Recent Developments in Disjunctive Programming Egon Balas Carnegie - PowerPoint PPT Presentation

Recent Developments in Disjunctive Programming Egon Balas Carnegie Mellon University Recent Developments in Disjunctive Programming 01.09.17 1 / 56

Recent Developments in Disjunctive Programming 1. Background and basic results 2. Convexification and extended formulations 3. L&P cuts from split disjunctions 4. General (non-split, multiple-term) disjunctions 5. The convex hull in R n (Parts 4-5 based on joint work with, respectively, Tamas Kis and Aleksandr Kazachkov) Recent Developments in Disjunctive Programming 01.09.17 2 / 56

1. Background and basic results • Convexity – the dividing line between “tame” and “wild” problems • Nonconvex sets can often be modeled via Integer Programming — which is NP -complete • Disjunctive programming: optimization over disjunctive sets , defined by inequalities joined by connectives ∧ , ∨ , ⇒ , ¬ (nonconvex because of ∨ ) • The largest known class of nonconvex sets convexifiable in polynomial time • Many equivalent forms, two extreme: • CNF: intersection of elementary disjunctive sets i ∈ Q j ( a i x ≥ a i 0 ) } , F = ∧ j ∈ T S j , S j = { x : ∨ j ∈ T • DNF: union of polyhedra P i = { x : A i x ≥ b i } , i ∈ Q P i , F = ∪ i ∈ Q Any disjunctive set can be brought to either form by a sequence of simple steps. Recent Developments in Disjunctive Programming 01.09.17 3 / 56

Disjunctive sets in DNF: Unions of polyhedra are a large class of nonconvex sets with a compact convex representation. The union of q polyhedra in R n (= the disjunction between q systems of linear inequalities in n variables) has a convex hull representation as a polyhedron in R ( qn ) (= a linear system in O ( qn ) variables). Disjunctive sets in CNF: If facial, any such set can be convexified sequentially , i.e. by imposing the elementary disjunctions one at a time, each time generating the convex hull of the current set: 0-1 MIP’s are facial , (general) MIP’s are not . This is the basic property distinguishing 0-1 programs from general integer programs. Recent Developments in Disjunctive Programming 01.09.17 4 / 56

The convex hull of a disjunctive set (E.B., Disjunctive Programming, CMU MSRR 348, 1974, Discrete Applied Math , 1998) Basic observation An inequality α x ≥ α 0 is valid for the disjunction i ∈ Q ( A i x ≥ b i ) ∨ if and only if it is valid for each system A i x ≥ b i , i ∈ Q . With this in mind, we have Theorem 1.1 Farkas’ Lemma for disjunctive sets i ∈ Q ( A i x ≥ b i ), and Q ∗ := { i ∈ Q : { x : A i x ≥ b i } � = ∅} . The inequality Let F = ∨ α x ≥ β is satisfied by all x ∈ F if and only if there exist vectors u i ∈ R m , u i ≥ 0, such that α = u i A i , β ≤ u i b i , i ∈ Q ∗ . Recent Developments in Disjunctive Programming 01.09.17 5 / 56

i ∈ Q ( a i x ≥ b i ), where a i ∈ R n , Thus, an elementary disjunction of the form ∨ b i > 0, i ∈ Q , and each term is feasible, implies the valid inequality n � a i � � x j ≥ 1 . max b i i ∈ Q j =1 which is a weakening of each a i x ≥ b i . Recent Developments in Disjunctive Programming 01.09.17 6 / 56

i ∈ Q P i , P i = { x ∈ R n : A i x ≥ b i } , i ∈ Q , and let Theorem 1.2 Let F = ∪ Q ∗ = { i ∈ Q : P i � = ∅} . Then conv F = { x ∈ R n : x − � i ∈ Q ∗ y i = 0 A i y i − b i y i ≥ 0 0 y i i ∈ Q ∗ ≥ 0 (1 . 1) 0 i ∈ Q ∗ y i � = 1 0 for some vectors ( y i , y i 0 ) , i ∈ Q ∗ } . Denoting C i := { x : A i x ≥ 0 } , i ∈ Q , Q ∗ can be replaced with Q if � � � � i ∈ Q \ Q ∗ C i i ∈ Q ∗ C i ∪ ⊆ ∪ Recent Developments in Disjunctive Programming 01.09.17 7 / 56

To obtain the convex hull in R n , project the set S ( Q ) defined by (1.1) onto the x -space. Projection cone:  ( α, β, { u i } i ∈ Q ) : − α + u i A i  = 0   β − u i b i ≥ i ∈ Q W = 0 u i ≥ 0   Theorem 1.3 Proj x S ( Q ) = { x ∈ R n : α x ≥ β for all ( α, β, { u i } i ∈ Q ) ∈ extr W } The inequalities α x ≥ β can be generated as the ( α, β )-components of extreme rays of W , i.e. by solving a system of O ( qn ) variables. Recent Developments in Disjunctive Programming 01.09.17 8 / 56

The above representation of conv F was derived by using the H -polyhedral representation of F . What about the V -polyhedral representation? P i = conv V i + cone W i , i ∈ Q P i , F = ∪ i ∈ Q Theorem 1.4 (M. Perregaard and E.B., IPCO 2001) The inequality γ x ≥ δ is valid for F if and only if γ p ≥ δ for all p ∈ V i � i ∈ Q (1 . 2) γ r ≥ 0 for all r ∈ W i Proof. γ x ≥ δ is valid for F if and only if it is valid for all P i , i ∈ Q . Thus conv F is defined by the inequalities γ x ≥ δ corresponding to basic solutions ( γ, δ ) of the system (1.2). The system (1.2) defines conv F in R n , but it has ∪ i ∈ Q ( | V i | + | W i ) inequalities. Recent Developments in Disjunctive Programming 01.09.17 9 / 56

Two consequences of the compact convex hull representation (1) The fact that a nonconvex set in R n can be described by a convex polyhedron in a higher dimensional space gave rise to extended formulations . Two types of benefits: (a) tighter LP relaxations (b) integrality of higher dimensional representation proves integrality of original polyhedron (2) The description of the convex hull of a disjunctive set by the inequalities corresponding to basic solutions of a CGLP has led to the study of lift-and-project cuts for MIP’s. First to be studied were cuts from split disjunctions . Recent Developments in Disjunctive Programming 01.09.17 10 / 56

2. Convexification and extended formulations (E.B., SIAM J on Algebraic Discrete Methods , 1985) • A general technique for tightening formulations of MIP’s: replace the “ big M” representation of some disjunctive subset by its convex hull representation • CNF and DNF are both intersections of unions of polyhedra F = ∩ j ∈ T S j , S j = ∪ i ∈ Q P i , j ∈ T (2 . 1) • Call (2.1) a regular form (RF) Theorem 2.1 Any disjunctive set F in RF(2.1) can be brought to DNF by | T | − 1 applications of the following basic step, which preserves regularity: For some k , ℓ ∈ T , bring S k ∩ S ℓ to DNF, i.e. replace it with S k ℓ = ∪ i ∈ Q k ( P i ∩ P j ) j ∈ Q ℓ Recent Developments in Disjunctive Programming 01.09.17 11 / 56

Given a disjunctive set in regular form (2.1), define the hull-relaxation of F = ∩ j ∈ T S j as h -rel F = ∩ j ∈ T conv S j Theorem 2.2 For i = 0 , 1 , . . ., t , let F i = ∩ j ∈ T i S j be a sequence of regular forms, such that (i) F 0 is in CNF (ii) F t is in DNF (iii) for i = 1 , . . . , t , F i is obtained from F i − 1 by a basic step. Then h -rel F 0 ⊇ h -rel F 1 ⊇ · · · ⊇ h -rel F t = conv F t Recent Developments in Disjunctive Programming 01.09.17 12 / 56

Clearly, conv S k ℓ ⊆ conv S k ∩ conv S ℓ . When to replace a disjunctive set in RF by its convex hull? –When it makes conv S k ℓ tighter than (conv S k ) ∩ (conv S ℓ ). Theorem 2.3 Let S j = ∪ i ∈ Q j P i , j = 1 , 2. Then conv ( S 1 ∩ S 2 ) = (conv S 1 ) ∩ (conv S 2 ) if and only if every extreme point (direction) of (conv S 1 ) ∩ (conv S 2 ) is an extreme point (direction) of P i ∩ P k for some ( i , k ) ∈ Q 1 × Q 2 . Recent Developments in Disjunctive Programming 01.09.17 13 / 56

Example. Let P i , i = 1 , . . . , 4 be as in the figure, and F = S 1 ∩ S 2 , S 1 = P 1 ∪ P 2 , S 2 = P 3 ∪ P 4 (0,1) (1,1) ( ½,1) P3 P1 P2 (0,½) (1,½) P4 (0,0) (1,0) (½,0) (½,1) (0,1) (0,½) (1,½) (0,½) (1,½) (1,0) (½,0) (conv S1) I (conv S2) conv (S1 I S2) Then conv ( S 1 ∩ S 2 ) � (conv S 1 ) ∩ (conv S 2 ) since vertices ( 1 2 , 0) and ( 1 2 , 1) of conv S 1 ∩ conv S 2 are not vertices of either P 1 ∩ P 3 , P 1 ∩ P 4 , P 2 ∩ P 3 or P 2 ∩ P 4 . Recent Developments in Disjunctive Programming 01.09.17 14 / 56

If a conjunct S j = ∪ i ∈ Q j P i of a RF F k = ∩ j ∈ T S j , is replaced with conv S j , we get x − � y i  = 0  i ∈ Q j   A i y i − b i y i  ≥ 0 i ∈ Q j   0  j ∈ T (2 . 2) y i ≥ 0 0    y i � = 1   0   i ∈ Q j In any solution nonbasic for the j -th subsystem, y i 0 �∈ { 0 , 1 } Imposing y i 0 ∈ { 0 , 1 } for i ∈ Q j , j ∈ T ? No: if the CNF is F 0 = r ∈ T 0 S r , ∧ S r = s ∈ Q r ( a s x ≥ b s ) , ∨ then replace the last equation of (2.2) by y r 0 − δ r ( M i = row index set of A i ) � = 0 , s ∈ Q r r ∈ T 0 i ∈ Q j | s ∈ M i � δ r = 1 r ∈ T 0 s s ∈ Q r δ r s ∈ { 0 , 1 } s ∈ Q r , r ∈ T 0 i.e. we need the same number of 0-1 variables as in CNF. Recent Developments in Disjunctive Programming 01.09.17 15 / 56

Application: job shop scheduling with sequence-dependent setup times min t j t j − t i ≥ ( i , j ) ∈ A d ij t j − t i ≥ d ij ∨ t i − t j ≥ d ji ( i , j ) ∈ E k , k ∈ M t i = start time of job i d ij = duration of i + setup time for j A = set of precedence arcs M = set of machines E k = set of disjunctive pairs of arcs for machine k Recent Developments in Disjunctive Programming 01.09.17 16 / 56