[PPT] - Pension fund ALM with Multivariate Second order Stochastic PowerPoint Presentation

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Sebastiano Vitali, Vittorio Moriggia, Miloš Kopa

University of Bergamo Charles University

Pension fund ALM with Multivariate Second order Stochastic Dominance constraints

Chemnitz, CMS2019

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Purpose of the work

To model and implement an Asset Liability Management problem of a Pension Fund in a Defined Benefit framework having:

a short-term profitability target,
a medium-term insurance risk-adjusted return
a long-term strategic objective

The multivariate Second order Stochastic Dominance (SSD) is formulated with three alternatives which we investigate Definition of multivariate second order stochastic dominance between the wealth of the Pension Fund and a benchmark wealth

2

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The multivariate SSD 3

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Univariate SSD

4

Let (Ω, 𝐺, 𝑄) denote a probability space and let 𝑌 and 𝑍 be two random variables having as cumulative distribution functions 𝐺

𝑌 and

𝐺

𝑍.

Let’s define the twice cumulative distribution function as 𝐺

𝑌 2 (𝜃) = න −∞ 𝜃

𝐺

𝑌 𝛽 d𝛽

We say that 𝑌 dominates 𝑍 in the Second order Stochastic Dominance (SSD) sense, 𝑌 ≻𝑇𝑇𝐸 𝑍, if 𝐺

𝑌 2 𝜃 ≤ 𝐺 𝑍 2 𝜃 ,

∀𝜃 ∈ 𝑆 If the random variables are discrete and then represented by random vectors X and Y, and if the realizations are equiprobable, then the SSD is equivalent to X ≤ WY where W is a double stochastic matrix.

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Multivariate SSD

5

When we consider multivariate random variables, we need to re-think the SSD relation. Assume that a multivariate random variable 𝒀 has 𝑈 dimensions and then we observe 𝑌𝑢, 𝑢 = 1, … , 𝑈 univariate random variables. If the random variable is discrete, each univariate random variable 𝑌𝑢 can be represented with with a vector X𝑢, then 𝒀 can be represented with a matrix having 𝑈 columns, one for each dimension: X1 … X𝑈 = 𝑦1,1 … 𝑦1,𝑈 … … … 𝑦𝑇,1 … 𝑦𝑇,𝑈 The meaning of 𝒀 ≻𝑇𝑇𝐸 𝒁 is not unique and can be declined in various ways. We analyze three of them.

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Component-wise Multivariate SSD (C-MSSD)

6 ≻𝑇𝑇𝐸1 ≻𝑇𝑇𝐸2 ≻𝑇𝑇𝐸3

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸𝑢 𝑍

𝑢

∀𝑢 (disjointly)

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Linear Multivariate SSD (Lin-MSSD)

7

ത 𝑌 = ෍

𝑢=1 𝑈

𝑌𝑢 ⋅ 𝑑𝑢, ∀𝑑𝑢≥ 0, ෍

𝑢=1 𝑈

𝑑𝑢 = 1

≻𝑇𝑇𝐸 ∀𝑑𝑢

𝑌 ≻𝑇𝑇𝐸 lin 𝑍 iff σ𝑢=1

𝑈

𝑑𝑢 𝑌𝑢 ≻𝑇𝑇𝐸 σ𝑢=1

𝑈

𝑑𝑢 𝑍

𝑢, ∀𝑑𝑢 ≥ 0| σ𝑢=1 𝑈

𝑑𝑢 = 1

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MultiDimension Multivariate SSD (MD-MSSD) 8

≻𝑇𝑇𝐸

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸 𝑍

𝑢

∀𝑢 (jointly)

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Multivariate SSD

9

The three possible definitions:

The Component-wise Multivariate SSD (C-MSSD):

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸𝑢 𝑍

𝑢

∀𝑢 (disjointly) X𝑢 ≤ W𝑢Y𝑢, ∀𝑢

The Linear Multivariate SSD (Lin-MSSD):

Dencheva and Ruszczynski (2009), Dentcheva and Wolfhagen (2015, 2016)

𝑌 ≻𝑇𝑇𝐸 lin 𝑍 iff σ𝑢=1

𝑈

𝑑𝑢 𝑌𝑢 ≻𝑇𝑇𝐸 σ𝑢=1

𝑈

𝑑𝑢 𝑍

𝑢, ∀𝑑𝑢 ≥ 0| σ𝑢=1 𝑈

𝑑𝑢 = 1 𝐝⊤X ≤ W(𝐝)𝐝⊤Y, ∀𝐝 ≥ 0, Σ𝑢=1

𝑈

𝑑𝑢 = 1

The MultiDimension Multivariate SSD (MD-MSSD):

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸 𝑍

𝑢

∀𝑢 (jointly) X𝑢 ≤ WY𝑢, a ∀𝑢

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Multivariate SSD

10

The three possible definitions:

The Component-wise Multivariate SSD (C-MSSD):

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸𝑢 𝑍

𝑢

∀𝑢 (disjointly) X𝑢 ≤ W𝑢Y𝑢, ∀𝑢

The Linear Multivariate SSD (Lin-MSSD):

Dencheva and Ruszczynski (2009), Dentcheva and Wolfhagen (2015, 2016)

𝑌 ≻𝑇𝑇𝐸 lin 𝑍 iff σ𝑢=1

𝑈

𝑑𝑢 𝑌𝑢 ≻𝑇𝑇𝐸 σ𝑢=1

𝑈

𝑑𝑢 𝑍

𝑢, ∀𝑑𝑢 ≥ 0| σ𝑢=1 𝑈

𝑑𝑢 = 1 𝐝⊤X ≤ W(𝐝)𝐝⊤Y, ∀𝐝 ≥ 0, Σ𝑢=1

𝑈

𝑑𝑢 = 1

The MultiDimension Multivariate SSD (MD-MSSD):

𝑌 ≻𝑇𝑇𝐸 𝑍 iff 𝑌𝑢 ≻𝑇𝑇𝐸 𝑍

𝑢

∀𝑢 (jointly) X𝑢 ≤ WY𝑢, a ∀𝑢 C-MSSD Lin-MSSD MD-MSSD

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The ALM model 11

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Approach structure

Financial Datafeed

Portfolio Universe
Risk factors

Simulation input

Econometric model definition
Econometric model estimation
Population model setting
Stochastic tree structure

Monte Carlo simulator

Nodal financial coefficient generation
Monte Carlo scenario generation
Population actuarial simulation

Stochastic Programming

Dynamic portfolio model
Stochastic program solution

Solution analysis

12

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Extended Asset Universe

Treasury 1-3y Treasury 3-5y Treasury 5-7y Treasury 7-10y Securitized Corporate Inv Grade Corporate High Yield Public Equity

Cash Treasuries Securitized Corporate Public Equity Real Estate

Real Estate

0% 0% 0% 0% 0% 0% 30% 100% 100% 100% 50% 20%

Asset Class Asset List Lower & Upper

13

Treasury 10+y Cash Floaters

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Notation for sets

Time partition from time 0 to year 20 Set of decision times Set of intermediate stages Set of scenarios Financial assets Bond asset classes Asset total set Public Equity, Real Estate and Derivatives

14

T = {𝑢0 = 0,1,2, … , 𝐼} } T𝑒 = {𝑢0 = 0,1,2,3, 5, 10, 𝐼 ൟ T𝑗𝑜𝑢 = T\{Td} = {4,6,7,8, 9, 11, … , 19 } 𝑡 = {1,2, … , 𝑇 𝑗 ∈ {1,2, … , 14} 𝐽1: 𝑗 = 1,2, 3, 4, 5 , 𝐽2: 𝑗 = {6,7,8} 𝐽3: 𝑗 = 9 , 𝐽4: 𝑗 = {10}, 𝐽5: 𝑗 = {11,12,13,14} 𝐽 = ራ

𝑙=1,…,5

𝐽𝑙

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Notation for investment variables

Buying decision in stage t, scenario s, of asset i Selling in stage t, scenario s, of asset i that was bought in h Expiry of a fixed-income asset in stage t, scenario s, of asset i that was bought in h Holding in stage t, scenario s, of asset i that was bought in h Cash account in stage t, scenario s

15

𝑦𝑗,𝑢,𝑡

+

𝑦𝑗,ℎ,𝑢,𝑡

−

𝑦𝑗,ℎ,𝑢,𝑡

𝑓𝑦𝑞

𝑦𝑗,ℎ,𝑢,𝑡 𝑨𝑢,𝑡 = 𝑨𝑢,𝑡

+ − 𝑨𝑢,𝑡 −

Sponsors’ unexpected contributions Φ𝑢,𝑡

𝑙

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Variable definitions

Net pension payments Defined benefit obligation (DBO) Asset value Asset portfolio value Net Defined benefit obligation Intermediate net payments

16

𝑗𝑜𝑞𝑣𝑢 𝑗𝑜𝑞𝑣𝑢 𝑌𝑗,𝑢𝑘,𝑡 = ෍

ℎ<𝑢𝑘

𝑦𝑗,ℎ,𝑢𝑘,𝑡 𝐷𝑌𝑢𝑘,𝑡 = ෍

𝑗∈𝐽

𝑌𝑗,𝑢𝑘,𝑡 +𝑨𝑢𝑘,𝑡

+

B𝑢𝑘,𝑡 = Λ𝑢𝑘,𝑡 − 𝐷𝑌𝑢𝑘,𝑡 𝑀𝑢𝑘,𝑡

𝑎

= ෍

ℎ<𝑢𝑘,ℎ𝜗𝑈

𝑦𝑗,ℎ,𝑢𝑘−1 ∙ 𝜊𝑗,𝑢𝑘,𝑡 + ෍

ℎ<𝑢𝑘,𝑢𝑘−ℎ≥𝑈𝑗

𝑦𝑗,ℎ,𝑢𝑘,𝑡

𝑓𝑦𝑞

−𝑀𝑢𝑘,𝑡

𝑂𝐹𝑈

𝑀𝑢,𝑡

𝑂𝐹𝑈

Λ𝑢,𝑡 𝑌𝑗,𝑢,𝑡 𝐷𝑌𝑢𝑘,𝑡 B𝑢𝑘,𝑡 𝑀𝑢𝑘,𝑡

𝑎

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Variable definitions

Liquidity gap

17

𝛻𝑢𝑘,𝑡 = L𝑢𝑘,𝑡

𝑂𝐹𝑈 −

෍

𝑢𝑘−1<ℎ<𝑢𝑘

𝑀ℎ,𝑡

𝑨

1 + 𝜂𝑢,𝑡 −Π𝑢𝑘,𝑡

1,𝐽𝑂𝑊 −

෍

ℎ<𝑢𝑘,𝑢𝑘−ℎ=𝑈𝑗

𝑦𝑗,ℎ,𝑢𝑘,𝑡

𝑓𝑦𝑞

𝛻𝑢𝑘,𝑡 𝛺𝑢𝑘,𝑡 = 𝛻𝑢𝑘,𝑡 + 𝐿𝑢𝑘,𝑡

1

+ 𝛺𝑢𝑘−1,𝑡, Ψ𝑢0,𝑡 = 0 Liquidity gap plus ALM risk 𝛺𝑢𝑘,𝑡 ALM risk K𝑢𝑘,𝑡

1

= 𝑒𝑠+ ∙ tj − tj−1 ⋅ Δ𝑦𝑢𝑘,𝑡 − ΔΛ𝑢𝑘,𝑡

+

− 𝑒𝑠− ∙ tj − tj−1 ⋅ Δ𝑦𝑢𝑘,𝑡 − ΔΛ𝑢𝑘,𝑡

−

K𝑢𝑘,𝑡

1

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Variable definitions

Realized portfolio return Coupon return Capital gain return Total portfolio return Unrealized gain and losses

18

Π𝑢𝑘,𝑡

𝐽𝑂𝑊 = Π𝑢𝑘,𝑡 1,𝐽𝑂𝑊 + 𝐻𝑢𝑘,𝑡

… 𝑉𝐻𝑀𝑢𝑘,𝑡 = ෍

𝑗∈𝐽

෍

ℎ<𝑢𝑘,ℎ𝜗 ෠ 𝑈

𝑦𝑗,ℎ,𝑢𝑘,𝑡 ∙ 𝜓𝑗,ℎ,𝑢𝑘,𝑡 Π𝑢𝑘,𝑡

𝐽𝑂𝑊

Π𝑢𝑘,𝑡

1,𝐽𝑂𝑊

𝐻𝑢𝑘,𝑡 Π𝑢𝑘,𝑡 𝑉𝐻𝑀𝑢𝑘,𝑡 … Π𝑢𝑘,𝑡 = Π𝑢𝑘,𝑡

𝐽𝑂𝑊 + 𝑉𝐻𝑀𝑢𝑘,𝑡 − 𝑉𝐻𝑀𝑢0,𝑡

Cumulated realized portfolio return Π𝑢,𝑡

𝐽𝑂𝑊,𝑑𝑣𝑛

Π𝑢,𝑡

𝐽𝑂𝑊,𝑑𝑣𝑛 = ෍ 𝑢𝑙≤𝑢𝑘

Π𝑢𝑙,𝑡

𝐽𝑂𝑊

Cumulated total portfolio return Π𝑢𝑘,𝑡

𝑑𝑣𝑛

Π𝑢𝑘,𝑡

𝑑𝑣𝑛 = Π𝑢,𝑡 𝐽𝑂𝑊,𝑑𝑣𝑛 + 𝑉𝐻𝑀𝑢𝑘,𝑡 − 𝑉𝐻𝑀𝑢0,𝑡

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Variable definitions

Total risk capital Actuarial risk capital Investment risk capital Market risk

19

𝐿𝑢𝑘,𝑡 = 𝐿𝑢𝑘,𝑡

𝑈𝐹𝐷 + 𝐿𝑢𝑘,𝑡 𝐽𝑂𝑊

𝐿𝑢𝑘,𝑡

𝑈𝐹𝐷 = 𝜚 ⋅ 𝛭𝑢,𝑡

𝐿𝑢𝑘,𝑡 𝐿𝑢𝑘,𝑡

𝑈𝐹𝐷

K𝑢𝑘,𝑡

𝐽𝑂𝑊

K𝑢𝑘,𝑡

𝑁

K𝑢𝑘,𝑡

𝐽𝑂𝑊 = K𝑢𝑘,𝑡 1

+ K𝑢𝑘,𝑡

𝑁 +K𝑢𝑘−1,𝑡 𝐽𝑂𝑊

K𝑢𝑘,𝑡

𝑁

= ෍

𝑜=2,…12

෍

ℎ<𝑢𝑘,ℎ∈𝑈𝑒

𝑦𝑜,ℎ,𝑢𝑘,𝑡 ⋅ 𝑙𝑜 ∙ 𝑢𝑘 − 𝑢𝑘−1 ALM risk K𝑢𝑘,𝑡

1

= 𝑒𝑠+ ∙ tj − tj−1 ⋅ Δ𝑦𝑢𝑘,𝑡 − ΔΛ𝑢𝑘,𝑡

+

− 𝑒𝑠− ∙ tj − tj−1 ⋅ Δ𝑦𝑢𝑘,𝑡 − ΔΛ𝑢𝑘,𝑡

−

K𝑢𝑘,𝑡

1

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Variable definitions

Total portfolio return per unit tail risk

20

𝑎𝑢𝑘,𝑡 = Π𝑢𝑘,𝑡

𝑑𝑣𝑛

K𝑢𝑘,𝑡

𝐽𝑂𝑊 + Φ𝑢𝑘,𝑡 𝑙

𝑎𝑢𝑘,𝑡 Total extraordinary plan sponsors’ contributions 𝛸𝑢𝑘,𝑡 = 𝛸𝑢𝑘,𝑡

𝑙

+ 𝛸𝑢𝑘−1,𝑡 𝛸𝑢𝑘,𝑡

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Other constraints

Inventory balance constraints at time 𝑢0 = 0, root node

21

Inventory balance constraints at time 𝑢𝑘 Cash balance constraints at time 𝑢0 = 0, root node Cash balance constraints at time 𝑢𝑘 Single asset lower bound Asset class upper bound Turnover constraint Liquidity constraint Maximum risk exposure Single asset upper bound Asset class lower bound … … … …

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Long-term sustainability Short-term profitability Industrial plan target

MAX

bjective function

MIN

Expected Value Expected Shortfall

Objective formulation

ALM Risk Net DBO RORAC Sponsor Injection Operating Profit Liquidity Gap

Liquidity Gap + ALM Risk

RORAC Sponsor Injection Net DBO

10% 30% 40% 20% H&N

1 − 𝛽 ෍ 𝜇𝑘 𝔽 𝑍

𝑘,𝑢𝑘 − 𝛽 ෍ 𝜇𝑘 𝔽 ෩

𝑍

𝑘 − 𝑍 𝑘,𝑢𝑘 𝑍 𝑘,𝑢𝑘 < ෩

𝑍

𝑘

22

Ψ𝑢,𝑡 K𝑢,𝑡

1

𝛻𝑢,𝑡 Z𝑢,𝑡 B𝑢,𝑡 𝛸𝑢,𝑡

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Dynamic Asset Allocation

Existent Portfolio H&N Optimal Solution

Real Estate Public Equity Corporates Securitized Treasuries Cash

Cutting-edge stochastic

ptimization framework

Dynamic Optimal Solution

23

Benchmark portfolio

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Results– Benchmark

32

SSD 7 C-MSSD 6-7 C-MSSD 5-7 MD-MSSD 6-7 MD-MSSD 5-7

248.066 243,017 9,190 1,644 Mean StdDev V@R AV@R ObjVal Time

no SSD

242,783 239,930 4,675 555

31,950

166

31,973

232 551,419 441,119 120,375 79,387

59,747

421 1,014,506 690,723 314,693 254,560

123,889

335 551,426 441,025 120,373 79,455

59.748

1,622 1,014,435 690,490 314,620 254,689

123,890

1,956

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Results – Benchmark 2

33

SSD 7 C-MSSD 6-7 C-MSSD 5-7 MD-MSSD 6-7 MD-MSSD 5-7

496,286 545,782 53,442 27,055 Mean StdDev V@R AV@R ObjVal Time

no SSD

242,783 239,930 4,675 555

31,950

166

47,892

771 1,310,899 897,851 377,758 302,775

163,579

859 2,398,575 1,412,045 867,990 760,485

350,516

650 1,310,899 897,851 377,758 302,775

163,579

2391 2,400,063 1,413,601 867,993 760,059

350,520

1001

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Results – Benchmark 3

34

SSD 7 C-MSSD 6-7 C-MSSD 5-7 MD-MSSD 6-7 MD-MSSD 5-7

745,815 848,462 95,059 55,136 Mean StdDev V@R AV@R ObjVal Time

no SSD

242,783 239,930 4,675 555

31,950

166

72,798

1,416 2,059,915 1,329,457 644,790 485,869

283,462

458 3,673,643 2,084,549 1,346,306 1,111,726

623,047

387 2,059,912 1,329,444 644,790 485,882

283,463

1097 3,673,642 2,084,574 1,346,306 1,111,726

623,047

1380

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Summary table

35

SSD 7 C 6-7 C 5-7 MD 6-7 MD 5-7

Benchmark Benchmark 2 Benchmark 3

SSD 7 C 6-7 C 5-7 MD 6-7 MD 5-7 SSD 7 C 6-7 C 5-7 MD 6-7 MD 5-7 no SSD

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Conclusions

The alternative versions of the Multivariate SSD are very close to each others The MD-MSSD is a stronger condition and its meaning is more clear and reasonable

36

The MD-MSSD is more computational demanding, but still tractable

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Bibliography – ALM Stochastic Programming 37

Kopa M., Moriggia V., Vitali S. (2018). Individual optimal pension allocation under stochastic dominance constraints, Annals of Operations Research, 260(1-2), pp. 255-291, DOI 10.1007/s10479-016-2387-x Vitali S., Moriggia V. and Kopa M. (2017). Optimal pension fund composition for an Italian private pension plan sponsor, Computational Management Science, 14(1), pp. 135- 160, DOI: 10.1007/s10287-016-0263-4 Consigli G., Moriggia V., Benincasa E., Landoni G., Petronio F., Vitali S., di Tria M., Skoric M., Uristani A., 2018. Optimal multistage defined-benefit pension fund management. In: Recent Advances in Commmodity and Financial Modeling: Quantitative methods in Banking, Finance, Insurance, Energy and Commodity markets. (Consigli, G., Stefani, S., Zambruno, G. Eds.). Springers International Series in Operations Research and Management Science. Moriggia V., Kopa M., Vitali S. (2018). Pension fund management with hedging derivatives, stochastic dominance and nodal contamination, Omega, DOI 10.1016/j.omega.2018.08.011

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Bibliography – Stochastic Dominance 38

Post T., Kopa M. (2013). General Linear Formulations of Stochastic Dominance Criteria, European Journal of Operational Research, 230, 2, 321-332 Kuosmanen T. (2004). Efficient diversification according to stochastic dominance

criteria. Management Science, 50(10):1390-1406

Post T., Kopa M. (2016). Portfolio choice based on third-degree stochastic dominance, Management Science, 63(10):3381-3392 Branda M., and Kopa M. (2016). DEA models equivalent to general Nth order stochastic dominance efficiency tests. Operations Research Letters, 44(2): 285-289. Dentcheva D., Wolfhagen E. (2015). Optimization with multivariate stochastic dominance constraints, SIAM Journal on Optimization, 25(1), 564-588 Dentcheva D., Ruszczynski A. (2009). Optimization with multivariate stochastic dominance constraints, Math. Program. Ser. B, 117(1), 111-127

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