A good recipe for solving MINLPs The Problem The Ingredients VNS - PowerPoint PPT Presentation

A good recipe for solving MINLPs L. Liberti, G. Nannicini, N. Mladenovic A good recipe for solving MINLPs The Problem The Ingredients VNS Local Branching Leo Liberti 1 , Giacomo Nannicini 1 , Nenad Mladenovi´ c 2 BB for MINLPs SQP The RECIPE MINLPLib 1 LIX, ´ Ecole Polytechnique, France Computational 2 Brunel University, London, UK , and Experiments Institute of Mathematics, Academy of Sciences, Belgrade, Serbia Conclusions 31 October 2008

A good recipe for solving Summary of Talk MINLPs L. Liberti, G. Nannicini, N. Mladenovic 1 The Problem The Problem The Ingredients 2 The Ingredients VNS VNS Local Branching BB for MINLPs Local Branching SQP The RECIPE Branch-and-Bound for cMINLPs MINLPLib Sequential Quadratic Programming Computational Experiments 3 The RECIPE Conclusions 4 MINLPLib 5 Computational Experiments 6 Conclusions

A good recipe for solving MINLPs 1 The Problem L. Liberti, G. Nannicini, N. Mladenovic 2 The Ingredients The Problem VNS The Ingredients Local Branching VNS Local Branching Branch-and-Bound for cMINLPs BB for MINLPs SQP Sequential Quadratic Programming The RECIPE MINLPLib 3 The RECIPE Computational Experiments Conclusions 4 MINLPLib 5 Computational Experiments 6 Conclusions

A good recipe for solving Problem Classes MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem The Ingredients VNS • We distinguish four classes of mathematical programs: Local Branching BB for MINLPs 1 Linear Programs (LPs) SQP 2 Mixed-Integer Linear Programs (MILPs) The RECIPE 3 Nonlinear Programs (NLPs) MINLPLib 4 Mixed-Integer Nonlinear Programs (MINLPs) Computational Experiments • For nonlinear programs, convexity of objective function Conclusions and constraints is also important to determine the problem’s difficulty

A good recipe for solving A Difficult Task MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem • We address the most difficult problem class: nonconvex The Ingredients VNS MINLPs Local Branching BB for MINLPs SQP • Obviously, this class is also the most expressive The RECIPE • Difficulties arise from both nonconvexity and integrality MINLPLib • We cannot aim for optimality, hence we would easily settle Computational Experiments for a fast and reliable heuristic (who would not?) Conclusions • Especially true for industrial applications: we need something that is easy to use and does not need hundreds of parameters

A good recipe for solving MINLPs 1 The Problem L. Liberti, G. Nannicini, N. Mladenovic 2 The Ingredients The Problem VNS The Ingredients Local Branching VNS Local Branching Branch-and-Bound for cMINLPs BB for MINLPs SQP Sequential Quadratic Programming The RECIPE MINLPLib 3 The RECIPE Computational Experiments Conclusions 4 MINLPLib 5 Computational Experiments 6 Conclusions

A good recipe for solving VNS: Introduction MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem The Ingredients VNS Local Branching • Applicable to discrete and continuous problems BB for MINLPs SQP • Uses any local search as a black-box The RECIPE • In its basic form, easy to implement MINLPLib Computational • Few configurable parameters Experiments • Used for a variety of problem, large amount of references Conclusions [Hansen and Mladenovi´ c, 2001]

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem The Ingredients VNS Local Branching BB for MINLPs SQP The RECIPE MINLPLib Computational Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic � random 1 � � � The Problem The Ingredients VNS Local Branching BB for MINLPs SQP The RECIPE MINLPLib Computational Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE � local minimum 1 � MINLPLib � � Computational Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE � local minimum 1 � MINLPLib � � Computational k=1 Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE local minimum 1 � � MINLPLib � � � � � � Computational random 2 k=1 Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE local search 2 local minimum 1,2 � � MINLPLib � � � � � � Computational random 2 k=1 Experiments Conclusions

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE local search 2 local minimum 1,2 � � MINLPLib � � � � � � Computational random 2 k=1 Experiments Conclusions k=2

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE local search 2 local minimum 1,2 � � MINLPLib � � � � � � Computational random 2 k=1 Experiments � � � � random 3 � � Conclusions k=2

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP The RECIPE local search 2 local minimum 1,2 � � MINLPLib � � � � � � Computational random 2 k=1 Experiments � � � � random 3 � � Conclusions k=2 � � � � � � local minimum 3

A good recipe for solving VNS: Algorithm (sketch) MINLPs L. Liberti, G. Nannicini, N. Mladenovic random 1 � � � � The Problem The Ingredients local search 1 VNS Local Branching BB for MINLPs SQP k=Kmax The RECIPE local search 2 local minimum 1,2 � � MINLPLib � � . � � � � . Computational random 2 k=1 . Experiments � � � � random 3 k=1 � � Conclusions k=2 � � � � � � local minimum 3

A good recipe for solving VNS: Configurable parameters MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem The Ingredients VNS Local Branching • Maximum neighbourhood size: k max BB for MINLPs SQP • Number of local searches in each neighbourhood L The RECIPE • Other parameters if further termination conditions MINLPLib included, e.g.: Computational Experiments • maximum running time Conclusions • desired objective function value • . . .

A good recipe for solving Local Branching MINLPs L. Liberti, G. Nannicini, N. Mladenovic The Problem • Efficient heuristic to find good solution for MILPs. The Ingredients VNS • We consider Local Branching for binary variables Local Branching BB for MINLPs [Fischetti and Lodi, 2005] SQP • Let B be the set of indices of binary variables, let x ∗ be a The RECIPE MINLPLib binary feasible solution, and let k ∈ Z , k > 0 Computational • We explore the k -neighbourhood of x ∗ by enforcing the Experiments Conclusions local branching constraint: � � (1 − x i ) + x i ≤ k i ∈ B : x ∗ i =1 i ∈ B : x ∗ i =0

A good recipe for solving Branch-and-Bound for cMINLPs MINLPs L. Liberti, G. Nannicini, N. Mladenovic • For convex problems, we can use Branch-and-Bound The Problem methods, since obtaining lower bounds is easy The Ingredients VNS • A typical approach is to relax integrality and solve the Local Branching BB for MINLPs resulting convex NLP to optimality SQP The RECIPE • Branching is done on integer variables MINLPLib • If the problem is nonconvex, BB can be used as an Computational Experiments heuristic (i.e., it finds a local optimum) Conclusions • The BB local NLP subsolver needs an initial feasible starting point • We supply such point computing a constraint feasible (possibly non integral!) solution with an SQP method; this increases our chances of finding an integer feasible point

A good recipe for solving Sequential Quadratic MINLPs L. Liberti, Programming G. Nannicini, N. Mladenovic The Problem The Ingredients • SQP methods find local optima to nonconvex MINLPs VNS Local Branching BB for MINLPs • Idea: solve a sequence of quadratic approximation of the SQP original problem subject to a linearization of the The RECIPE constraints MINLPLib Computational • The quadratic approximation is obtained by a convex Experiments model of the objective function Hessian at a current Conclusions solution point • SQP are now at a very advanced stage: there are good chances of finding a constraint feasible point starting from any given initial point, even for large NLPs