Introducing Adaptive Algorithmic Behavior of Primal Heuristics in - PowerPoint PPT Presentation

joint work with Matthias Miltenberger and Jakob Witzig Monash University, Melbourne, Australia, 22 March, 2019 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 1/32 Introducing Adaptive Algorithmic Behavior of Primal Heuristics in SCIP for Solving Mixed Integer Programs Gregor Hendel , hendel@zib.de

A research institute and computing center of the State of Berlin with research units: • Numerical Analysis and Modeling • Visualization and Data Analysis • Optimization: • Scientific Information Systems • Computer Science and High Performance Computing Gregor Hendel – SCIP’s Adaptive Primal Heuristics 2/32 Zuse Institute Berlin – Fast Algorithms, Fast Computers Energy – Transportation – Health – Mathematical Optimization Methods

3/32 Many international contributors and users Gregor Hendel – SCIP’s Adaptive Primal Heuristics FICO Xpress, Gurobi, MOSEK, and GAMS • 7 former developers are now building commercial optimization soħtware at CPLEX, • 10 awards for Masters and PhD theses: MOS, EURO, GOR, DMV Careers • more than 10 000 downloads per year from 100+ countries • RWTH Aachen: GCG 26 active developers • FAU Erlangen-Nürnberg: SCIP • TU Darmstadt: SCIP and SCIP-SDP • ZIB: SCIP, SoPlex, UG, ZIMPL 4 development centers in Germany • 8 postdocs and professors • 14 running PhD projects • 4 running Bachelor and Master projects Meet the SCIP Team

Introduction Large Neighborhood Search for MIP Multi-Armed Bandit Selection SCIP’s Adaptive LNS Reward Function for LNS Computational Results Diving & Adaptive Diving Outlook: Adaptive LP Pricing Gregor Hendel – SCIP’s Adaptive Primal Heuristics 4/32 Overview

Introduction

c T x s.t. (MIP) Solution method: • typically solved with branch-and-cut • at each node, an LP relaxation is (re-)solved with the dual Simplex algorithm • primal heuristics, e.g., Large Neighborhood Search and diving methods, support the solution process Gregor Hendel – SCIP’s Adaptive Primal Heuristics 5/32 Mixed Integer Programs min Ax ≥ b ℓ ≤ x ≤ u x ∈ { 0 , 1 } n b × Z n i − n b × Q n − n i

Introduction Large Neighborhood Search for MIP

Large Neighborhood Search (LNS) heuristics solve auxiliary MIPs and can be c T distinguished by their respective neighborhoods. Gregor Hendel – SCIP’s Adaptive Primal Heuristics 6/32 LNS and the Auxiliary MIP Auxiliary MIP Let P be a MIP with solution set F P . For a polyhedron N ⊆ Q n and objective coeffjcients c aux ∈ Q n , a MIP P aux defined as { } aux x | x ∈ F P ∩ N min is called an auxiliary MIP of P , and N is called neighborhood.

• Improvement neighborhood n c T x 7/32 x inc Gregor Hendel – SCIP’s Adaptive Primal Heuristics c dual c T x inc 1 x obj Typical LNS Neighborhoods Let M ⊆ { 1 , . . . , n i } , x ∗ ∈ Q n . • Fixing neighborhood x ∈ Q n | x j = x ∗ N fix ( M , x ∗ ) := { } j ∀ j ∈ M

Gregor Hendel – SCIP’s Adaptive Primal Heuristics 7/32 Typical LNS Neighborhoods Let M ⊆ { 1 , . . . , n i } , x ∗ ∈ Q n . • Fixing neighborhood x ∈ Q n | x j = x ∗ N fix ( M , x ∗ ) := { } j ∀ j ∈ M • Improvement neighborhood x ∈ Q n | c T x ≤ ( 1 − δ ) · c T x inc + δ · c dual } { N obj ( δ, x inc ) :=

8/32 x Gregor Hendel – SCIP’s Adaptive Primal Heuristics x inc obj d max b x inc x n LBranch Relaxation Induced Neighborhood Search (RINS) [Danna et al., 2005] Local Branching [Fischetti and Lodi, 2003] Examples of LNS Heuristics N RINS := N fix ( M = ({ x lp , x inc }) , x inc ) ∩ N obj ( δ, x inc ) .

8/32 Local Branching [Fischetti and Lodi, 2003] Gregor Hendel – SCIP’s Adaptive Primal Heuristics Relaxation Induced Neighborhood Search (RINS) [Danna et al., 2005] Examples of LNS Heuristics N RINS := N fix ( M = ({ x lp , x inc }) , x inc ) ∩ N obj ( δ, x inc ) . x ∈ Q n | { � � x − x inc � } N LBranch := b ≤ d max ∩ N obj ( δ, x inc ) � � �

• RENS [Berthold, 2014] • Proximity [Fischetti and Monaci, 2014] • DINS [Ghosh, 2007] • Zeroobjective [in SCIP, Gurobi, XPress,…] • Analytic Center Search [Berthold et al., 2017] • Relaxation Induced Neighborhood Search (RINS) [Danna et al., 2005] • Local Branching [Fischetti and Lodi, 2003] • Crossover, Mutation [Rothberg, 2007] • … Gregor Hendel – SCIP’s Adaptive Primal Heuristics 9/32 Famous LNS Heuristics

• Relaxation Induced Neighborhood Search (RINS) [Danna et al., 2005] • Local Branching [Fischetti and Lodi, 2003] • Crossover, Mutation [Rothberg, 2007] • Proximity [Fischetti and Monaci, 2014] • Zeroobjective [in SCIP, Gurobi, XPress,…] • Analytic Center Search [Berthold et al., 2017] • … Gregor Hendel – SCIP’s Adaptive Primal Heuristics 9/32 Famous LNS Heuristics • RENS [Berthold, 2014] • DINS [Ghosh, 2007]

Introduction Multi-Armed Bandit Selection

• stochastic i.i.d. rewards for each action over time • adversarial an opponent tries to maximize the player’s regret. Two main scenarios: Literature: [Bubeck and Cesa-Bianchi, 2012] Gregor Hendel – SCIP’s Adaptive Primal Heuristics 10/32 The Multi-Armed Bandit Problem • Discrete time steps t = 1 , 2 , . . . • Finite set of actions H 1. Choose h t ∈ H 2. Observe reward r ( h t , t ) ∈ [ 0 , 1 ] 3. Goal: Maximize ∑ t r ( h t , t )

• adversarial an opponent tries to maximize the player’s regret. Two main scenarios: Literature: [Bubeck and Cesa-Bianchi, 2012] Gregor Hendel – SCIP’s Adaptive Primal Heuristics 10/32 The Multi-Armed Bandit Problem • Discrete time steps t = 1 , 2 , . . . • Finite set of actions H 1. Choose h t ∈ H 2. Observe reward r ( h t , t ) ∈ [ 0 , 1 ] 3. Goal: Maximize ∑ t r ( h t , t ) • stochastic i.i.d. rewards for each action over time

Literature: [Bubeck and Cesa-Bianchi, 2012] Two main scenarios: Gregor Hendel – SCIP’s Adaptive Primal Heuristics 10/32 The Multi-Armed Bandit Problem • Discrete time steps t = 1 , 2 , . . . • Finite set of actions H 1. Choose h t ∈ H 2. Observe reward r ( h t , t ) ∈ [ 0 , 1 ] 3. Goal: Maximize ∑ t r ( h t , t ) • stochastic i.i.d. rewards for each action over time • adversarial an opponent tries to maximize the player’s regret.

Upper Confidence Bound (UCB) r h t T h t Exp.3 p h t 1 if t , H t if t 11/32 t w h t h w h t 1 Individual parameters 0 can be calibrated to the problem at hand. Gregor Hendel – SCIP’s Adaptive Primal Heuristics 1 1 1 1 h h t and Bandit Algorithms ¯ Let T h ( t ) = ∑ r h ( t ) = ∑ 1 h = h t r h , t 1 h = h t T h ( t ) t ′ ≤ t t ′ ≤ t ε -greedy √ |H| Select heuristic at random with probability ε t = ε t , otherwise use best.

Exp.3 11/32 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 0 can be calibrated to the problem at hand. and Individual parameters 1 1 t w h h w h t 1 p h t Bandit Algorithms ¯ Let T h ( t ) = ∑ r h ( t ) = ∑ 1 h = h t r h , t 1 h = h t T h ( t ) t ′ ≤ t t ′ ≤ t ε -greedy √ |H| Select heuristic at random with probability ε t = ε t , otherwise use best. Upper Confidence Bound (UCB) { √ }  α ln( 1 + t ) ¯ argmax r h ( t − 1 ) + if t > |H| ,  T h ( t − 1 ) h t ∈ h ∈H { H t } if t ≤ |H| . 

11/32 1 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 0 can be calibrated to the problem at hand. and Individual parameters 1 Bandit Algorithms ¯ Let T h ( t ) = ∑ r h ( t ) = ∑ 1 h = h t r h , t 1 h = h t T h ( t ) t ′ ≤ t t ′ ≤ t ε -greedy √ |H| Select heuristic at random with probability ε t = ε t , otherwise use best. Upper Confidence Bound (UCB) { √ }  α ln( 1 + t ) ¯ argmax r h ( t − 1 ) + if t > |H| ,  T h ( t − 1 ) h t ∈ h ∈H { H t } if t ≤ |H| .  Exp.3 exp( w h , t ) p h , t = ( 1 − γ ) · h ′ exp( w h ′ , t ) + γ · ∑ |H|

11/32 1 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 1 and Bandit Algorithms ¯ Let T h ( t ) = ∑ r h ( t ) = ∑ 1 h = h t r h , t 1 h = h t T h ( t ) t ′ ≤ t t ′ ≤ t ε -greedy √ |H| Select heuristic at random with probability ε t = ε t , otherwise use best. Upper Confidence Bound (UCB) { √ }  α ln( 1 + t ) ¯ argmax r h ( t − 1 ) + if t > |H| ,  T h ( t − 1 ) h t ∈ h ∈H { H t } if t ≤ |H| .  Exp.3 exp( w h , t ) p h , t = ( 1 − γ ) · h ′ exp( w h ′ , t ) + γ · ∑ |H| Individual parameters α, ε, γ ≥ 0 can be calibrated to the problem at hand.

SCIP’s Adaptive LNS

• controls 8 neighborhoods • further algorithmic steps: generic fixings, adaptive fixing rate • released with SCIP 5.0, improved in SCIP 6.0 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 12/32 Adaptive Large Neighborhood Search • new primal heuristic plugin heur_alns.c • neighborhoods are bandit-selected based on their reward

SCIP’s Adaptive LNS Reward Function for LNS

r gap r sol Solution Reward r sol h t t Gap Reward r fail r gap h t t Failure Penalty r fail h t t h t t n h t 13/32 1 1 scaling (opt.) 2 2 1 Default settings in ALNS: 1 0 8 2 0 5 Gregor Hendel – SCIP’s Adaptive Primal Heuristics 1 1 r alns n lim x new if x old 1 c dual c T x old c T x new c T x old , else 0 x new , if x old 1 Rewarding Neighborhoods Goal A suitable reward function r alns ( h t , t ) ∈ [ 0 , 1 ]