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
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
Mixed-Integer Nonlinear Programs (MINLPs) min c T x s.t. g k ( x ) ≤ 0 ∀ k ∈ [ m ] x i ∈ Z ∀ i ∈ I ⊆ [ n ] x i ∈ [ ℓ i , u i ] ∀ i ∈ [ n ] The functions g k ∈ C 1 ([ ℓ, u ] , R ) can be 0 100 200 300 200 10 200 0 0 5 − 200 − 1 1 − 1 1 convex or nonconvex Gleixner, Serrano · MINLPs in SCIP 2 / 24
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
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
• 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 0 x l u g x 0 • If g x 0 we can separate. 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. Gleixner, Serrano · MINLPs in SCIP 5 / 24
• 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 0 x l u g x 0 • If g x 0 we can separate. 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 Gleixner, Serrano · MINLPs in SCIP 5 / 24
• Idea: find convex underestimator g of g x • Then x l u g x 0 x l u g x 0 • If g x 0 we can separate. 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 Gleixner, Serrano · MINLPs in SCIP 5 / 24
• Then x l u g x 0 x l u g x 0 • If g x 0 we can separate. 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 ) Gleixner, Serrano · MINLPs in SCIP 5 / 24
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
x 2 y 2 x 2 y 2 xy 3 • 0 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! Building polyhedral relaxations: an example 2 1 • x 2 + y 2 + 2 exp( xy 3 ) ≤ 3 with x , y ∈ [ − 2 , 2 ] 0 - 1 - 2 - 2 - 1 0 1 2 Gleixner, Serrano · MINLPs in SCIP 6 / 24
• 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! Building polyhedral relaxations: an example 2 1 • x 2 + y 2 + 2 exp( xy 3 ) ≤ 3 with x , y ∈ [ − 2 , 2 ] 0 • exp( · ) > 0 ⇒ x 2 + y 2 ≤ x 2 + y 2 + 2 exp( xy 3 ) - 1 - 2 - 2 - 1 0 1 2 Gleixner, Serrano · MINLPs in SCIP 6 / 24
Building polyhedral relaxations: an example 2 1 • x 2 + y 2 + 2 exp( xy 3 ) ≤ 3 with x , y ∈ [ − 2 , 2 ] 0 • exp( · ) > 0 ⇒ x 2 + y 2 ≤ x 2 + y 2 + 2 exp( xy 3 ) - 1 - 2 - 2 - 1 0 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
Building polyhedral relaxation in SCIP • x 2 + y 2 + 2 exp( xy 3 ) ≤ 3 • Introduce auxiliary variables • z 1 = y 3 • z 2 = xz 1 • x 2 + y 2 + 2 exp( z 2 ) ≤ 3 • We can find polyhedral relaxations of z 1 = y 3 • For z 2 = xz 1 we have McCormick inequalities: max { xz 1 + z 1 x − xz 1 , x ¯ z 1 + z 1 ¯ x − ¯ x ¯ z 1 } ≤ z 2 ≤ min { x ¯ z 1 + z 1 x − x ¯ z 1 , xz 1 + z 1 ¯ x − ¯ xz 1 } • Finally, x 2 + y 2 + 2 exp( z 2 ) is convex Gleixner, Serrano · MINLPs in SCIP 7 / 24
Branching on a nonlinear variable in a nonconvex constraint allows for tighter relaxations: Spatial Branch and bound Solutions might not be separable: conv { ( x , y ) : x 2 = y , x ∈ [ ℓ, u ] } is x 2 ≤ y ≤ ℓ 2 + u 2 − ℓ 2 u − ℓ ( x − ℓ ) ∀ x ∈ [ ℓ, u ] . 1.0 0.8 0.6 0.4 0.2 � 1.0 � 0.5 0.5 1.0 Gleixner, Serrano · MINLPs in SCIP 8 / 24
Spatial Branch and bound Solutions might not be separable: conv { ( x , y ) : x 2 = y , x ∈ [ ℓ, u ] } is x 2 ≤ y ≤ ℓ 2 + u 2 − ℓ 2 u − ℓ ( x − ℓ ) ∀ x ∈ [ ℓ, u ] . Branching on a nonlinear variable in a nonconvex constraint allows for tighter relaxations: 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 � 1.0 � 0.5 0.5 1.0 � 1.0 � 0.5 0.5 1.0 Gleixner, Serrano · MINLPs in SCIP 8 / 24
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
Extends MIP heuristics to MINLP • SCIP runs its MIP heuristics on MIP relaxation of MINLP • use heuristic’s proposed solution as x 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 ¯ x i / ∈ Z → STOP • Fix x i to ¯ x i . • Solve remaining NLP to local optimality using ¯ x as initial point. min Gleixner, Serrano · MINLPs in SCIP 10 / 24
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 ¯ x i / ∈ Z → STOP • Fix x i to ¯ x i . • 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
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
Multi-start Heuristic [Chinneck et al. 2013] • Idea: use different starting points for NLP solver • Good for: NLPs without too many integer variables 4 ≤ x 2 + y 2 ≤ 9 4 ≤ ( x − 2 ) 2 + y 2 ≤ 9 x ∈ [ − 4 , 6 ] y ∈ [ − 4 , 4 ] Gleixner, Serrano · MINLPs in SCIP 12 / 24
2. for each point x k : g i x k • s k g i x k i i 2 g i x k • Update: n 1 s k x k x k i n i 1 3. identify clusters of points C 1 C p 4. for each cluster • build convex combination 1 y C j x x C j • round each fractional integer variable to closest integer • use resulting point as starting point for NLP Multi-start in SCIP Input: Nonlinear Constraints g i ( x ) ≤ 0 1. generate random points in [ ℓ, u ] Gleixner, Serrano · MINLPs in SCIP 13 / 24
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