Improved strategies for branching on general disjunctions G. Cornu´ ejols, Improved strategies for branching L. Liberti, G. Nannicini on general disjunctions Introduction Theoretical foundations ejols 1 , Leo Liberti 2 , Giacomo Nannicini 2 G´ erard Cornu´ Improved general disjunctions Computational 1 Carnegie Mellon University, Pittsburgh, PA, USA , and experiments (1) LIF, Facult´ e de Sciences de Luminy, Marseille, France 2 LIX, ´ A combined Ecole Polytechnique, France branching algorithm Computational experiments (2) January 12, 2009
Improved strategies for Summary of talk branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini 1 Introduction Introduction Theoretical 2 Theoretical foundations foundations Improved general 3 Improved general disjunctions disjunctions Computational experiments (1) 4 Computational experiments (1) A combined branching algorithm 5 A combined branching algorithm Computational experiments (2) 6 Computational experiments (2)
Improved strategies for Mixed-Integer Linear Programs branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • A mathematical program with linear objective function, foundations linear constraints and both continuous and integer Improved general variables is a Mixed-Integer Linear Program (MILP) disjunctions Computational • MILPs arise in several real-life situations experiments (1) • MILP solvers are often used as a tool in the context of A combined solving Mixed-Integer Nonlinear Programs branching algorithm Computational experiments (2)
Improved strategies for Mixed-Integer Linear Programs branching on general disjunctions G. Cornu´ ejols, L. Liberti, • Consider the following MILP in standard form: G. Nannicini Introduction c ⊤ x min Theoretical Ax = b foundations P x ≥ 0 Improved general ∀ j ∈ N I x j ∈ Z , disjunctions Computational experiments where c ∈ R n , b ∈ R m , A ∈ R m × n and (1) N I ⊂ N = { 1 , . . . , n } . A combined branching algorithm • The Linear Program (LP) relaxation of P is obtained by dropping the integrality constraints, and we denote it by ¯ Computational P experiments (2) x to ¯ • If the optimal solution ¯ P is integral, then it is optimal for P
Improved strategies for Branch-and-Bound branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • The standard method to solve MILPs is with a foundations Branch-and-Bound (BB) algorithm Improved general • There are three basic necessary ingredients in the BB disjunctions algorithm: Computational experiments 1 Obtaining lower bounds (1) 2 Obtaining upper bounds A combined branching 3 Dividing a subproblem algorithm Computational experiments (2)
Improved strategies for Branch-and-Bound branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • The standard method to solve MILPs is with a foundations Branch-and-Bound (BB) algorithm Improved general • There are three basic necessary ingredients in the BB disjunctions algorithm: Computational experiments 1 Obtaining lower bounds ← LP relaxation (1) 2 Obtaining upper bounds A combined branching 3 Dividing a subproblem algorithm Computational experiments (2)
Improved strategies for Branch-and-Bound branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • The standard method to solve MILPs is with a foundations Branch-and-Bound (BB) algorithm Improved general • There are three basic necessary ingredients in the BB disjunctions algorithm: Computational experiments 1 Obtaining lower bounds ← LP relaxation (1) 2 Obtaining upper bounds ← LP relaxation, heuristics A combined branching 3 Dividing a subproblem algorithm Computational experiments (2)
Improved strategies for Branch-and-Bound branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • The standard method to solve MILPs is with a foundations Branch-and-Bound (BB) algorithm Improved general • There are three basic necessary ingredients in the BB disjunctions algorithm: Computational experiments 1 Obtaining lower bounds ← LP relaxation (1) 2 Obtaining upper bounds ← LP relaxation, heuristics A combined branching 3 Dividing a subproblem ← our focus algorithm Computational experiments (2)
Improved strategies for Branching on single variables branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction • Branching is usually done by changing the bounds of an Theoretical foundations integer constrained variable: Improved x be the optimal solution to ¯ • Let ¯ P , and let i ∈ N I such general disjunctions that ¯ x i is fractional Computational • We divide P into P 1 and P 2 adding the constraints experiments x i ≤ ⌊ ¯ x i ⌋ (left branch) and x i ≥ ⌈ ¯ x i ⌉ (right branch) to (1) the two subproblems, respectively A combined branching • Very easy and fast approach algorithm Computational experiments (2)
Improved strategies for Branching on general disjunctions branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical • Branching can occur with respect to any direction π ∈ R n foundations by adding the constraints π ⊤ x ≤ β 1 and π ⊤ x ≥ β 2 with Improved general β 1 < β 2 to P 1 and P 2 respectively, as long as no integer disjunctions Computational feasible point is cut off experiments (1) • Can this be profitable with respect to branching on single A combined variables? branching algorithm Computational experiments (2)
Improved strategies for Example branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical foundations Improved general disjunctions Computational experiments (1) A combined branching algorithm Computational experiments (2)
Improved strategies for Example branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical foundations Improved general disjunctions Computational experiments (1) A combined branching algorithm Computational experiments (2)
Improved strategies for Example branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical foundations Improved general disjunctions Computational experiments (1) A combined branching algorithm Computational experiments (2)
Improved strategies for Preliminaries branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical foundations • Let D ( π, π 0 ) define the split disjunction Improved π ⊤ x ≤ π 0 ∨ π ⊤ x ≥ π 0 + 1 , where π ∈ Z n , π 0 ∈ Z , π j = 0 general disjunctions π ⊤ ¯ � � for i / ∈ N I , π 0 = x Computational experiments • By integrality of ( π, π 0 ) , any feasible solution to P (1) satisfies every split disjunction A combined branching algorithm Computational experiments (2)
Improved strategies for Definitions branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini • Let B ⊂ N be an optimal basis of ¯ P , let J = N \ B be Introduction the set of nonbasic variables Theoretical foundations • The corresponding simplex tableau is given by: Improved general � disjunctions x i = ¯ x i − ¯ a ij x j ∀ i ∈ B Computational j ∈ J experiments (1) • For j ∈ J , let r j ∈ R n be the extreme ray (associated with A combined branching x j ) of the cone { x ∈ R n | Ax = b ∧ ( x j ≥ 0 ∀ j ∈ J ) } with algorithm apex ¯ x Computational experiments (2) • The r j ’s can be read directly from the simplex tableau
Improved strategies for Intersection cuts branching on general disjunctions G. Cornu´ ejols, • Let ǫ ( π, π 0 ) = π ⊤ ¯ x − π 0 L. Liberti, G. Nannicini • Assume that the disjunction D ( π, π 0 ) is violated by ¯ x , Introduction i.e. 0 < ǫ ( π, π 0 ) < 1 Theoretical • The intersection cut associated with a basis B and a split foundations disjunction D ( π, π 0 ) is Improved general disjunctions x j � Computational α j ( π, π 0 ) ≥ 1 , experiments (1) j ∈ J A combined branching where ∀ j ∈ J we define algorithm Computational if π ⊤ r j < 0 − ǫ ( π,π 0 ) experiments (2) π ⊤ r j 1 − ǫ ( π,π 0 ) if π ⊤ r j > 0 α j ( π, π 0 ) = π ⊤ r j + ∞ otherwise
Improved strategies for Intersection cuts and branching branching on general disjunctions G. Cornu´ ejols, L. Liberti, G. Nannicini Introduction Theoretical foundations Improved general disjunctions Computational experiments (1) A combined branching algorithm Computational experiments (2)
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