Lecture 3. Stochastic-Equilibrium in Network Expansion ...

Lecture 3. Stochastic-Equilibrium in NetworkExpansion Planning under Uncertainty,

SE-NEP

Laureano F. EscuderoUniversidad Rey Juan Carlos (URJC), Mostoles (Madrid),

[email protected]

Laureano F. Escudero Universidad Rey Juan Carlos (URJC), Mostoles (Madrid), [email protected]

Program. Course on Stochastic-Equilibrium in NetworkExpansion Planning under uncertainty, SE-NEP

Lecture 1. Introduction to Network Expansion Planning.Deterministic version.Lecture 2. Stochastic multistage NEP: Strategic scenariotrees, multi-period Tactical scenario graphs andOperational two-stage trees. Scenario reduction. RiskNeutral.Lecture 3. Stochastic-Equilibrium in NetworkExpansion Planning under Uncertainty, SE-NEP.Lecture 4. Risk averse policies in stochastic optimization.Lecture 5. Matheuristic Nested Stochastic Decomposition.


Outline

Aim and ScopeMath Optim under uncertaintyPilot case: An extension of the classical static Bilevel TollAssignment Problem (TAP)Single-level equivalent multi-period stochasticequilibrium-based optimization model for networkexpansion planning. Primal-dual modelComputational experienceConclusions


Stochastic Optimization. Some elements

A period of a given time horizon is a time unit where therealization of the uncertain parameters takes place.

A scenario is a realization of the uncertain parameters alongthe periods of a given time horizon.

A scenario group for a given period is the set of scenarios withthe same realization of the uncertain parameters up to theperiod:Non-anticipativity principle should be satisfied.

A scenario group has one-to-one correspondence with a nodein the scenario tree.


t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

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33

T = 1, . . . , 7

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

N = 1, . . . , 33

N6 = 10 . . . , 17

Ω7 = 22, . . . , 25

Figura: Multi-period scenario tree


Scenario tree notation

T , set of the time periods along the horizon, T = |T |.Ω, set of scenarios.N , set of nodes in the tree.

Ωn ⊆ Ω, set of scenarios in a group with one-to-onecorrespondence with node n, for n ∈ N .

Nt ⊂ N , set of nodes that belong to period t , for t ∈ T .An, set including node n and its ancestor nodes, for n ∈ N .Sn, set including its successor nodes to node n, for n ∈ N .

Note that Sn = ∅ for n ∈ NT .Sn

1 ⊆ Sn, immediate successor nodes to node n, for n ∈ N .σn, immediate ancestor node of node n, for n ∈ N . Note:

σn =null, for n ∈ N1.tn, period to which node n belong to, for n ∈ N .

wω, probability assigned to scenario ω ∈ Ω.wn =

∑ω∈Ωn wω, weight or probability of node n ∈ N .


Multiperiod Mixed 0-1 DEMfor Network Expansion Planning (NEP)

z = max∑n∈N

wn(anxn + bnyn)

s.t. xn ∈ 0,1nx(n), xσn ≤ xn ∀n ∈ N

Anxn + Bnyn = hn ∀n ∈ Nyn ∈ Y n ∀n ∈ N ,

where xn is a vector included by the so-named 0-1 step vars,and Y n is a feasible set of mixed 0-1 vectors.

Let xn, yn: Value vectors of the vectors xn, yn, resp., in a (full orpartial) solution, if any, for n ∈ N .


Stochastic Equilibrium-based NEP optimization. TAP:Sets

N , nodes in the network.

K, commodities to transport.

A1, potential arcs in the network nodes by its own means.

A2, arcs in the network nodes by alternative means.

A = A1 ∪ A2, arcs in the network.

Γn+ ⊂ A, arcs from node n in the network, for n ∈ N .

Γn− ⊂ A, arcs to node n in the network, for n ∈ N .


Stochastic Equilibrium-based NEP optimization. TAP:Deterministic parameters

θa, upper bound on the tariff that is allowed for commoditytransporting through network arc a in any period, fora ∈ A1.

Ca, maintenance cost of network arc a, provided that it isavailable, for a ∈ A1.

ok , dk , origin and destination nodes for commodity ktransportation, resp., through the network nodeset N , fork ∈ K.


Stochastic Equilibrium-based NEP optimization. TAP:Stochastic parameters for scenario tree node n ∈ N

Una , investment cost for building network arc a, for a ∈ A1.

Un, available budget for network expansion planning.

V nk , commodity k volume to be transported for origin to

destination, for k ∈ K.

Bna , commodity transport unit cost through non-network arc a,

for a ∈ A2.


Stochastic Equilibrium-based NEP optimization. TAP:Variables xn

a in the upper-level model for scenarionode n, for n ∈ N

xna , 0-1 step variable whose value 1 means that arc a is part of

the network by node n (and, then, by period tn) andotherwise, 0, for a ∈ A1.

xna ∈ 0,1, xσ

n

a ≤ xna

step variables help to tightening the model in a strongermanner than their counterparts, so-named impulse ones.

xna = 1: arc a is made available by scenario node n, i.e.,

either at the node or at any of its ancestors in the scenariotree, such that:for the former case, xn

a − xσn

a = 1 and,for the latter one, xn

a = xσn

a = 1.


Stochastic Equilibrium-based NEP optimization. TAP:Variables θn

a in the upper-level model for scenario noden, for n ∈ N

θna , unit toll price (i.e., tariff) for commodity transport through

network arc a, for a ∈ A1.

0 ≤ θna ≤ θa

It is not active in case that arc a is not available, i.e.,xn

a = 0.


Variables in (each) dupla (commodity k , scenario noden)-based lower-level model, for k ∈ K, n ∈ N

The variables in (each) lower-level model are related to the flowof the commodities in set K through the nodes of the network,by using either the network available arcs or other meansdepending on the transport cost of the alternatives in thescenario nodes of the tree along the time horizon.

yna,k , 0-1 variable whose value 1 means that network arc a is

used for transporting commodity k in scenario node n andotherwise, 0, for a ∈ A1.

yna,k ∈ 0,1, yn

a,k ≤ xna

zna,k , 0-1 variable whose value 1 means that alternative arc a is

used for transporting commodity k in scenario node n andotherwise, 0, for a ∈ A2.Note: Its integrality can be relaxed.


Goal: Looking for the Stackelberg (1934) equilibriumbetween the Leaders expected profit and the Followersexpected transport cost.

That is, the solution of the upper-level model is influencedby the optimization of the (each) lower-level model, andvice versa.

The maximization of the expected profit in the scenarios ofthe upper-level model has an additional type of constraint:

In each (k ,n)-based lower level model, minimizing inscenario node n the cost of commodity k transportationthrough the available network links and the cost of usingalternative means, for k ∈ K, n ∈ N .

Modeling scheme: A tariff-based network’s single model tobe solved up to optimality for each scenario node:Coordinated primal-dual submodels.


Upper-level model

p = max∑n∈N

wn

[∑k∈K

∑a∈A1

V nk θ

nayn

a,k −∑a∈A1

(Un

a (xna − xσ

n

a ) + Caxna)]

(1a)

s.t. 0 ≤ θna ≤ θa ∀a ∈ A1, n ∈ N (1b)

xna ∈ 0,1, xσ

n

a ≤ xna ∀a ∈ A1, n ∈ N (1c)∑

a∈A1

Una (xn

a − xσn

a ) ≤ Un ∀n ∈ N (1d)

yna,k ∈ 0,1, yn

a,k ≤ xna ∀a ∈ A1, k ∈ K, n ∈ N (1e)


(k ,n)-scenario node-based lower-level model, fork ∈ K, n ∈ N

cnk = min

(∑a∈A1

V nk θ

nayn

a,k +∑a∈A2

V nk Bn

azna,k)

(2a)

s.t.∑

a∈Γn+∩A1

yna,k +

∑a∈Γn

+∩A2

zna,k −

∑a∈Γn

−∩A1

yna,k −

∑a∈Γn

−∩A2

zna,k

= ρn,k ∀n ∈ N (2b)

0 ≤ yna,k ∀a ∈ A1 (2c)

0 ≤ zna,k ∀a ∈ A2, (2d)

where for n ∈ N : ρn,k = 1 for n = ok , ρn,k = −1 for n = dk andotherwise, ρn,k = 0.


It is well-known that there is an optimal solution for model(2) where the (y , z)-variables take 0-1 values.

So, no upper bound should be imposed on those variables,but bound 1 could reinforce the model.

Notice that there is a submodel for each dupla (scenarionode, commodity) in TAP.

In other types of applications there is only a submodel foreach scenario node.

As an Illustrative example, the lower-level reacts to each(θ, x)-offering from the upper-level solution until getting atariff equilibrium.

Applications: Open market in energy, good selling,transportation, etc. where both type of agents havealternatives for pursuing their aims.


The primal-dual approach is chosen in this work, sinceprovisional results on the KKT approach requiredunaffordable computing effort for instances underconsideration.

That big effort, probably, was due to the big M-parametersthat are required for the linearly representation of thequadratic complementary slackness conditions for eachscenario node.

Let δnn,k be the free dual variable of the conservation flow

constraint (2b) of the primal-dual model below, for thetriplet given by network node n, commodity k and scenarionode n.


Variable f na,k : Mixed 0-1 bilinear function θn

ayna,k , for

a ∈ A1, k ∈ K, n ∈ N , to be equivalently replaced with theFortet inequalities:

f na,k ≤ θayn

a,kθn

a − θa(1− yna,k ) ≤ f n

a,k ≤ θna + θa(1− yn

a,k )(3)


Single deterministic equivalent model tothe multi-period stochastic equilibrium-basedprimal-dual optimization model for network expansionplanning


Upper level primal model

p = max∑n∈N

wn

[∑k∈K

∑a∈A1

V nk f n

a,k −∑a∈A1

(Un

a (xna − xσ

n

a ) + Caxna)]

(4a)

s.t. 0 ≤ θna ≤ θa ∀a ∈ A1, n ∈ N (4b)

f na,k ≤ θayn

a,k , θna − θa(1− yn

a,k ) ≤ f na,k ,

f na,k ≤ θ

na + θa(1− yn

a,k )

∀a ∈ A1, k ∈ K, n ∈ N (4c)xn

a ∈ 0,1, xσn

a ≤ xna ∀a ∈ A1, n ∈ N (4d)∑

a∈A1

Una (xn

a − xσn

a ) ≤ Un ∀n ∈ N (4e)

yna,k ∈ 0,1, yn

a,k ≤ xna ∀a ∈ A1, k ∈ K, n ∈ N (4f)


s.t. constraints of the (k ,n)-based lower models, for k ∈ K, n ∈ N

Primal model constraints:∑a∈Γn

+∩A1

yna,k +

∑a∈Γn

+∩A2

zna,k −

∑a∈Γn

−∩A1

yna,k −

∑a∈Γn

−∩A2

zna,k

= ρn,k ∀n ∈ N , k ∈ K (4g)

0 ≤ zna,k ≤ 1 ∀a ∈ A2, k ∈ K (4h)

Dual model constraints:V n

k θna − δn

n,k + δnn′,k≥ 0 ∀a = (n, n′) ∈ A1, k ∈ K (4i)

V nk Bn

a − δnn,k + δn

n′,k≥ 0 ∀a = (n, n′) ∈ A2, k ∈ K (4j)

Equating the (k ,n)-based primal and dual objective functions:(∑a∈A1

V nk f n

a,k +∑a∈A2

V nk Bn

azna,k)

= δnok ,k − δ

ndk ,k (4k)


Nested Stochastic Decomposition methodologoy.Some refs.

NSD versions for stage-independent uncertainties:SDDP for SLP, for solving large-sized LP instances,providing lower & upper bounds:Pereira & Pinto WRR’85, MP’91 for Risk Neutral, and manyothers.Philpott & de Matos EJOR’12; Philpott, de Matos & FinardiOR’13 with time-consistent and coherent measuresSDDiP for SMIP, for solving large-sized mixed 0-1instances with 0-1 state vars (finite convergence withprobability one).Zou, Ahmed & Sun MPB’18

Finite convergence with probability one NSD version forstage-dependent uncertainties:

NSD for MSMIP, with 0-1 state vars.Zou, Ahmed & Sun, OptimOnline’16.


Some refs. on NSD methodology (c)

Matheuristic NSD versions for mixed 0-1 state vars withstage-dependent uncertainties, for MSMIP:

Case w/ influential vars from immediate ancestor node:NSD-RN. Aldasoro, LFE, Merino, Monge & Perez TOP’14,among others. Continuous state vars.NSD-BILEVEL-NEP, for Stochastic Equilibrium in NetworkExpansion Planning. 0-1 state vars.LFE, Monge & Rodrigez-Chıa. Submitted, 2019.

Case w/ influential var from ancestor nodes:NSD-TSD, with Time in-consistent Stochastic Dominancefunctional.LFE, Monge & RomeroMorales COR’15NSD-ECSD, with Time consistent Expected ConditionalStochastic Dominance functional.LFE, Monge & RomeroMorales COR’18


NSD matheuristic, Nested Stochastic Decomposition

NSD starts by partitioning the set of time periods intomodeler-driven so-named stages each one is a subset ofconsecutive periods, creating thus a collection ofsubtrees/subproblems.The subproblems are not independent but each one islinked by the 0-1 step vars to the immediate ancestorsubproblem.NSD solves the subproblems at each iteration, byexecuting first a front-to-back scheme and after aback-to-from scheme.


Front-to-Back (FtB)

Back-to-Front (BtF)

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

e = 1 e = 2 e = 3

First stage Intermediate stages Last stage

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33

T = 1, . . . , 7,E = 1, 2, 3

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

G2 = 4, 5, 6, 7, 8, 9

R2 = 4, 5

C4 = 4, 6, 7

L15 = 28, 29

Figura: Partitioning periods in stages


Scenario subtrees supporting the NSD subproblems.Sets

E , stages in the time horizon.Te, set of periods in stage e, such that

T =∑

e∈E Te, Te ∩ Te′ = ∅ : e 6= e′.Ge ⊆ N , set of nodes in stage e, for e ∈ E .Re ⊆ Ge, root nodes of the subtrees in stage e, for e ∈ E .Cr ⊆ Ge, nodes that belong to the subtree rooted with node r , for

r ∈ Re,e ∈ E .Lr ⊆ Cr , leaf nodes in subtree related to set Cr , for r ∈ Re, e ∈ E .

Note: The 0-1 step x`-vars are the state ones, i.e., they arethe only vars in a subproblem supported by the subtreerooted with node r that have nonzero elements inconstraints associated with the root nodes in theimmediate successor subproblems to leaf node `

Those subtrees are defined by the nodeset⋃

r ′∈S`′1Cr ′ , for

` ∈ Lr , r ∈ Re, e ∈ E : e < E .Laureano F. Escudero Universidad Rey Juan Carlos (URJC), Mostoles (Madrid), [email protected]

NSD matheuristic. EFV curves

Let the so-named Expected Future Value (EFV) curve ofthe subproblem supported by the subtree given by nodesetCr ′ , whose root node r ′ is an immediate successor of leafnode `, for r ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E , whereremember Lr ⊆ Cr .

EFV curves estimate the impact of the state vars(decisions) of the leaf nodes in the objective function ofsuccessor subproblems.

(.), current solution of the variables’ vector (.).


For easing the exposition, let a set of variables in the originalmodel (4) be notated as vector Z n, for n ∈ N . It can beexpressed

Z n ≡(yg ∀g ∈ An; xg ∀g ∈ An \ n

)(5)


EFV curve λ′r ′(x `), forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

λ′r′(x`), (assumed convex) curve of the expected future

objective function value (EFV) in the set of scenarios Ωr ′ ,due to the leaf node-related variable x`. It is difficult tocompute.NSD approach approximates λ′r

′(x`) with the piecewise

linear convex EFV function λr ′(x`).

xℓ1b

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xℓ2

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

xℓ


Objective function value F ′r of subproblem defined bynode set Cr , r ∈ Re, e ∈ E

F ′r = max∑`∈Lr

w `[∑

n∈A`

(anxn + bnyn) +∑

r ′∈S`1

w r ′

w `λ′r′(

x`)]

s.t. Anxn + Bnyn = hn, yn ∈ Y n ∀n ∈ Cr

xn ∈ 0,1nx , xσn ≤ xn ∀n ∈ Cr

Zσr= Zσr

xσr − xσ

r= 0

(6)


Elements definition of EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

Let Q` denote the set of indices q, such that the referencelevels indexed with q define the EFV function λr ′ .Each qth reference level is composed by vector(

x`q; µr ′q , πr ′q ∀r ′ ∈ S`1

),

where vector x`q

is the value of the vars in vector x`q, for

q ∈ Q`, retrieved from the subproblem related to scenario noder , andconstant µr ′q and vector πr ′q define a linear function of vector x`

obtained from the solution of the subproblem related to theimmediate successor node r ′ of scenario leaf node `, i.e.,q ∈ Q`, r ′ ∈ S`1, ` ∈ Lr .

λr ′ = minq∈Q`

µr ′q + πr ′q x`

.


EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

λr ′ = minq∈Q`

µr ′q + πr ′q x`

.

It is required that it is a valid cut for EFV curve λ′r′(·) as well as

a tight one, such that λ′r′(x`

q) = µr ′q + πr ′q x`

q.


Objective function value F r of subproblem defined bynode set Cr , r ∈ Re, e ∈ E : e < E , where EFV curveλ′r(x `) is replaced with EFV (piecewise linear) functionλr ′(x `), for r ′ ∈ S`1, ` ∈ Lr

F r = max∑`∈Lr

w `[∑

n∈A`

(anxn + bnyn) +∑

r ′∈S`1

w r ′

w `λr ′] (7a)

s.t. Anxn + Bnyn = hn, yn ∈ Y n ∀n ∈ Cr (7b)xn ∈ 0,1nx , xσ

n ≤ xn ∀n ∈ Cr (7c)

Zσr= Zσr q

(7d)

x` − xσr q

= 0 (7e)λr ′ ≤ µr ′q + πr ′q x` ∀q ∈ Q`, r ′ ∈ S`1, ` ∈ Lr . (7f)


xℓ1b

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xℓ2

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

xℓ


NSD approach. Front-to-Back scheme from stagee = 1 to last stage e = E

Subproblems (7) from stage 1 to last one E are solved bypassing the values of linking x-vars onto the subproblemsin the next stage.

λ-values are zero in iteration q = 1.

In case of infeasibility of subproblem (7) for r ∈ Re, e ∈ E :merging scheme of stages e-1 and e.

Building a feas solution for original model (4)(xn, yn ∀n ∈ N ).

Obj fun value F =∑

n∈N wn(anxn + bnyn)

In that case, a testing is performed to analyze whether itimproves the incumbent solution.


Front-to-Back (FtB)

Back-to-Front (BtF)

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

e = 1 e = 2 e = 3

First stage Intermediate stages Last stage

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32

33

T = 1, . . . , 7,E = 1, 2, 3

Ω = Ω1 = 18, 19, . . . , 33

A6 = 6, 4, 3, 2, 1

t6 = 5

σ6 = 4

S6 = 10, 11, 18, 19, 20, 21

S61 = 10, 11

G2 = 4, 5, 6, 7, 8, 9

R2 = 4, 5

C4 = 4, 6, 7

L15 = 28, 29


NSD approach. Front-to-Back scheme from stagee = 1 to last stage e = E (c)

Upper bound of the solution value of the original model (4):solution value F r of model (7) for r = 0 (i.e., e = 1).

OG: Optimality gap of a given feasible solution value F ofthe original model (4) with respect to F 0 at iteration, say q.


NSD approach. Stopping criteria

1 OG is not greater than a given tolerance, say ε12 The relative change in the upper bounds bound F 0 and

the relative change in the obj fun value F ,both between the last two consecutive iterations q − 1 andq are not greater than a given tolerance, say ε2.

3 An upper bound on the number of iterations is reached,say q.

4 Computing time limit is reached.


NSD approach. Back-to-Front scheme from last stagee = E to stage e = 2

It aims to refining the EFV functions λr ′ in the subproblemssupported by the subtrees rooted with the nodeset r ′ inRe, e ∈ E \ 1 around the feasible soln (Z n, xn ∀n ∈ NT )built in the front-to-back scheme at the current iteration q.

The subproblems in a given stage are solved, passing therefinement of the EFV functions onto the subproblems inthe previous stage.


NSD approach. On EFV function λr ′, forr ′ ∈ S`1, ` ∈ Lr , r ∈ Re, e ∈ E : e < E

Generating and appending a new set S`1 of linear functionsλr ′q

λr ′ ≤ µr ′q + πr ′q x` (8)

to the collection (7f):

λr ′ ≤ µr ′q + πr ′q x` ∀q ∈ Q` \ q, r ′ ∈ S`1, ` ∈ Lr

in model (7) related to the solution x`q, retrieved from the

front-to-back scheme, where q is the last iteration.


On obtaining vector (µr ′q , πr ′q)

It is worth to pointing out that some state-of-the-art optimizers(as CPLEX and others) do not provide the duals of theconstraints for fixed vars in MIP models.So, some alternatives:

Benders cut, B (Benders NM’62).

Lagrangean cut, L(Zou, Ahmed & Sun OptimOnline’16, MPB’18).

Strengthened Benders cut, SB(Zou, Ahmed & Sun OptimOnline’16, MPB’18).

Heuristic Benders cut, HB.

A mixture of SB cuts for later sages and L cuts for earlierstages, in particular, stage e = 1. Work-in-Progress.


Approximation schemes for EFV curve λ′r ′(x `): L-,SB-, B- and HB-based cuts for obtaining EFV functionλr ′(x `)

L: Valid and tight

SB: Valid and no-guarantee it is tight.

B: Valid and ’almost’ sure it is not tight.

HB: No guarantee it is valid, and sure it is tighter than B.

That is, there it not any guarantee that the EFV functionλr ′(x`) does not cut any feasible solution of the originalmodel (4).

So, no guarantee that the NSD incumbent solution is theoptimal one for, nor an upper bound is obtained.


HB-based cut for approximating EFV curve λ′r ′(x `) viaEFV function λr ′(x `)

No guarantee that the EFV function λr ′(x`) does not cutany feasible solution of the original model (4).

So, no guarantee that the NSD incumbent solution is theoptimal one for, nor an upper bound is obtained.

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xℓ2

λr′3(x) = µr′3 + πr′3x

λr′1 (x) = µr′1 + πr′1x

λr′2(x) = µr′2 + πr′2x

λr′

xℓ


Computational experience. Computing HW/SW

A processor Intel(R) Xeon(R) CPU E5-2650 v3 @2.30GHz, 20 cores, RAM 62 GiB.

A C++ experimental code.

CPLEX v12.8, using between 15 and 20 cores.

NSD parameters: ε1 = 0,0001, ε2 = 0,0001,iters limit q = 100,Lagrangean iters limit q = 4.

Given the difficulty of the instances, E = T and, so,|Te| = 1, Te = t r, |Cr | = 1, for r ∈ Re, e ∈ E ,

t r , period which scenario node r belongs to.


4 Testbeds

|N|, |A1| and |A2|, number of nodes, network links andlinks by other means, resp., for the transport network.Scenario tree dimensions:

|N|, number of scenario nodes.b, number of immediate successor nodes for any scenarionode (i.e., |Sn

1 | = b, n ∈ N ).|Ω|, number of scenarios (i.e., |NT |).


Cuadro: Problem and model (4) dimensions. Testbed 1

Ins |N| |A1| |A2| |T | b |N | |Ω| m nc n01i1 12 4 13 2 2 3 2 1950 474 822i2 12 4 13 2 3 4 3 2600 632 1096i3 15 8 18 2 2 3 2 2946 675 1212i4 12 4 13 2 4 5 4 3250 790 1370i5 15 8 18 2 3 4 3 3928 900 1616i6 12 4 13 2 5 6 5 3900 948 1644i7 12 4 13 3 2 7 4 4550 1106 1918i8 15 8 18 2 4 5 4 4910 1125 2020i9 16 6 18 2 2 3 2 4797 1125 2034i10 15 8 18 2 5 6 5 5892 1350 2424i11 16 6 18 2 3 4 3 6396 1500 2712i12 15 8 18 3 2 7 4 6874 1575 2828i13 20 12 25 2 2 3 2 7329 1623 3012i14 16 6 18 2 4 5 4 7995 1875 3390i15 12 4 13 3 3 13 9 8450 2054 3562i16 20 12 25 2 3 4 3 9772 2164 4016


Results. CPLEX plain use and NSD in in L4, SB, HB

of i and ti , incumbent soln value for original model (4) andcomputing time (secs) for i = 1,3, wherei = 1 (CPLEX plain use w/ computing time as NSD;i = 3 (NSD).

of 1, CPLEX upper bound of opt sol, in L4, SB.

OG1 %, CPLEX optimality gap, 100.of 1−of 1of 1

, in L4, SB.

of 3, NSD estimation of upper bound of opt sol in L4, SB.

OG3 %, NSD estimation of the optimality gap, as

100. of 3−of 3of 3

in HB.

GR1, goodness ratio of NSD over CPLEX plain use, as of 3of 1

.


Cuadro: Lagrangean cut results, q = 4. Testebed 3

Inst of 1 of 1 OG1 % t1 of 3 of 3 t3 q GR1 OG3 %i33 72631 94099 29.56 7294 74931 75683 7216 21 1.03 1.00i34 81026 97104 19.84 14147 81669 81669 14056 24 1.01 0.00i35 92979 95707 2.93 1625 93137 94666 1605 12 1.00 1.64i36 170443 201916 18.47 85696 170483 171246 85281 53 1.00 0.45i37 176306 215347 22.14 21835 174842 181464 21725 36 0.99 3.79i38 381177 386851 1.49 1546 381128 381310 1518 7 1.00 0.05i39 335975 338966 0.89 7155 335915 350352 7083 22 1.00 4.30i40 103367 106136 2.68 2722 103499 103934 2685 13 1.00 0.42i41 85825 115113 34.13 26162 91885 91885 25976 23 1.07 0.00i42 196408 201730 2.71 2376 197388 197818 2144 7 1.00 0.22i43 603138 725302 20.25 34472 611307 698282 34262 27 1.01 14.23i44 220452 229088 3.92 7823 221316 222287 7741 21 1.00 0.44i45 369220 462554 25.28 132712 371832 480146 132712 24 1.01 29.13i46 92660 135512 46.25 35109 104510 111248 34927 22 1.13 6.45i47 387549 401969 3.72 2992 392588 392790 2944 7 1.01 0.05i48 274256 345959 26.14 58593 280734 281914 58264 45 1.02 0.42


Cuadro: Lagrangean cut results, q = 4. Testebed 4

Inst of 1 of 1 OG1 % t1 of 3 of 3 t3 q GR1 OG3 %i49 790692 833771 5.45 3058 813920 814102 3004 7 1.03 0.02i50 239974 318446 32.70 133110 246084 278280 132360 29 1.03 13.08i51 708116 734970 3.79 7457 719978 772717 7348 11 1.02 7.33i52 225544 309677 37.30 4017 298530 299633 3957 7 1.32 0.37i53 572181 734620 28.39 84470 588962 694614 84043 39 1.03 17.94i54 1335404 1591737 19.20 131115 1378708 1415879 131115 51 1.03 2.70i55 340948 363425 6.59 20946 348936 349436 20682 26 1.02 0.14i56 404471 531153 31.32 102128 424072 432277 101547 45 1.05 1.93i57 1055410 139126 830580 1582535 139126 11 90.53i58 650253 733051 12.73 47672 707648 710406 47152 37 1.09 0.39i59 301048 507319 68.52 134385 384553 503137 133665 16 1.28 30.84i60 1366863 1595141 16.70 30628 1556647 1565923 30190 24 1.14 0.59i61 449334 542291 20.69 47286 515729 515729 46644 32 1.15 0.00i62 523257 1062376 103.03 133379 825055 1729673 132765 5 1.58 109.64i63 2387036 142238 1877392 7209842 142238 6 284.03i64 336794 779913 131.57 136197 595635 921531 135417 7 1.77 54.71


Cuadro: Strengthened Benders cut results. Testebed 3



Cuadro: Strengthened Benders cut results. Testebed 4

Inst of 1 of 1 OG1 % t1 of 3 of 3 t3 q GR1 OG3 %i49 744545 834413 12.07 1082 814102 819551 1056 4 1.09 0.67i50 196564 320987 63.30 33648 246084 347831 33435 16 1.25 41.35i51 669308 735607 9.91 3973 719978 780192 3904 9 1.08 8.36i52 222571 309679 39.14 3228 299164 306235 3173 8 1.34 2.36i53 556944 745055 33.78 2086 588962 877015 2058 2 1.06 48.91i54 1082307 1621786 49.85 2875 1378708 1554399 2830 2 1.27 12.74i55 339253 363568 7.17 11144 348936 405605 10985 24 1.03 16.24i56 402302 532103 32.26 8126 423602 621672 8015 7 1.05 46.76i57 1077435 14943 830581 1197268 14250 2 44.15i58 476227 764845 60.61 1507 705704 922217 1451 2 1.48 30.68i59 233723 507319 117.06 55427 384553 567263 55099 14 1.65 47.51i60 1654596 8265 1556647 1616861 8053 9 3.87i61 379202 564082 48.75 1674 513926 721819 1605 2 1.36 40.45i62 582338 1062812 82.51 32982 825055 1238422 32745 3 1.42 50.10i63 2411878 32193 1877393 2779655 31248 2 48.06i64 336794 779913 131.57 75776 595635 904444 75291 11 1.77 51.85


Cuadro: Heuristic Benders cut results. Testebed 3. q = 25



Cuadro: Heuristic Benders cut results. Testebed 4. q = 25

Inst of 1 of 1 OG1 % t1 of 3 of 3 t3 q GR1 OG3 %i49 600285 905787 50.89 310 814102 819383 299 2 1.36 0.65i50 164519 320987 95.11 13554 246084 261950 13471 25 1.50 6.45i51 643257 735774 14.38 2269 719978 726927 2223 10 1.12 0.97i52 220727 337634 52.96 790 299630 301331 769 5 1.36 0.57i53 563372 734684 30.41 12775 588962 645348 12660 25 1.05 9.57i54 1103812 1621704 46.92 18188 1379860 1402625 18020 25 1.25 1.65i55 312301 364179 16.61 3360 348936 350386 3285 17 1.12 0.42i56 403435 531861 31.83 11822 424438 448284 11683 25 1.05 5.62i57 1061630 23621 830581 1048290 22682 11 26.21i58 476227 733565 54.04 6358 705704 724172 6232 17 1.48 2.62i59 233723 507319 117.06 25437 384553 433195 25256 24 1.65 12.65i60 1654694 4082 1556647 1568197 3934 8 0.74i61 383953 542841 41.38 6458 513926 530043 6329 18 1.34 3.14i62 593396 1062702 79.09 34211 825055 1050935 33980 12 1.39 27.38i63 2411878 24304 1877393 2585925 23877 4 37.74i64 370189 780051 110.72 53183 595635 673192 52767 25 1.61 13.02


L4, SB, HB cut option results in NSD.Summary for SE-NEP

SE-NEP: Very difficult M01QP multistage problem w/large-scale model dimensions.

Weakness of original model (4): high range that is allowedto tariff θn

a , given the big θa-based constraints that arerequired for equivalently linearising θn

ayna,k .

Sharing the incumbent solution value of 3 in 23 out of the32 instances of Testbeds 3 and 4 (including the largestone, i64).

There are small differences in those values for most of theother instances.

The EFV functions λr ′ in SB and HB only differ in theconstant µr ′q .


Cuadro: Comparison of L4, SB and HB cut options. Testbed 3

Inst of HB of L4 GRL4 GRL4t of SB GRSB GRSB

t

i33 74931 74931 1.000 6.29 74931 1.000 2.75i34 81669 81669 1.000 7.78 80615 0.987 3.19i35 93137 93137 1.000 5.90 93137 1.000 3.62i36 170483 170483 1.000 15.01 170483 1.000 1.00i37 172836 174842 1.012 22.15 172836 1.000 2.92i38 381310 381128 1.000 2.67 381310 1.000 1.90i39 335915 335915 1.000 6.43 335915 1.000 1.59i40 103499 103499 1.000 6.58 103499 1.000 3.73i41 91503 91885 1.004 8.33 91194 0.997 4.46i42 197818 197388 0.998 5.00 197388 0.998 3.61i43 615374 611307 0.993 4.40 611307 0.993 0.17i44 221316 221316 1.000 6.55 221316 1.000 4.25i45 371832 371832 1.000 6.26 371832 1.000 0.33i46 106293 104510 0.983 5.99 104510 0.983 2.98i47 392628 392588 1.000 5.33 392790 1.000 3.60i48 277899 280734 1.010 8.76 274950 0.989 0.86


Cuadro: Comparison of L4, SB and HB cut options. Testbed 4

Inst of HB of L4 GRL4 GRL4t of SB GRSB GRSB

ti49 814102 813920 1.000 10.05 814102 1.000 3.53i50 246084 246084 1.000 9.83 246084 1.000 2.48i51 719978 719978 1.000 3.31 719978 1.000 1.76i52 299630 298530 0.996 5.15 299164 0.998 4.13i53 588962 588962 1.000 6.64 588962 1.000 0.16i54 1379860 1378708 0.999 7.28 1378708 0.999 0.16i55 348936 348936 1.000 6.30 348936 1.000 3.34i56 424438 424072 0.999 8.69 423602 0.998 0.69i57 830581 830580 1.000 6.13 830581 1.000 0.63i58 705704 707648 1.003 7.57 705704 1.000 0.23i59 384553 384553 1.000 5.29 384553 1.000 2.18i60 1556647 1556647 1.000 7.67 1556647 1.000 2.05i61 513926 515729 1.004 7.37 513926 1.000 0.25i62 825055 825055 1.000 3.91 825055 1.000 0.96i63 1877393 1877392 1.000 5.96 1877393 1.000 1.31i64 595635 595635 1.000 2.57 595635 1.000 1.43


Headings of tables for L4, SB, HB comparison

of c , incumbent solution of cut option c, resp, forc ∈ L4,SB,HB.GRc and GRc

t , goodness ratio of of c and tc of cup option cover HB, computed as of c

of HB, for c ∈ L4,SB;

Note: Obviously, a ratio above 1 means that the cut optionc expected profit or computing time is higher that the onein HB.


Observations in tables for L4, SB, HB comparison

L4, SB and HB share the incumbent solution value of 3 inmany of the 32 instances (including the largest ones).

The goodness ratios GRL4 and GRSB over HB vary from0.983 and 1.012.

So, it seems that OG3 is a good estimation of the optimalitygap in the difficult original SE-NEP model (4).

The goodness ratios GRL4t and GRSB

t in the computingtime have a high variability.

In fact, the ratio goes from 2.67 to 22.15 in L4 and from0.16 to 4.46 in SB, both over HB.


Cuadro: Average results for Testbeds 1 and 2 (small-middle instancesi1-i32)

Cut option t3 q GR1

Lagrangean 1572 10.4 1.00Strengthened Benders 425 7.9 1.01Heuristic Benders 328 8.2 1.04


Boxplot Radarchat

Heuristic Benders cut Lagrangean cut Strengthened Benders cut

010

2030

4050

OG3Time

Iter

GR1

GR2

Heuristic Benders cutStrengthened Benders cutLagrangean cut

Figura: Intances i1-i32. Boxplot: NSD optimality gap OG3 % in the cutoptions HB, L4 and SB. Radarchart for the NSD computing time t3,iterations q and goodness ratios GR1 and GR2 in L4, SB and HB


Cuadro: Average results for Testbeds 3 and 4 (large instances i33-i64)

Cut option t3 q GR1 GR2

Lagrangean 54106 22.4 1.094 1.172Strengthened Benders 11313 9.5 1.174 1.164Heuristic Benders 9256 15.4 1.240 1.159


Boxplot Radarchat

Heuristic Benders cut Lagrangean cut Strengthened Benders cut

020

4060

8010

0

OG3Time

Iter

GR1

GR2

Heuristic Benders cutStrengthened Benders cutLagrangean cut

Figura: Intances i33-i64. Boxplot: NSD optimality gap OG3 % in thecut options HB, L4 and SB. Radarchart for the NSD computing timet3, iterations q and goodness ratios GR1 and GR2 in L4, SB and HB.Instance i63 was removed

Except for the outliers, the optimality gap OG3 is smaller inL4 than SB, what is not a surprise.OG3 for HB: A good estimation of OG3


Conclusions. SE-NEP, Stochastic-Equilibrium inmulti-period Network Expansion Planning

E-NEP, Multi-period planning under uncertainty

Difficult problem: Uncertainty, mixed 0-1 bilinear NEP,1 upper-level and as many lower-levels as scenario nodes.

Single primal-dual modeling scheme of choice.

Strong step 0-1 vars type(i.e., by occurring in scenario node n):xn ∈ 0,1 where xσ

n ≤ xn for n ∈ N versusimpulse 0-1 ones (i.e., at occurring in scenario node n):xn ∈ 0,1 where

∑n∈Aω

xn ≤ 1 for ω ∈ Ω.

Decomposition methodology, a must for problem solving inlarge-sized instances,due to dimensions of problem network and scenario tree.


Conclusions. NSD, Nested Stochastic Decomposition

For single primal-dual model, NSD proposal: Good resultsversus CPLEX, a M01LP/M01QP state-of-the-art solver.

Better for state variables only in consecutive two periods .

NSD performance improving: only step 0-1 vars

NSD: A good tool for helping to solve large-sized verydifficult problems, by considering,as a future research work,an `2-norm Regularized scenario Clustering SimplicialDecomposition Progressive algorithm (RCSDPA), see LFE,Garın, Monge & Unzueta Sub’19, for solving theF ′r -related subproblem (7).


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