Undercover A primal heuristic for MINLP based on sub-MIPs generated - PowerPoint PPT Presentation

Undercover A primal heuristic for MINLP based on sub-MIPs generated by set covering Ambros M. Gleixner joint work with Timo Berthold Zuse Institute Berlin M ATHEON Berlin Mathematical School Aussois International Workshop on Combinatorial Optimization, 6 January 2010

Outline 1 Introduction: primal solutions for MINLP 2 A generic algorithm for Undercover 3 Finding minimum covers Covering MIQCPs General covering problems 4 First experiments with MIQCPs 5 Extensions: fix-and-propagate etc. 6 Variations: convexification & domain reduction 7 Conclusion 2 / 35

Mixed-integer nonlinear programming An MINLP is an optimisation problem of the form d T x minimise subject to g i ( x ) � 0 for i = 1 , . . . , m , (1) L k � x k � U k for k = 1 , . . . , n , x k ∈ Z for k ∈ I , with I ⊆ { 1 , ..., n } , d ∈ R n , g i : R n → R , L k ∈ R ∪ {−∞} , U k ∈ R ∪ {∞} . 3 / 35

Mixed-integer nonlinear programming An MINLP is an optimisation problem of the form d T x minimise subject to g i ( x ) � 0 for i = 1 , . . . , m , (1) L k � x k � U k for k = 1 , . . . , n , x k ∈ Z for k ∈ I , with I ⊆ { 1 , ..., n } , d ∈ R n , g i : R n → R , L k ∈ R ∪ {−∞} , U k ∈ R ∪ {∞} . ⊲ Special case MIQCP: g i ( x ) = x T A i x + b i T x + c i (2) with A i ∈ R n × n symmetric, b i ∈ R n , c i ∈ R . 3 / 35

Mixed-integer nonlinear programming An MINLP is an optimisation problem of the form d T x minimise subject to g i ( x ) � 0 for i = 1 , . . . , m , (1) L k � x k � U k for k = 1 , . . . , n , x k ∈ Z for k ∈ I , with I ⊆ { 1 , ..., n } , d ∈ R n , g i : R n → R , L k ∈ R ∪ {−∞} , U k ∈ R ∪ {∞} . ⊲ Special case MIQCP: g i ( x ) = x T A i x + b i T x + c i (2) with A i ∈ R n × n symmetric, b i ∈ R n , c i ∈ R . ⊲ Main classification: def convex ⇐ ⇒ g i convex for all i = 1 , . . . , m (3) vs. nonconvex MINLPs. 3 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � 4 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � Simple NLP Rounding � � Fractional Diving & Vectorlength Diving BonamiGon¸ calves08 ( � ) � Iterative Rounding NanniciniBelotti � � 4 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � Simple NLP Rounding � � Fractional Diving & Vectorlength Diving BonamiGon¸ calves08 ( � ) � Iterative Rounding NanniciniBelotti � � nonconvex obj. Feasibility Pump BonamiCornu´ ejolsLodiMargot08 � convex feas. region D’AmbrosioFrangioniLibertiLodi09 � � LinderothAbhishekLeyfferSartenaer08 � 4 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � Simple NLP Rounding � � Fractional Diving & Vectorlength Diving BonamiGon¸ calves08 ( � ) � Iterative Rounding NanniciniBelotti � � nonconvex obj. Feasibility Pump BonamiCornu´ ejolsLodiMargot08 � convex feas. region D’AmbrosioFrangioniLibertiLodi09 � � LinderothAbhishekLeyfferSartenaer08 � Local Branching NanniciniBelottiLiberti08 � � 4 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � Simple NLP Rounding � � Fractional Diving & Vectorlength Diving BonamiGon¸ calves08 ( � ) � Iterative Rounding NanniciniBelotti � � nonconvex obj. Feasibility Pump BonamiCornu´ ejolsLodiMargot08 � convex feas. region D’AmbrosioFrangioniLibertiLodi09 � � LinderothAbhishekLeyfferSartenaer08 � Local Branching NanniciniBelottiLiberti08 � � RECIPE LibertiNanniciniMladenovi´ c08 � � 4 / 35

Primal solutions for generic MINLP Source convex nonconvex Feasible relaxation solution � � MIP heuristics for linear outer approximations � � NLP local search with fixed integralities � � Simple NLP Rounding � � Fractional Diving & Vectorlength Diving BonamiGon¸ calves08 ( � ) � Iterative Rounding NanniciniBelotti � � nonconvex obj. Feasibility Pump BonamiCornu´ ejolsLodiMargot08 � convex feas. region D’AmbrosioFrangioniLibertiLodi09 � � LinderothAbhishekLeyfferSartenaer08 � Local Branching NanniciniBelottiLiberti08 � � RECIPE LibertiNanniciniMladenovi´ c08 � � RENS BertholdHeinzVigerske09 (for MIQCPs) � � . . . 4 / 35

Outline 1 Introduction: primal solutions for MINLP 2 A generic algorithm for Undercover 3 Finding minimum covers Covering MIQCPs General covering problems 4 First experiments with MIQCPs 5 Extensions: fix-and-propagate etc. 6 Variations: convexification & domain reduction 7 Conclusion 5 / 35

Motivation ⊲ Common paradigm in MIP heuristics (e.g. RINS, DINS, RENS): fix a subset of variables � easy subproblem � solve “easy” in MIP context: few integralities “easy” in MINLP context rather: few nonlinearities 6 / 35

Motivation ⊲ Common paradigm in MIP heuristics (e.g. RINS, DINS, RENS): fix a subset of variables � easy subproblem � solve “easy” in MIP context: few integralities “easy” in MINLP context rather: few nonlinearities ⊲ Observation: Any MINLP can be reduced to a MIP by fixing (only sufficiently many) variables. Experience: For several practically relevant MIQCPs comparatively few fixings are sufficient! 6 / 35

Motivation ⊲ Common paradigm in MIP heuristics (e.g. RINS, DINS, RENS): fix a subset of variables � easy subproblem � solve “easy” in MIP context: few integralities “easy” in MINLP context rather: few nonlinearities ⊲ Observation: Any MINLP can be reduced to a MIP by fixing (only sufficiently many) variables. Experience: For several practically relevant MIQCPs comparatively few fixings are sufficient! ⊲ Idea: try to identify a small subset of variables to fix in order to obtain a mixed-integer linear subproblem. 6 / 35

Definitions Definition (cover of a function) Let ⊲ a function g : D → R , x �→ g ( x ) on a domain D ⊆ R n , ⊲ a point x ⋆ ∈ D , and ⊲ a set C ⊆ { 1 , . . . , n } of variable indices be given. We call C an x ⋆ -cover of g if and only if the set { ( x , g ( x )) | x ∈ D , x k = x ⋆ k for all k ∈ C} (4) is affine. We call C a (global) cover of g if and only if C is an x ⋆ -cover of g for all x ⋆ ∈ D . 7 / 35

Definitions Definition (cover of an MINLP) Let ⊲ P be an MINLP of form (1), ⊲ x ⋆ ∈ [ L , U ] , and ⊲ C ⊆ { 1 , . . . , n } be a set of variable indices of P . We call C an x ⋆ -cover of P if and only if C is an x ⋆ -cover for g 1 , . . . , g m . We call C a (global) cover of P if and only if C is an x ⋆ -cover of P for all x ⋆ ∈ [ L , U ] . 8 / 35

A generic algorithm Input : MINLP P as in (1) 1 begin 2 compute a solution x ⋆ 3 of an approximation of P round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35

A generic algorithm Input : MINLP P as in (1) 1 begin 2 compute a solution x ⋆ 3 of an approximation of P round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35

A generic algorithm Input : MINLP P as in (1) 1 begin 2 compute a solution x ⋆ 3 of an approximation of P round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35

A generic algorithm Input : MINLP P as in (1) 1 begin 2 compute a solution x ⋆ 3 of an approximation of P round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35

A generic algorithm Input : MINLP P as in (1) 1 begin 2 compute a solution x ⋆ 3 of an approximation of P round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35

A generic algorithm Input : MINLP P as in (1) 1 Remarks: begin 2 compute a solution x ⋆ ⊲ As an approximation e.g. use an 3 of an approximation of P LP or NLP relaxation within a branch-and-bound solver. round x ⋆ k for all k ∈ I 4 determine an 5 x ⋆ -cover C of P solve the sub-MIP of P 6 given by fixing x k = x ⋆ k for all k ∈ C end 7 9 / 35