[PPT] - Primal heuristic for MINLPs in SCIP Ambros Gleixner, and Felipe PowerPoint Presentation

SLIDE 1

Primal heuristic for MINLPs in SCIP

Ambros Gleixner, and Felipe Serrano

Zuse Institute Berlin · serrano@zib.de SCIP Optimization Suite · http://scip.zib.de Workshop on Discrepancy Theory and Integer Programming Amsterdam · June 12, 2018

SLIDE 2

Outline

Introduction: LP-based Branch and Bound Spatial Branch and bound Heuristics Sub-NLP NLP diving Multi-start MPEC Undercover RENS Conclusion

Gleixner, Serrano · MINLPs in SCIP 1 / 24

SLIDE 3

Mixed-Integer Nonlinear Programs (MINLPs)

min cTx s.t. gk(x) ≤ 0 ∀k ∈ [m] xi ∈ Z ∀i ∈ I ⊆ [n] xi ∈ [ℓi, ui] ∀i ∈ [n] The functions gk ∈ C1([ℓ, u], R) can be

−1 1 −1 1 5 10

convex or

100 200 300 200 −200 200

nonconvex

Gleixner, Serrano · MINLPs in SCIP 2 / 24

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One way of solving MINLPs to global optimality

Methods for finding (good) feasible solutions

Primal heuristics

Proof that there is no better solution

LP-based spatial branch and bound

Gleixner, Serrano · MINLPs in SCIP 3 / 24

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LP based spatial Branch & Bound

Build a (extended formulation of a) polyhedral relaxation R
Solve R and get solution x∗
If x∗ is feasible we are done. If not,
Try to strengthen R by separating x∗
When not possible, branch possibly on continuous variables (spatially)

Gleixner, Serrano · MINLPs in SCIP 4 / 24

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Building polyhedral relaxations: the problem

We can start with the variable’s bounds as our relaxation.
Then we have to solve the separation problem: Given

{x ∈ [l, u] : g(x) ≤ 0} and ¯ x s.t. g(¯ x) > 0 either

Find a separating inequality or
prove that none exists.
Very expensive for general g
However, when g is convex, it is as easy as computing a gradient:

g x g x x x

Idea: find convex underestimator g of g x
Then x

l u g x x l u g x

If g x

0 we can separate.

Gleixner, Serrano · MINLPs in SCIP 5 / 24

SLIDE 7

Building polyhedral relaxations: the problem

We can start with the variable’s bounds as our relaxation.
Then we have to solve the separation problem: Given

{x ∈ [l, u] : g(x) ≤ 0} and ¯ x s.t. g(¯ x) > 0 either

Find a separating inequality or
prove that none exists.
Very expensive for general g
However, when g is convex, it is as easy as computing a gradient:

g x g x x x

Idea: find convex underestimator g of g x
Then x

l u g x x l u g x

If g x

0 we can separate.

Gleixner, Serrano · MINLPs in SCIP 5 / 24

SLIDE 8

Building polyhedral relaxations: the problem

We can start with the variable’s bounds as our relaxation.
Then we have to solve the separation problem: Given

{x ∈ [l, u] : g(x) ≤ 0} and ¯ x s.t. g(¯ x) > 0 either

Find a separating inequality or
prove that none exists.
Very expensive for general g
However, when g is convex, it is as easy as computing a gradient:

g(¯ x) + ∇g(¯ x)(x − ¯ x) ≤ 0

Idea: find convex underestimator g of g x
Then x

l u g x x l u g x

If g x

0 we can separate.

Gleixner, Serrano · MINLPs in SCIP 5 / 24

SLIDE 9

Building polyhedral relaxations: the problem

We can start with the variable’s bounds as our relaxation.
Then we have to solve the separation problem: Given

{x ∈ [l, u] : g(x) ≤ 0} and ¯ x s.t. g(¯ x) > 0 either

Find a separating inequality or
prove that none exists.
Very expensive for general g
However, when g is convex, it is as easy as computing a gradient:

g(¯ x) + ∇g(¯ x)(x − ¯ x) ≤ 0

Idea: find convex underestimator ˆ

g of g(x)

Then x

l u g x x l u g x

If g x

0 we can separate.

Gleixner, Serrano · MINLPs in SCIP 5 / 24

SLIDE 10

Building polyhedral relaxations: the problem

We can start with the variable’s bounds as our relaxation.
Then we have to solve the separation problem: Given

{x ∈ [l, u] : g(x) ≤ 0} and ¯ x s.t. g(¯ x) > 0 either

Find a separating inequality or
prove that none exists.
Very expensive for general g
However, when g is convex, it is as easy as computing a gradient:

g(¯ x) + ∇g(¯ x)(x − ¯ x) ≤ 0

Idea: find convex underestimator ˆ

g of g(x)

Then {x ∈ [l, u] : g(x) ≤ 0} ⊆ {x ∈ [l, u] : ˆ

g(x) ≤ 0}

If ˆ

g(¯ x) > 0 we can separate.

Gleixner, Serrano · MINLPs in SCIP 5 / 24

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Building polyhedral relaxations: an example

x2 + y2 + 2 exp(xy3) ≤ 3 with x, y ∈ [−2, 2]
x2

y2 x2 y2 2 xy3

2
1

1 2

2
1

1 2

Admittedly, ad-hoc argument
In practice: if functions are simple enough, we know convex/concave

envelopes

If function is not simple enough, make it simpler!

Gleixner, Serrano · MINLPs in SCIP 6 / 24

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Building polyhedral relaxations: an example

x2 + y2 + 2 exp(xy3) ≤ 3 with x, y ∈ [−2, 2]
exp(·) > 0 ⇒ x2 + y2 ≤ x2 + y2 + 2 exp(xy3)
2
1

1 2

2
1

1 2

Admittedly, ad-hoc argument
In practice: if functions are simple enough, we know convex/concave

envelopes

If function is not simple enough, make it simpler!

Gleixner, Serrano · MINLPs in SCIP 6 / 24

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Building polyhedral relaxations: an example

x2 + y2 + 2 exp(xy3) ≤ 3 with x, y ∈ [−2, 2]
exp(·) > 0 ⇒ x2 + y2 ≤ x2 + y2 + 2 exp(xy3)
2
1

1 2

2
1

1 2

Admittedly, ad-hoc argument
In practice: if functions are simple enough, we know convex/concave

envelopes

If function is not simple enough, make it simpler!

Gleixner, Serrano · MINLPs in SCIP 6 / 24

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Building polyhedral relaxation in SCIP

x2 + y2 + 2 exp(xy3) ≤ 3
Introduce auxiliary variables
z1 = y3
z2 = xz1
x2 + y2 + 2 exp(z2) ≤ 3
We can find polyhedral relaxations of z1 = y3
For z2 = xz1 we have McCormick inequalities:

max{xz1 +z1x−xz1, x¯ z1 +z1¯ x−¯ x¯ z1} ≤ z2 ≤ min{x¯ z1 +z1x−x¯ z1, xz1 +z1¯ x−¯ xz1}

Finally, x2 + y2 + 2 exp(z2) is convex

Gleixner, Serrano · MINLPs in SCIP 7 / 24

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Spatial Branch and bound

Solutions might not be separable: conv{(x, y) : x2 = y, x ∈ [ℓ, u]} is x2 ≤ y ≤ ℓ2 + u2 − ℓ2 u − ℓ (x − ℓ) ∀x ∈ [ℓ, u]. Branching on a nonlinear variable in a nonconvex constraint allows for tighter relaxations:

1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0

Gleixner, Serrano · MINLPs in SCIP 8 / 24

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Spatial Branch and bound

Solutions might not be separable: conv{(x, y) : x2 = y, x ∈ [ℓ, u]} is x2 ≤ y ≤ ℓ2 + u2 − ℓ2 u − ℓ (x − ℓ) ∀x ∈ [ℓ, u]. Branching on a nonlinear variable in a nonconvex constraint allows for tighter relaxations:

1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0

Gleixner, Serrano · MINLPs in SCIP 8 / 24

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Outline

Introduction: LP-based Branch and Bound Spatial Branch and bound Heuristics Sub-NLP NLP diving Multi-start MPEC Undercover RENS Conclusion

Gleixner, Serrano · MINLPs in SCIP 9 / 24

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Sub-NLP

Idea: fix integer variables to integer values and run a local NLP solver
Good for: MINLP

Let ¯ x be LP-optimum of the current node’s relaxation

If there is a i ∈ I such that ¯

xi / ∈ Z → STOP

Fix xi to ¯

xi.

Solve remaining NLP to local optimality using ¯

x as initial point.

min

Extends MIP heuristics to MINLP

SCIP runs its MIP heuristics on MIP relaxation of MINLP
use heuristic’s proposed solution as x

Gleixner, Serrano · MINLPs in SCIP 10 / 24

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Sub-NLP

Idea: fix integer variables to integer values and run a local NLP solver
Good for: MINLP

Let ¯ x be LP-optimum of the current node’s relaxation

If there is a i ∈ I such that ¯

xi / ∈ Z → STOP

Fix xi to ¯

xi.

Solve remaining NLP to local optimality using ¯

x as initial point.

min

Extends MIP heuristics to MINLP

SCIP runs its MIP heuristics on MIP relaxation of MINLP
use heuristic’s proposed solution as ¯

x

Gleixner, Serrano · MINLPs in SCIP 10 / 24

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NLP diving

Idea: Solve NLP relaxations, fixing an integer variable after each NLP
Good for: MINLP
solve NLP relaxation
fix an integer variable
propagate
repeat
if fixing is infeasible, backtrack
undo last fixing and fix to another value
if infeasible again, abort

Gleixner, Serrano · MINLPs in SCIP 11 / 24

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Multi-start Heuristic [Chinneck et al. 2013]

Idea: use different starting points for NLP solver
Good for: NLPs without too many integer variables

4 ≤ x2 + y2 ≤ 9 4 ≤ (x − 2)2 + y2 ≤ 9 x ∈ [−4, 6] y ∈ [−4, 4]

Gleixner, Serrano · MINLPs in SCIP 12 / 24

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Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i gi xk gi xk

2

gi xk i

Update:

xk xk 1 n

n i 1

sk

i

3. identify clusters of points C1

Cp

4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

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Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i gi xk gi xk

2

gi xk i

Update:

xk xk 1 n

n i 1

sk

i

3. identify clusters of points C1

Cp

4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

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Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i := −gi(xk) ||∇gi(xk)||2 ∇gi(xk) ∀i

Update:

x+

k = xk + 1

n

∑

i=1

sk

i

3. identify clusters of points C1

Cp

4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

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Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i := −gi(xk) ||∇gi(xk)||2 ∇gi(xk) ∀i

Update:

x+

k = xk + 1

n

∑

i=1

sk

i

3. identify clusters of points C1

Cp

4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

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Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i := −gi(xk) ||∇gi(xk)||2 ∇gi(xk) ∀i

Update:

x+

k = xk + 1

n

∑

i=1

sk

i

3. identify clusters of points C1

Cp

4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

SLIDE 27

Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i := −gi(xk) ||∇gi(xk)||2 ∇gi(xk) ∀i

Update:

x+

k = xk + 1

n

∑

i=1

sk

i

3. identify clusters of points C1, . . . , Cp
4. for each cluster
build convex combination

y

1 Cj x Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

SLIDE 28

Multi-start in SCIP

Input: Nonlinear Constraints gi(x) ≤ 0

1. generate random points in [ℓ, u]
2. for each point xk:
sk

i := −gi(xk) ||∇gi(xk)||2 ∇gi(xk) ∀i

Update:

x+

k = xk + 1

n

∑

i=1

sk

i

3. identify clusters of points C1, . . . , Cp
4. for each cluster
build convex combination

y :=

1 |Cj|

∑

x∈Cj x

round each fractional integer

variable to closest integer

use resulting point as starting point

for NLP

Gleixner, Serrano · MINLPs in SCIP 13 / 24

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MPEC Heuristic [Schewe and Schmidt 2016]

Idea: write x ∈ {0, 1} as x(1 − x) = 0, relax to x(1 − x) ≤ ε and solve

sequences of NLP with ε → 0

Good for: Mixed Binary NLPs
MPEC stands for Mathematical Program with Equilibrium Constraints

min f(x) s.t. gj(x) ≤ 0 ∀j ∈ {1, . . . , m} xi ∈ [ℓi, ui] ∀i ∈ {1, . . . , k} xi ∈ {0, 1} ∀i ∈ {k + 1, . . . , n} xi(1 − xi) = 0

Gleixner, Serrano · MINLPs in SCIP 14 / 24

SLIDE 30

MPEC Heuristic [Schewe and Schmidt 2016]

Idea: write x ∈ {0, 1} as x(1 − x) = 0, relax to x(1 − x) ≤ ε and solve

sequences of NLP with ε → 0

Good for: Mixed Binary NLPs
MPEC stands for Mathematical Program with Equilibrium Constraints

min f(x) s.t. gj(x) ≤ 0 ∀j ∈ {1, . . . , m} xi ∈ [ℓi, ui] ∀i ∈ {1, . . . , k} xi ∈ [0, 1] ∀i ∈ {k + 1, . . . , n} xi(1 − xi) ≤ 0 ∀i ∈ {k + 1, . . . , n} (NLP)

Gleixner, Serrano · MINLPs in SCIP 14 / 24

SLIDE 31

MPEC Heuristic [Schewe and Schmidt 2016]

Idea: write x ∈ {0, 1} as x(1 − x) = 0, relax to x(1 − x) ≤ ε and solve

sequences of NLP with ε → 0

Good for: Mixed Binary NLPs
MPEC stands for Mathematical Program with Equilibrium Constraints

min f(x) s.t. gj(x) ≤ 0 ∀j ∈ {1, . . . , m} xi ∈ [ℓi, ui] ∀i ∈ {1, . . . , k} xi ∈ [0, 1] ∀i ∈ {k + 1, . . . , n} xi(1 − xi) ≤ ε ∀i ∈ {k + 1, . . . , n} (NLPε)

Gleixner, Serrano · MINLPs in SCIP 14 / 24

SLIDE 32

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x

solution of NLP using x as starting point

2. if x is feasible for NLP and xi

0 1 STOP

3. if x is feasible for NLP but not binary:
2 ; GOTO 1. with x

x

4. x is infeasible for NLP but satisfies xi 1

xi STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

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MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x is feasible for NLP and xi

0 1 STOP

3. if x is feasible for NLP but not binary:
2 ; GOTO 1. with x

x

4. x is infeasible for NLP but satisfies xi 1

xi STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 34

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x is feasible for NLP but not binary:
2 ; GOTO 1. with x

x

4. x is infeasible for NLP but satisfies xi 1

xi STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 35

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
2 ; GOTO 1. with x

x

4. x is infeasible for NLP but satisfies xi 1

xi STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 36

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x is infeasible for NLP but satisfies xi 1

xi STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 37

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x∗ is infeasible for NLPε but satisfies x∗

i (1 − x∗ i ) ≤ ε → STOP

5. x is infeasible for NLP and there are xi 1

xi

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 38

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x∗ is infeasible for NLPε but satisfies x∗

i (1 − x∗ i ) ≤ ε → STOP

5. x∗ is infeasible for NLPε and there are x∗

i (1 − x∗ i ) > ε

reset or fix xi for binary variables with xi 1

xi

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 39

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x∗ is infeasible for NLPε but satisfies x∗

i (1 − x∗ i ) ≤ ε → STOP

5. x∗ is infeasible for NLPε and there are x∗

i (1 − x∗ i ) > ε

reset or fix xi for binary variables with x∗

i (1 − x∗ i ) > ε

xi

xi 0 5 1

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 40

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x∗ is infeasible for NLPε but satisfies x∗

i (1 − x∗ i ) ≤ ε → STOP

5. x∗ is infeasible for NLPε and there are x∗

i (1 − x∗ i ) > ε

reset or fix xi for binary variables with x∗

i (1 − x∗ i ) > ε

xi :=

{ 0, x∗

i > 0.5

1,

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 41

MPEC in SCIP

Algorithm

0. choose ε ∈ [0, 1

4] and x ∈ Rn

1. x∗ := solution of NLPε using x as starting point
2. if x∗ is feasible for NLPε and x∗

i ∈ {0, 1} → STOP

3. if x∗ is feasible for NLPε but not binary:
ε := ε

2 ; GOTO 1. with x := x∗

4. x∗ is infeasible for NLPε but satisfies x∗

i (1 − x∗ i ) ≤ ε → STOP

5. x∗ is infeasible for NLPε and there are x∗

i (1 − x∗ i ) > ε

reset or fix xi for binary variables with x∗

i (1 − x∗ i ) > ε

xi :=

{ 0, x∗

i > 0.5

1,

therwise
GOTO 1.

Gleixner, Serrano · MINLPs in SCIP 15 / 24

SLIDE 42

Undercover [Berthold and Gleixner 2014]

Idea: fix variables so that MINLP transforms into a MIP
Good for: MIQCQP
Solve a relaxation (LP, NLP)
Identify minimum set C of variables to be fixed in
rder to obtain a MIP
Fix variables in C to relaxation’s solution.
Solve MIP
Postprocess: Fix integer values and solve NLP

Gleixner, Serrano · MINLPs in SCIP 16 / 24

SLIDE 43

Undercover in SCIP

Identifying minimum set C of variables to be fixed
If

∂2 ∂xixj gk(x) ̸= 0 then xi or xj must be fixed

Build graph with nodes xi and arcs {xi, xj} if

∂ ∂xixj gk(x) ̸= 0

Minimum vertex cover yields the set we are looking for.
Fix variables in C to relaxation’s solution
Fix a variable, if variable must be integer, fix to rounded value.
Propagate bounds: to avoid ”obvious” infeasible fixings
If next fixing value is outside bounds, choose closest bound or resolve

relaxation

If infeasible, backtrack: undo last fixing and try new value

Gleixner, Serrano · MINLPs in SCIP 17 / 24

SLIDE 44

RENS: The optimal rounding [Berthold 2014]

RENS stands for Relaxation Enforced Neighborhood Search
Idea: restrict integer variables around LP-relaxation’s solution and solve

remaining MINLP

Good for: MIP, MINLP
Let ¯

x be the LP solution

If |{i ∈ I : ¯

xi ∈ Z}| > p|I| for some p ∈ [0, 1] then

Change bounds of xi to {⌊¯

xi⌋, ⌈¯ xi⌉}

Solve smaller MINLP
This gives the feasible rounding with best
bjective value

Gleixner, Serrano · MINLPs in SCIP 18 / 24

SLIDE 45

Computational results for RENS

Basic LP solutions often show high integrality
Success rate seems to decreases when more integer variables are fixed
Roundability of MIP and MINLP are similar

Gleixner, Serrano · MINLPs in SCIP 19 / 24

SLIDE 46

Computational results for RENS

Basic LP solutions often show high integrality
Success rate seems to decreases when more integer variables are fixed
Roundability of MIP and MINLP are similar

Gleixner, Serrano · MINLPs in SCIP 20 / 24

SLIDE 47

Outline

Introduction: LP-based Branch and Bound Spatial Branch and bound Heuristics Sub-NLP NLP diving Multi-start MPEC Undercover RENS Conclusion

Gleixner, Serrano · MINLPs in SCIP 21 / 24

SLIDE 48

Conclusion

Summary

very brief introduction to LP-based branch and bound
overview of MINLP heuristics in SCIP

What is missing in SCIP

Non-linear Feasibility Pump
Problem specific heuristics

Thank you for your attention

Gleixner, Serrano · MINLPs in SCIP 22 / 24

SLIDE 49

Conclusion

Summary

very brief introduction to LP-based branch and bound
overview of MINLP heuristics in SCIP

What is missing in SCIP

Non-linear Feasibility Pump
Problem specific heuristics

Thank you for your attention

Gleixner, Serrano · MINLPs in SCIP 22 / 24

SLIDE 50

Conclusion

Summary

very brief introduction to LP-based branch and bound
overview of MINLP heuristics in SCIP

What is missing in SCIP

Non-linear Feasibility Pump
Problem specific heuristics

Thank you for your attention

Gleixner, Serrano · MINLPs in SCIP 22 / 24

SLIDE 51

Primal heuristic for MINLPs in SCIP

Ambros Gleixner, and Felipe Serrano

Zuse Institute Berlin · serrano@zib.de SCIP Optimization Suite · http://scip.zib.de Workshop on Discrepancy Theory and Integer Programming Amsterdam · June 12, 2018

SLIDE 52

Literature I

Timo Berthold. RENS – the optimal rounding. Mathematical Programming Computation, 6(1):33–54, 2014.

doi: 10.1007/s12532-013-0060-9.

Timo Berthold and Ambros M. Gleixner. Undercover: a primal MINLP heuristic exploring a largest sub-MIP. Mathematical Programming, 144(1-2):315–346, 2014.

doi: 10.1007/s10107-013-0635-2.

Lars Schewe and Martin Schmidt. Computing feasible points for minlps with mpecs. Technical report, Technical report, Optimization Online, 2016. URL http://www. optimization-online. org/DB_HTML/2016/12/5778. html, 2016.

Gleixner, Serrano · MINLPs in SCIP 23 / 24

SLIDE 53

Literature II

Laurence Smith, John Chinneck, and Victor Aitken. Improved constraint consensus methods for seeking feasibility in nonlinear programs. Computational Optimization and Applications, 54(3):555–578, 2013.

doi: 10.1007/s10589-012-9473-z.

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