text: whether

text: or

text: not

text: the

text: prob-

text: ij

text: ability

text: mass

text: of

text: atom

text: ξi

text: in

text: tQhe

text: empirical

text: distribution

text: Pˆn

text: is

text: shifted

text: to

text: the

text: atom

text: ζj

text: in

text: the

text: reduced

text: distribution

text: whereas

text: the

text: decision

text: variable

text: c

text: records

text: the

text: cost

text: of

text: i

text: moving

text: the

text: atom

text: ξ

text: under

text: the

text: transportation

text: plan

text: Π.

text: i

text: Proof

text: of

text: Theorem

text: We

text: prove

text: the

text: statement

text: by

text: showing

text: that

text: optimal

text: solutions

text: to

text: problem

text: correspond

text: to

text: feasible

text: solutions

text: in

text: problem

text: with

text: the

text: same

text: objective

text: function

text: value

text: in

text: their

text: respective

text: problems

text: and

text: vice

text: versa.

text: Fix

text: a

text: minimizer

text: to

text: problem

text: which

text: corresponds

text: to

text: a

text: m

text: feasible

text: solution

text: I

text: in

text: problem

text: if

text: we

text: set

text: I

text: i

text: I

text: j

text: m−

text: j

text: ij

text: for

text: all

text: j

text: J.

text: Note

text: that

text: ξ

text: l

text: for

text: j

text: J

text: and

text: i

text: I

text: Thus,

text: both

text: solutions

text: i

text: i

text: j

text: j

text: adopt

text: the

text: same

text: objective

text: value.

text: Conversely,

text: ﬁx

text: a

text: minimizer

text: to

text: problem

text: This

text: solution

text: j

text: m

text: corresponds

text: to

text: a

text: feasible

text: solution

text: (Π,

text: c,

text: to

text: problem

text: if

text: we

text: set

text: π

text: ∈1

text: m

text: ij

text: if

text: i

text: and

text: π

text: otherwise

text: for

text: all

text: j

text: J,

text: as

text: well

text: as

text: c

text: ξ

text: l

text: for

text: all

text: i

text: I

text: ∈j

text: ij

text: i

text: i

text: j

text: and

text: j

text: J.

text: By

text: construction,

text: both

text: solutions

text: adopt

text: the

text: same

text: objective

text: value.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 30 Napat Rujeerapaiboon, Kilian Schindler, Daniel Kuhn, Wolfram Wiesemann Appendix: Auxiliary Results The proofs of Theorem 2 relies on the following two lemmas. Lemma 1 The semideﬁnite program (6) admits an optimal solution (τ, S) with S = I (cid:62) ∈ R α + β11 for some α, β . Proof Let (τ, S(cid:63)) be any optimal solution to (6), which exists because (6) has a continuous objective function and a compact feasible set, and denote by S the ∈ set of all permutations of I. For any σ S, the permuted solution (τ, Sσ), with (cid:32) (cid:33) sσ = s(cid:63) (cid:88)is also op(cid:88)timal in (6). Note ﬁr(cid:88)st that (τ,(cid:88)Sσ) is feasib(cid:88)le in (6) because ij σ(i)σ(j) ≤ 1 | | − | | τ | | I 2sσ 2 I sσ + sσ + sσ (cid:48) ∈ Ij 2 ∈ (cid:32) j ii j ∈ ik ∈ kk (cid:48)∈ kk (cid:33) j(cid:88)J i(cid:88)Ij k (cid:88)Ij k (cid:88)Ij k,k(cid:54) (cid:88)I(cid:48)j k=k ⇐⇒ ≤ 1 | | − | | τ | | Iσ 2s(cid:63) 2 Iσ s(cid:63) + s(cid:63) + s(cid:63) (cid:48) , j∈J Ijσ 2 i∈Ijσ j ii j k∈Ij ik k∈Ij kk k,k(cid:54) (cid:48)∈I(cid:48)j kk k=k { ∈ } ∈ where the index sets Iσ = σ(i) : i I for j J form an m-set partition j (cid:23) j ≤ ∈ from within P(I, m), and because Sσ 0 and sσ = s(cid:63) 1 for all i I by ii σ(i)σ(i) (cid:80) construction. Moreover, it is clear that (τ, Sσ) and (τ, S(cid:63)) share the same objective ∈ value in (6). Thus, (τ, Sσ) is optimal in (6) for every σ S. The convexity of problem (6) implies that (τ, S) with S = 1 ∈ Sσ is also n! σ S optimal in (6). The claim follows by noting that S is invariant under permutations I (cid:62) ∈ R (cid:117)(cid:116) of the coordinates and thus representable as α + β11 for some α, β . ∈ R I (cid:62) ∈ S Lemma 2 For α, β the eigenvalues of S = α + β11 n are given by α + nβ − (with multiplicity 1) and α (with multiplicity n 1). Proof Note that S is a circulant matrix, meaning that each of its rows coincides with the preceding row rotated by one element to the right. Thus, the eigenvalues − − of S are given by α+β(1+ρ1 +. . . ρn 1), j = 0, . . . , n 1, where ρ = e2πij/n and i j (cid:80) j j denotes the imaginary unit; see e.g. Gray (2006). For j = 0 we then obtain the − − eigenvalue α + nβ, and for j = 1, . . . , n 1 we obtain the other n 1 eigenvalues, − − − (cid:117)(cid:116) all of which equal α because n 1 e2πijk/n = (1 e2πij)/(1 e2πij/n) = 0. k=0 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Napat

text: Rujeerapaiboon,

text: Kilian

text: Schindler,

text: Daniel

text: Kuhn,

text: Wolfram

text: Wiesemann

text: Appendix:

label: section

text: Auxiliary

text: Results

text: The

text: proofs

text: of

text: Theorem

text: relies

text: on

text: the

text: following

text: two

text: lemmas.

text: Lemma

text: The

text: semideﬁnite

text: program

text: admits

text: an

text: optimal

text: solution

text: (τ,

text: S)

text: with

text: S

text: I

text: R

text: α

text: for

text: some

text: α,

text: β

text: Proof

text: Let

text: (τ,

text: be

text: any

text: optimal

text: solution

text: to

text: which

text: exists

text: because

text: has

text: a

text: continuous

text: objective

text: function

text: and

text: a

text: compact

text: feasible

text: set,

text: and

text: denote

text: by

text: S

text: the

text: set

text: of

text: all

text: permutations

text: of

text: I.

text: For

text: any

text: σ

text: S,

text: the

text: permuted

text: solution

text: (τ,

text: Sσ),

text: with

text: sσ

text: also

text: in

text: Note

text: that

text: is

text: in

text: because

text: ij

text: σ(i)σ(j)

text: τ

text: I

text: I

text: sσ

text: sσ

text: sσ

text: Ij

text: j

text: ii

text: j

text: ik

text: kk

text: kk

text: k

text: k

text: k=k

text: τ

text: Iσ

text: Iσ

text: j∈J

text: Ijσ

text: i∈Ijσ

text: j

text: ii

text: j

text: k∈Ij

text: ik

text: k∈Ij

text: kk

text: kk

text: k=k

text: where

text: the

text: index

text: sets

text: Iσ

text: σ(i)

text: i

text: I

text: for

text: j

text: J

text: form

text: an

text: m-set

text: partition

text: j

text: j

text: from

text: within

text: P(I,

text: m),

text: and

text: because

text: Sσ

text: and

text: sσ

text: for

text: all

text: i

text: I

text: by

text: ii

text: σ(i)σ(i)

text: construction.

text: Moreover,

text: it

text: is

text: clear

text: that

text: (τ,

text: Sσ)

text: and

text: (τ,

text: share

text: the

text: same

text: objective

text: value

text: in

text: Thus,

text: (τ,

text: Sσ)

text: is

text: optimal

text: in

text: for

text: every

text: σ

text: S.

text: The

text: convexity

text: of

text: problem

text: implies

text: that

text: (τ,

text: S)

text: with

text: S

text: Sσ

text: is

text: also

text: n!

text: σ

text: S

text: optimal

text: in

text: The

text: claim

text: follows

text: by

text: noting

text: that

text: S

text: is

text: invariant

text: under

text: permutations

text: I

text: R

text: of

text: the

text: coordinates

text: and

text: thus

text: representable

text: as

text: α

text: for

text: some

text: α,

text: β

text: R

text: I

text: S

text: Lemma

text: For

text: α,

text: β

text: the

text: eigenvalues

text: of

text: S

text: α

text: n

text: are

text: given

text: by

text: α

text: nβ

text: (with

text: multiplicity

text: and

text: α

text: (with

text: multiplicity

text: n

text: Proof

text: Note

text: that

text: S

text: is

text: a

text: circulant

text: matrix,

text: meaning

text: that

text: each

text: of

text: its

text: rows

text: coincides

text: with

text: the

text: preceding

text: row

text: rotated

text: by

text: one

text: element

text: to

text: the

text: right.

text: Thus,

text: the

text: eigenvalues

text: of

text: S

text: are

text: given

text: by

text: ρn

text: j

text: n

text: where

text: ρ

text: and

text: i

text: j

text: j

text: j

text: denotes

text: the

text: imaginary

text: unit;

text: see

text: e.g.

text: Gray

text: For

text: j

text: we

text: then

text: obtain

text: the

text: eigenvalue

text: α

text: nβ,

text: and

text: for

text: j

text: n

text: we

text: obtain

text: the

text: other

text: n

text: eigenvalues,

text: all

text: of

text: which

text: equal

text: α

text: because

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Scenario Reduction Revisited: Fundamental Limits and Guarantees 7 ∈ ∈ For each j J and i I , the squared norm in the ﬁrst constraint of (4) can be exp(cid:13)(cid:13)ressed in term(cid:13)(cid:13)s of the injn(cid:13)(cid:13)(cid:13)er produc(cid:88)ts betw(cid:13)(cid:13)(cid:13)een pairs of empirical observations: ξ −mean(I )(cid:32)2 = | 1| |I |ξ − ξ 2 (cid:33) i j 2 Ij 2 j i(cid:88)∈ k 2 (cid:88) (cid:88) k Ij 1 | | (cid:62) − | | (cid:62) (cid:62) (cid:62) = | | I 2ξ ξ 2 I ξ ξ + ξ ξ + ξ ξ (cid:48) . Ij 2 j i i j ∈ i k ∈ k k (cid:48)∈ k k k Ij k Ij k,k(cid:54) I(cid:48)j k=k Introducing the Gram matrix S = [ξ1, . . . , ξn](cid:62)[ξ1, . . . , ξn] ∈ Sn, S (cid:23) 0 and rank(S) ≤ min{n, d} (5) (cid:32) (cid:33) then allo(cid:88)ws us to (cid:88)simplify the ﬁrst con(cid:88)straint in(cid:88)(4) to (cid:88) ≤ 1 | | − | | τ | | I 2s 2 I s + s + s (cid:48) . ∈ Ij 2 ∈ j ii j ∈ ik ∈ kk (cid:48)∈ kk j J i Ij k Ij k Ij k,k(cid:54) I(cid:48)j k=k ≤ Note that the second constraint in problem (4) can now be expressed as s 1, ii and hence all constraints in (4) are linear in the Gram matrix S. Our discussion implies that we obtain an upper bound on C (n, m) by refor- 2 mulating problem (4) as a semideﬁnite program in terms of the Gram matrix S (cid:32) (cid:33) 1 (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) τ∈Rm,aSx∈Sn nτ ≤ 1 | | − | | s.t. τ | | I 2s 2 I s + s + s (cid:48) ∈ Ij 2 ∈ j ii j ∈ ik ∈ kk (cid:48)∈ kk j J i Ij k Ij k Ij k,k(cid:54) I(cid:48)j ∀{ }k∈=k I P(I, m) j (cid:23) ≤ ∀ ∈ S 0, s 1 i I, ii (6) where we have relaxed the rank condition in the deﬁnition of the Gram ma- trix (5). Lemma 1 in the appendix shows that (6) has an optimal solution (τ(cid:63), S(cid:63)) I (cid:62) ∈ R that satisﬁes S(cid:63) = α + βee for some α, β . Moreover, Lemma 2 in the ap- I (cid:62) pendix shows that any matrix of the form S = α +β11 is positive semideﬁnite if ≥ ≥ and only if α 0 and α + nβ 0. We thus conclude that (6) can be reformulated as 1 τ,mα,aβx∈R n≤τ − ≤ (7) s.t. τ (n m)α, α + β 1 ≥ ≥ α 0, α + nβ 0, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Scenario

text: Reduction

text: Revisited:

text: Fundamental

text: Limits

text: and

text: Guarantees

text: For

text: each

text: j

text: J

text: and

text: i

text: I

text: the

text: squared

text: norm

text: in

text: the

text: ﬁrst

text: constraint

text: of

text: can

text: be

text: in

text: of

text: the

text: pairs

text: of

text: empirical

text: observations:

text: ξ

text: −mean(I

text: |I

text: |ξ

text: ξ

text: i

text: j

text: Ij

text: j

text: k

text: k

text: Ij

text: I

text: 2ξ

text: ξ

text: I

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: ξ

text: Ij

text: j

text: i

text: i

text: j

text: i

text: k

text: k

text: k

text: k

text: k

text: k

text: Ij

text: k

text: Ij

text: k=k

text: Introducing

text: the

text: Gram

text: matrix

text: S

text: ξn]

text: Sn,

text: S

text: and

text: rank(S)

text: min{n,

text: d}

text: then

text: us

text: to

text: the

text: ﬁrst

text: to

text: τ

text: I

text: I

text: s

text: s

text: s

text: Ij

text: j

text: ii

text: j

text: ik

text: kk

text: kk

text: j

text: J

text: i

text: Ij

text: k

text: Ij

text: k

text: Ij

text: k=k

text: Note

text: that

text: the

text: second

text: constraint

text: in

text: problem

text: can

text: now

text: be

text: expressed

text: as

text: s

text: ii

text: and

text: hence

text: all

text: constraints

text: in

text: are

text: linear

text: in

text: the

text: Gram

text: matrix

text: S.

text: Our

text: discussion

text: implies

text: that

text: we

text: obtain

text: an

text: upper

text: bound

text: on

text: C

text: (n,

text: m)

text: by

text: refor-

text: mulating

text: problem

text: as

text: a

text: semideﬁnite

text: program

text: in

text: terms

text: of

text: the

text: Gram

text: matrix

text: S

text: τ∈Rm,aSx∈Sn

text: nτ

text: s.t.

text: τ

text: I

text: I

text: s

text: s

text: s

text: Ij

text: j

text: ii

text: j

text: ik

text: kk

text: kk

text: j

text: J

text: i

text: Ij

text: k

text: Ij

text: k

text: Ij

text: }k∈=k

text: I

text: P(I,

text: m)

text: j

text: S

text: s

text: i

text: I,

text: ii

text: where

text: we

text: have

text: relaxed

text: the

text: rank

text: condition

text: in

text: the

text: deﬁnition

text: of

text: the

text: Gram

text: ma-

text: trix

text: Lemma

text: in

text: the

text: appendix

text: shows

text: that

text: has

text: an

text: optimal

text: solution

text: I

text: R

text: that

text: satisﬁes

text: α

text: βee

text: for

text: some

text: α,

text: β

text: Moreover,

text: Lemma

text: in

text: the

text: ap-

text: I

text: pendix

text: shows

text: that

text: any

text: matrix

text: of

text: the

text: form

text: S

text: α

text: is

text: positive

text: semideﬁnite

text: if

text: and

text: only

text: if

text: α

text: and

text: α

text: nβ

text: We

text: thus

text: conclude

text: that

text: can

text: be

text: reformulated

text: as

text: τ,mα,aβx∈R

text: n≤τ

text: s.t.

text: τ

text: (n

text: m)α,

text: α

text: β

text: α

text: α

text: nβ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##