Transcription of MixedIntegerLinearProgramming
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MixedIntegerLinearProgramming
Mixed Integer Linear ProgrammingCombinatorial Problem Solving (CPS)Javier Larrosa Albert Oliveras Enric Rodr guez-CarbonellMay 3, 2021 Mixed Integer Linear Programs2 / 61 Amixed integer linear program(MILP, MIP) is of the formmincTxAx=bx 0xi Z i I If all variables need to be integer,it is called a(pure) integer linear program(ILP, IP) If all variables need to be0or1(binary, boolean),it is called a0 1linear programComplexity: LP vs. IP3 / 61 Including integer variables increases enourmously the modeling power,at the expense of more complexity LP s can be solved inpolynomial timewith interior-point methods(ellipsoid method, Karmarkar s algorithm) integer programming is anNP-hardproblem. So: There isno known polynomial-time algorithm There arelittle chancesthat one will ever be found Even small problems may be hard to solve What follows is one of the many approaches(and one of the most successful) for attacking IP sLP Relaxation of a MIP4 / 61 Given a MIP(IP)mincTxAx=bx 0xi Z i Iitslinear relaxationis the LP obtained by dropping integrality constraints:(LP)mincTxAx=bx 0 Can we solveIPby solvi
Complexity: LPvs. IP 3/61 Including integer variables increases enourmously the modeling power, at the expense of more complexity LP’s can be solved in polynomial time with interior-point methods (ellipsoid method, Karmarkar’s algorithm) Integer Programming is an NP-hard problem. So: There is no known polynomial-time algorithm There are little chances that one will ever be …
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