Global Optimization of ODE constrained network problems Marc - - PowerPoint PPT Presentation

▶
global optimization of ode constrained network problems
SMART_READER_LITE
LIVE PREVIEW

Global Optimization of ODE constrained network problems Marc - - PowerPoint PPT Presentation

Dec 29, 2023 177 likes •457 views

Global Optimization of ODE constrained network problems Marc Pfetsch Joint work with: Oliver Habeck and Stefan Ulbrich TU Darmstadt Supported by the SFB Transregio 154: Mathematical Modelling, Simulation and Optimization using the Example of

slide-1
SLIDE 1

Global Optimization of ODE constrained network problems

Marc Pfetsch Joint work with: Oliver Habeck and Stefan Ulbrich TU Darmstadt

Supported by the SFB Transregio 154: Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 1

slide-2
SLIDE 2

Outline

1

General Theory Abstract Model and Relaxations Solving Strategies

2

Application – Stationary Gas Transport Model Relaxation of the ODE

3

Implementation – for Stationary Gas Transport The Algorithm Results

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 2

slide-3
SLIDE 3

Optimization Problem

We consider the ODE constrained optimization problem: min C(x, y0, yS, z) s.t. G(x, y0, yS, z) ≤ 0,

∂sy(s) = f

  • s, x, y(s)
  • ,

s ∈ [0, S] y0 = y(0), yS = y(S), x ∈ X, y0 ∈ Y 0, yS ∈ Y S, z ∈ Z, (Pode)

⊲ Y 0, Y S ⊂ Rn and X ⊂ Rk are polytopes and Z ⊂ Zm is bounded. ⊲ C : X × Y 0 × Y S × Z → R and G: X × Y 0 × Y S × Z → Rl are continuously

differentiable and possibly nonlinear.

⊲ Values of y only need to be known for finitely many points y(0), y(S) ∈ Rn. ⊲ Examples: Stationary networks (gas, water, (electricity), . . . )

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 3

slide-4
SLIDE 4

Idea of our Approach

⊲ Standard approach is “Discretize then optimize”:

discretize ODE (add N variables), use numerical approximation scheme and

  • btain MINLP

. Solve it and possibly refine.

⊲ Our goal: compute globally optimal solution without explicitly discretizing. ⊲ Workhorse: good convex/concave under/over-estimators. ⊲ Will show: Easy to construct in certain cases, e.g., gas optimization. ⊲ Yields globally convergent solution algorithm.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 4

slide-5
SLIDE 5

Equivalent Optimization Problem

Equivalent reformulation with the solution mapping F : (x, y0) → yS: min C(x, y0, yS, z) s.t. G(x, y0, yS, z) ≤ 0, yS − F

  • x, y0

= 0, x ∈ X, y0 ∈ Y 0, yS ∈ Y S, z ∈ Z. (P)

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 5

slide-6
SLIDE 6

Nonconvex Relaxation of (P)

Assume that there exist F ℓ : X × Y 0 × Nn → Rn and F u : X × Y 0 × Nn → Rn, with F ℓ x, y0, N

  • ≤ F
  • x, y0

≤ F u

x, y0, N

  • for all x ∈ X and y0 ∈ Y 0. Furthermore, on the polytopes X, Y 0 the functions F ℓ

and F u converge uniformly to F for N → ∞, i = 1, ... , n. This yields the relaxation min C(x, y0, yS, z) s.t. G(x, y0, yS, z) ≤ 0, F ℓ x, y0, N

  • ≤ yS ≤ F u

x, y0, N

  • ,

x ∈ X, y0 ∈ Y 0, yS ∈ Y S, z ∈ Z. (Pr(N))

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 6

slide-7
SLIDE 7

Convex Relaxation of (Pr(N))

Assume that there exist

⊲ convex underestimators ˇ

C ≤ C, ˇ G ≤ G, and ˇ F ℓ ≤ F ℓ

⊲ and a concave overestimator F u ≤ ˆ

F u. Hence, min ˇ C(x, y0, yS, z) s.t. ˇ G(x, y0, yS, z) ≤ 0, ˇ F ℓ x, y0, N

  • ≤ yS ≤ ˆ

F u x, y0, N

  • ,

x ∈ X, y0 ∈ Y 0, yS ∈ Y S, z ∈ conv(Z) (Pcv(N)) is a convex relaxation of (Pr(N)).

we can solve (Pr(N)) with spatial branch-and-bound.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 7

slide-8
SLIDE 8

(ǫ, δ)-optimal solutions Definition

A vector (x, y0, yS, z) ∈ X × Y 0 × Y S × Z is (ǫ, δ)-optimal, if

⊲ each constraint is violated by a maximum of δ ⊲ and the objective function satisfies C(x, y0, yS, z) ≤ C∗ + ǫ.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 8

slide-9
SLIDE 9

Solving Strategy

Goal: Find (ǫ, δ1 + δ2)-optimal solution of (P).

⊲ Start spatial branch-and-bound with some initial N ∈ Nn and feasibility

tolerance δ1.

⊲ Solve the convex relaxation (Pcv(N)) in a node. ⊲ Check if F u

x, y0, N

  • − F ℓ

x, y0, N

  • ∞ ≤ δ2

holds for the current solution of the relaxation.

⊲ Increase N if necessary, improve the under- and overestimators ˇ

F ℓ, ˆ F u, and solve the convex relaxation again.

⊲ Continue like in the “normal” spatial branch-and-bound algorithm.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 9

slide-10
SLIDE 10

Solving Strategy Theorem

For a nested sequence of nodes Fk = Xk × Y 0

k × Y T k × Zk with Fk+1 ⊆ Fk let

⊲ limk→∞ diam Fk = 0 ⊲ and the estimators become exact in the limit, i.e., the estimators converge to

the original functions on Fk for k → ∞. Then the solving strategy produces an (ǫ, δ1 + δ2) feasible point of (P) or shows that it is infeasible in finite time.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 10

slide-11
SLIDE 11

Outline

1

General Theory Abstract Model and Relaxations Solving Strategies

2

Application – Stationary Gas Transport Model Relaxation of the ODE

3

Implementation – for Stationary Gas Transport The Algorithm Results

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 11

slide-12
SLIDE 12

Model

⊲ Gas networks are given by directed graphs G = (V, A). ⊲ The nodes are junctions of the network. ⊲ The arcs are

(control)valves: linear constraints with binary variables resistors: (non)linear constraints with indicator variables compressors: (here) polyhedral approximation of operating range pipes: ordinary differential equation

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 12

slide-13
SLIDE 13

The Differential Equation

Stationary isothermal Euler equation with no inclination and |v|

c = c|q| Ap ≤ 0.8:

∂xp(x) = − λc2 q|q| p(x)

2D

  • A2p(x)2 − c2q2 =: φ
  • p(x), q
  • .

Variables:

⊲ pressure p(x) ⊲ mass flow q

Constants:

⊲ friction coefficient λ ⊲ speed of sound c ⊲ diameter D ⊲ cross sectional area A ⊲ length of the pipe L

For q ≥ 0 the solution p(x) is concave and non increasing.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 13

slide-14
SLIDE 14

Optimization Problem

Thus, we want to solve the following optimization problem: min C(p, q, z) s.t. G(p, q, z) ≤ 0,

∂xpa(x) = φa

  • pa(x), qa
  • ∀ a ∈ Apipe ⊆ A,

pu = pa(0), pv = pa(La)

∀ a = (u, v) ∈ Apipe,

qa ≤ qa ≤ qa

∀ a ∈ A,

pu ≤ pu ≤ pu

∀ u ∈ V,

z ∈ Z ⊂ Zm with spatial branch-and-bound.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 14

slide-15
SLIDE 15

Bounding Methods for Gas Transport Theorem

Given a discretization 0 = xN < xN−1 < · · · < x0 = L with hi+1 = xi − xi+1 for i = 0, 1, ... , N − 1. If q ≥ 0, bounds on p(0) are given by the following methods:

⊲ Upper bounds through the trapezoidal rule:

pu

0 = pout,

pu

i+1 + 1 2hi+1φ

  • pu

i+1, q

  • = pu

i − 1 2hi+1φ

  • pu

i , q

  • , ∀i = 0, 1, ... , N − 1.

⊲ Lower bounds through the explicit midpoint method:

pℓ

0 = pout,

pℓ

i+1 = pℓ i − hi+1 φ

  • pℓ

i − 1 2hi+1 φ

  • pℓ

i , q

  • , q
  • , ∀i = 0, 1, ... , N − 1.

yields (nonconvex) relaxation of Fa.

Critical property: nonnegative local truncation error.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 15

slide-16
SLIDE 16

Illustration

pin pout pout pout pin pin

Fa

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 16

slide-17
SLIDE 17

Illustration

pin pout pout pout pin pin pu(pout, q, N) pℓ(pout, q, N)

Fa

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 16

slide-18
SLIDE 18

Illustration

pin pout pout pout pin pin

R

Fa

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 16

slide-19
SLIDE 19

Illustration

pin pout pout pout pin pin conv(R)

Fa

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 16

slide-20
SLIDE 20

Outline

1

General Theory Abstract Model and Relaxations Solving Strategies

2

Application – Stationary Gas Transport Model Relaxation of the ODE

3

Implementation – for Stationary Gas Transport The Algorithm Results

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 17

slide-21
SLIDE 21

Some Informations

⊲ We use the branch-and-bound framework SCIP

.

⊲ We implemented a constraint handler for the ODEs. ⊲ Test instances are available at http://gaslib.zib.de/.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 18

slide-22
SLIDE 22

Basic Algorithm

We perform the following steps in the nodes of the branch-and-bound tree:

  • 1. Perform bound propagation (use estimators)
  • 2. Solve LP-relaxation
  • 3. Enforce feasibility of “most infeasible” pipe by one of the following steps:

3.1 Fixing the direction of the flow by branching. 3.2 Adding (parts of) the concave overestimator. 3.3 Branching w.r.t. input pressure, output pressure or massflow. 3.4 Adding a gradient cut (convex underestimator).

  • 4. Resolve new LP-relaxation or choose a new node.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 19

slide-23
SLIDE 23

Illustration

pin pout pout pout pin pin pin pout pout pout pin pin

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 20

slide-24
SLIDE 24

Example: GasLib-40

⊲ 40 nodes ⊲ 39 pipes ⊲ 6 compressors ⊲ 12 binary variables for compressors

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 21

slide-25
SLIDE 25

Results

  • bjective: minimize

−

v∈V pv

  • v∈V q±

v pv

  • a∈Acs za

solving time (seconds) 877.74 542.48 65.08 processed Nodes 12,444 2790 263 branchings on flow (%) 46.2 81.0 84.2 branchings on pressure (%) 52.9 0.0 8.8 branchings on binary variables (%) 0.9 19.0 7.0 leaves 5900 1412 73 cut offs by propagation 5297 1149 38 bound changes by propagation 32,444 63,317 6051 added overestimators 7362 2385 985 added underestimators 6360 4281 488

δ1 = ǫ = 10−6, δ2 = 10−4

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 22

slide-26
SLIDE 26

Conclusions

Future theoretical goals:

⊲ Extend to case with inclination. ⊲ Include temperature? ⊲ Transient case (much more difficult).

Future implementation issues:

⊲ Better branching rules ⊲ Additional primal heuristics ⊲ Avoid cycle flows ⊲ Cuts on binary variables ⊲ Larger instances

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 23

slide-27
SLIDE 27

Literature

  • O. Habeck, S. Ulbrich, and M. E. Pfetsch.

Global Optimization of ODE constrained network problems. Preprint, TRR 154, 2017.

  • pus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/186.
  • M. E. Pfetsch, T. Koch, L. Schewe, and B. Hiller, editors.

Evaluating Gas Network Capacities. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2015.

  • M. Schmidt, D. Aßmann, R. Burlacu, J. Humpola, I. Joormann, N. Kanelakis,
  • T. Koch, D. Oucherif, M. E. Pfetsch, L. Schewe, R. Schwarz, and M. Sirvent.

GasLib – A Library of Gas Network Instances. Data, 2: Article 40, 2017.

March 7, 2018 | SCIP Workshop 2018 | M. Pfetsch | 24