Cutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs

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1 Cutting Planes for First Level RLT Relaxations of Mixed 0-1 Programs 1 Cambridge, July Joint work with Franklin Djeumou Fomeni and Adam N. Letchford

2 Outline 1. Introduction 2. Literature Review 3. Cutting Planes 4. Separation 5. Computational Experiments 6. Concluding Remarks

3 Introduction The Reformulation-Linearization Technique (RLT), due to Adams and Sherali, is a technique for constructing hierarchies of LP relaxations for mixed-integer nonlinear programs. In practice, the first-level relaxation is usually the most useful. One can construct a first-level relaxation of any mixed 0-1 program with a linear or quadratic objective and any mixture of convex and quadratic constraints. We present several families of cutting planes, which can be used to strengthen such first-level relaxations.

4 Literature Review The seminal paper, Adams & Sherali (1986), was concerned with 0-1 quadratic programs of the form: { max x T Qx : Ax b, x {0,1} n}, where Q Q n n, A Q m n and b Q m. (Without loss of generality, we assume that Q is symmetric.) We will let V denote {1,...,n} and E denote {S V : S = 2}.

5 Literature Review (cont.) Let us introduce a new variable y ij for all {i,j} E, taking the value 1 if and only if both x i and x j do. The 0-1 QP can then be reformulated as the 0-1 LP: max s.t. i V Q ii x i + 2 e E Q e y e Ax b y ij x i ({i,j} E) y ij x j ({i,j} E) x i + x j y ij 1 ({i,j} E) (x,y) {0,1} V + E (In fact, this was already noted by Fortet, 1959.)

6 Literature Review (cont.) Replacing the condition (x,y) {0,1} V +E with the condition (x,y) [0,1] V +E yields an LP relaxation of the 0-1 QP. Adams and Sherali propose to strengthen the relaxation by adding 2mn additional inequalities as follows. Let α T x β be a constraint in the linear system Ax b, and let x k be one of the original variables. Then we have: (α T x)x k βx k = α i y ik (β α k )x k i V \{k} (α T x)(1 x k ) β(1 x k ) = α i (x i y ik ) β(1 x k ). i V \{k}

7 Literature Review (cont.) Sherali & Adams (1990) showed that, if the same approach is applied to a 0-1 LP, the LP relaxation of the reformulated problem dominates that of the original. They called the procedure the Reformulation-Linearization Technique. They also defined an entire hierarchy of relaxations, where the kth level RLT relaxation has variables representing products of up to k + 1 variables, and constraints that are derived by multiplying k + 1 of the original inequalities together.

8 Literature Review (cont.) There have been many developments since those early days: Lovász & Schrijver (1991) adapted the first-level relaxation to 0-1 problems with convex constraints, and also showed how to derive SDP relaxations that dominate the LP relaxation. In a series of papers, Adams & Sherali extended RLT to any bounded MINLP having a polynomial objective function and a mixture of convex and polynomial constraints. Padberg and others have studied the polytopes associated with various specific 0-1 QPs.

9 The New Cutting Planes The number of variables grows very rapidly with the level k. To keep the number of variables manageable, we concentrate on the first-level relaxation. We derive cutting planes using a two-phase approach: 1. Construct a valid cubic inequality; 2. Weaken it to make it valid for the first-level relaxation.

10 The New Cutting Planes: Phase 1 To construct valid cubic inequalities, we simply multiply triples of valid linear inequalities together. Suppose that the linear inequalities α i x β i, for i = 1,2,3, are valid for the original problem. We write them as β i α i x 0, and thereby derive the valid cubic inequality: (β 1 α 1 x)(β 2 α 2 x)(β 3 α 3 x) 0.

11 The New Cutting Planes: Phase 2 The key to phase 2 is the following lemma: Lemma 1 Suppose x i [0,1] for all i V, and y ij = x i x j for all e E. Then, for any triple {i,j,k} V, we have the following bounds on x i x j x k : x i x j x k min{y ij, y ik, y jk, 1 x i x j x k + y ij + y ik + y jk } x i x j x k max{0, y ij + y ik x i, y ij + y jk x j, y ik + y jk x k }.

12 The New Cutting Planes: Phase 2 (cont.) Due to the above lemma, we can convert a valid cubic inequality into a valid inequality for the first-level relaxation by replacing each term of the form x i x j x k with: any of 0, y ij + y ik x i, y ij + y jk x j or y ik + y jk x k (if it has a positive LHS coefficient). any of y ij, y ik, y jk or 1 x i x j x k + y ij + y ik + y jk (if the term has a negative LHS coefficient);

13 A special case: (s, t) inequalities Consider a single linear inequality α T x β and two binary variables x s and x t. We can form the following cubic inequality: (β α T x)x s x t 0. Since x 2 s x t = x s x 2 t = y st, we can re-write this as: α i x i x s x t (β α s α t )y st. i V \{s,t} To make further progress, we need more notation: for any S V, we let S + = {i S : α i > 0} and S = {i S : α i < 0}.

14 A special case: (s, t) inequalities (cont.) Then, for any partition of V into sets S,T,W and R, we can derive the (s, t) inequality: i S W α i y is + i T W α i y it α i x i α(w ) + α(s + W )x s i W + α(t + W )x t + ( β α({s,t} S + T + W R ) ) y st. Here, membership of i in S, T, W or R determines which of the weakenings x i x s x t min{y is, y it, y st, 1 x i x s x t + y st + y is + y it } x i x s x t max{0, y is + y st x s, y it + y st x t, y is + y it x i } we choose for the term x i x s x t.

15 More special cases: mixed and reverse (s, t) inequalities Following the same line of argument, we can derive: mixed (s, t) inequalities, by applying Lemma 1 to the cubic inequalities of form (β α T x)(1 x s )x t 0. reverse (s, t) inequalities, by applying Lemma 1 to the cubic inequalities of form (β α T x)(1 x s )(1 x t ) 0.

16 Remark I Using a simple disjunctive argument, it is possible to strengthen all three kinds of (s, t) inequalities. Basically, we compute upper bounds on the value that the LHS can take under the following four (mutually exclusive) conditions: x s = x t = y st = 0, x s = y st = 0 and x t = 1, x t = y st = 0 and x s = 1 x s = x t = y st = 1. We then use those bounds to tighten the RHS coefficients of x s, x t and y st.

17 Remark II The method can be applied to problems having: 1. A mixture of linear and quadratic constraints. 2. Non-linear constraints as long as they are convex.

18 Separation All three families of (s, t) inequalities are exponentially-large. So we need to think about separation. If the original problem has only linear constraints, then the separation problems for all three families can be solved in O(n 3 m) time.

19 Separation of standard (s, t) inequalities There are m ways to choose an inequality from Ax b. There are O(n 2 ) ways to select s and t. Then, for all i V +, we place i in S, T, W or R according to which of the following is largest: y is + y st x s y it + y st x t y is + y it x i 0. And, for all i V, we place i in S, T, W or R according to which of the following is largest: y is y it 1 x i x s x t + y is + y it + y st y st.

20 Separation (cont.) The separation problem for the mixed and reverse (s, t) inequalities can be tackled in a similar way. And the separation problems for the strengthened versions of all three families can be solved in O(n 3 m) time as well. For the cutting planes derived by multiplying together two linear inequalities and a variable or its complement, the separation time increases to O(n 3 m 2 ). For the ones derived by multiplying three linear inequalities, the separation time increases further to O(n 3 m 3 ).

21 Separation (cont.) Now consider the case in which there is a convex constraint x C, and we have a polynomial-time separation oracle for C (à la Lovász-Schrijver). We can show that the separation problems for all three kinds of strengthened (s, t) inequalities can be solved in polynomial time also in this case. On the other hand, we do not know how to solve the separation problem for the other inequalities in this case.

22 Computational Experiments We apply our scheme to the Quadratic Knapsack Problem: { max x T Qx : w T x c, x {0,1} n}, where Q Z n n +, w Zn + and c Z +. We randomly generate 5 instances for each different combination of size n {10,20,30,40,50} and density {25,50,75,100}. We separate all three families of strong (s, t) inequalities. We compare the resulting bounds against the benchmark upper bound given by the first-level RLT relaxation.

23 Computational Results Instances Gap (%) n RLTs RLTs + STs Improvement (%)

24 Computational Results Instances Gap (%) n RLTs RLTs + STs Improvement (%)

25 Concluding Remarks We have presented a general procedure for generating cutting planes for first-level RLT relaxations of mixed 0-1 programs. For the case in which all constraints are either linear or quadratic, we have polynomial-time separation routines for all families. For the case of convex constraints, we have polynomial-time separation routines only for the strengthened (s, t) inequalities. The computational results so far are quite encouraging.

Cutting Planes for RLT Relaxations of Mixed 0-1 Polynomial Programs

Cutting Planes for RLT Relaxations of Mixed 0-1 Polynomial Programs Franklin Djeumou Fomeni Konstantinos Kaparis Adam N. Letchford December 2013 Abstract The Reformulation-Linearization Technique (RLT),