In this work we have introduced a new data structure, in the form of a modified binary tree, that is able to efficiently store sets of nondominated solutions of BOMILPs. Until now similar structures have only been used in the pure integer case. We provide an extension for the more difficult mixed-integer case. We showed this structure performs with a worst case guarantee of O(t2log t) where t is the number of stored nodes. We tested the practical value of our data structure
with three experiments. The results show that our structure provides a more efficient method for storing solutions to BOMILP than other prevalent techniques. They also show that our structure is a very useful tool when used alongside branch-and-bound methods for solving BOMILPs.
Generalizing the ideas proposed in this chapter to the multiobjective mixed integer case is one line of future research, although we remark that this extension is not immediate because of the difficulty of doing comparisons and bound dominance in dimensions greater than two. We also recognize that within the scope of biobjective programming, there may be ways to increase the efficiency of our data structure, as explained next. Recall that each node of our structure may store either a point or a line segment. It is possible that in certain cases our structure stores several segments that all belong to a single piecewise linear curve. Therefore it may be beneficial to extend the functionality of our structure so that entire piecewise linear curves can be stored in a single node. Notice that in some cases this may allow for a significant reduction of the size of the tree and thus allow the structure to be populated and maintained more quickly. The reason that we did not implement our structure in this fashion is that for the BOMILP solution techniques we are familiar with, segments are generated one at a time and in general connecting segments are not generated sequentially. Also, for the specific instances we solved, it was not often that a significant number of connected line segments generated from the same slice problem were Pareto optimal. Implementing such refinements to the proposed data structure remains an avenue of future work.
2.A
Appendix A: Detailed results for Experiment 1
Table 2.6: Time and depth of the tree for Experiment 1. Rebal Time (s) Depth µ N Type Min Avg Max Min Avg Max 0 104 A0 0.22 0.25 0.33 36 43.8 55 A1 1.77 2.14 3.54 16 16.4 17 A2 0.28 0.35 0.68 16 17.7 20 A3 0.27 0.34 0.60 16 17.1 18 A4 0.27 0.34 0.61 16 17.1 18 L 252 309 438 – – – 105 A0 11.8 13.1 16.3 36 43.8 55 A1 489 623 1,010 19 20.1 21 A2 13.0 18.7 63.5 20 21.4 24 A3 13.1 19.6 70.8 20 20.7 22 A4 13.2 19.6 79.7 20 20.7 22 L – – – – – – 106 A0 374 385 393 57 61.0 65 A1 – – – – – – A2 351 403 680 22 24.8 27 A3 381 432 698 22 23.0 23 A4 385 435 678 22 23.0 24 L – – – – – – 107 A0 7,316 7,781 9,920 66 93.0 96 A1 – – – – – – A2 5,804 6,644 8,444 24 28.3 44 A3 6,312 7,322 9,326 24 24.6 26 A4 6,400 7,424 38,709 24 24.6 25 L – – – – – – 0.001 104 A0 0.06 0.06 0.08 24 31.1 42 A1 0.33 0.34 0.40 12 12.2 13 A2 0.08 0.09 0.12 12 13.8 17 A3 0.08 0.09 0.14 12 13.1 14 A4 0.08 0.09 0.14 12 13.1 14 L 2.72 2.83 2.94 – – – 105 A0 1.41 1.65 2.13 26 33.7 44 A1 9.8 11.6 17.9 13 14.0 15 A2 1.56 1.77 2.34 14 17.0 21 A3 1.69 1.97 2.87 14 14.9 16 A4 1.68 1.97 2.89 14 15.0 16 L 205 309 824 – – – 106 A0 36.5 38.6 50.9 35 35.2 54 A1 291 343 641 15 15.9 17 A2 34.9 36.8 46.4 16 21.5 25 A3 38.3 40.5 51.6 16 16.8 18 A4 38.3 40.2 52.0 16 16.8 18 L – – – – – – 107 A0 908 1,016 1,5000 33 43.3 55 A1~ 24,166 25,245 27,624 17 17.7 18 A2 866 935 1,500 21 27.0 33 A3 963 1,031 1,646 18 18.7 20 A4 967 1,038 1,595 18 18.7 20 L – – – – – –
Table 2.7: (Continuation of Table 2.6.)
Rebal Time (s) Depth µ N Type Min Avg Max Min Avg Max 0.01 104 A0 0.03 0.03 0.04 14 19.9 37 A1 0.13 0.13 0.14 10 10.3 12 A2 0.04 0.05 0.05 10 11.9 14 A3 0.05 0.05 0.05 11 11.2 12 A4 0.04 0.04 0.05 11 11.2 12 L 0.49 0.50 0.53 – – – 105 A0 0.66 0.69 0.77 18 22.4 34 A1 3.54 3.60 3.86 12 12.1 14 A2 0.74 0.75 0.88 12 15.8 20 A3 0.85 0.88 0.93 12 13.1 14 A4 0.85 0.88 0.93 13 13.1 14 L 30.1 30.7 33.1 – – – 106 A0 14.0 15.7 23.6 21 25.3 37 A1 99.6 117 217 13 14.0 15 A2 14.4 15.5 19.5 15 20.1 24 A3 16.8 18.2 24.9 14 14.9 16 A4 16.8 18.3 24.8 14 14.9 16 L 2,079 2,926 6,602 – – – 107 A0 334 352 403 24 28.1 37 A1 2,935 3,109 3,607 15 15.5 17 A2 332 342 382 20 24.1 30 A3 373 384 438 16 16.5 17 A4 377 386 451 16 16.5 18 L – – – – – – 0.1 104 A0 0.02 0.02 0.02 11 12.7 17 A1 0.06 0.06 0.06 8 8.7 10 A2 0.01 0.01 0.03 8 10.3 13 A3 0.02 0.02 0.02 9 9.4 10 A4 0.01 0.01 0.02 9 9.4 10 L 0.13 0.14 0.15 – – – 105 A0 0.36 0.36 0.38 13 15.3 18 A1 1.38 1.39 1.45 10 10.2 11 A2 0.41 0.41 0.43 11 14.3 18 A3 0.51 0.52 0.53 10 11.2 12 A4 0.51 0.51 0.53 10 11.2 12 L 5.22 5.28 5.37 – – – 106 A0 6.43 6.81 8.55 16 18.1 23 A1 35.7 38.5 54.1 12 12.1 14 A2 6.75 7.03 8.20 14 17.8 21 A3 8.59 8.98 10.7 12 13.1 14 A4 8.56 8.68 10.7 12 13.1 14 L 306 360 856 – – – 107 A0 139 140 144 20 20.9 22 A1 997 1,012 1,086 13 14.0 15 A2 140 141 144 18 21.2 26 A3 168 169 178 14 14.9 16 A4 168 170 176 14 15.0 16 L~ 20,903 24,326 29,097 – – –
Table 2.8: (Second continuation of Table 2.6.) Rebal Time (s) Depth µ N Type Min Avg Max Min Avg Max
1 104 A0 0.02 0.02 0.02 9 11.1 15 A1 0.04 0.04 0.04 7 8.0 9 A2 0.02 0.02 0.02 8 9.2 13 A3 0.02 0.02 0.02 8 8.6 10 A4 0.02 0.02 0.02 8 8.6 10 L 0.09 0.09 0.10 – – – 105 A0 0.24 0.24 0.25 11 12.7 16 A1 0.63 0.63 0.65 8 8.7 10 A2 0.26 0.26 0.27 8 12.2 15 A3 0.33 0.33 0.34 9 9.4 10 A4 0.32 0.32 0.34 9 9.4 10 L 1.42 1.44 1.57 – – – 106 A0 3.63 3.96 4.32 13 15.3 20 A1 13.8 15.7 17.7 10 10.3 12 A2 3.77 4.09 4.44 12 15.0 18 A3 5.21 5.70 6.23 10 11.2 12 A4 5.18 5.72 6.31 10 11.2 12 L 52.7 65.9 81.7 – – – 107 A0 64.2 65.1 77.7 17 17.9 20 A1 357 365 494 12 12.1 14 A2 65.7 66.8 78.2 16 18.2 22 A3 86.1 87.7 104 13 13.1 14 A4 85.7 87.3 106 13 13.1 14 L 1,481 3,229 6,211 – – – 10 104 A0 0.01 0.01 0.02 8 9.9 13 A1 0.03 0.03 0.03 7 7.3 8 A2 0.01 0.02 0.02 7 8.7 13 A3 0.02 0.02 0.02 7 8.0 9 A4 0.02 0.02 0.02 7 8.0 9 L 0.06 0.06 0.07 – – – 105 A0 0.21 0.22 0.25 9 10.8 14 A1 0.40 0.45 0.57 8 8.1 9 A2 0.22 0.23 0.26 8 10.5 13 A3 0.26 0.27 0.31 8 8.8 10 A4 0.25 0.28 0.31 8 8.8 10 L 0.92 1.03 1.17 – – – 106 A0 2.46 2.58 2.91 11 12.6 15 A1 6.39 6.78 8.02 8 8.7 10 A2 2.52 2.66 2.94 10 12.5 16 A3 3.37 3.52 3.93 9 9.5 11 A4 3.31 3.48 3.97 9 9.5 11 L 14.3 15.9 20.0 – – – 107 A0 36.2 36.6 42.8 13 15.1 18 A1 139 141 177 10 10.2 12 A2 37.2 37.6 43.1 13 15.1 19 A3 52.2 52.9 61.7 10 11.1 12 A4 51.7 52.2 62.3 10 11.1 12 L 528 531 561 – – –
Chapter 3
A Branch-and-Bound Algorithm
for BOMILP
[The contents of this chapter include material from a paper entitled “A branch-and-bound method for biobjective mixed integer linear programs,” which will be submitted to the journal Mathematical Programming Computation in August of 2016; the authors are N. Adelgren and A. Gupte.]
3.1
Introduction
Biobjective mixed integer linear programs (BOMIP) have the following form,
min x f1(x) := (c1)>x f2(x) := (c2)>x s.t. x ∈ XI :=x ∈ Rm+ × Z n +: Ax ≤ b, li≤ xi≤ ui .
A BOMIP is considered solved when the set of so-called Pareto optimal points has been discovered; see [27] for details. Applications of multiobjective programming can be found in a variety of disci- plines, including engineering, business, and management. Many problems require the use of discrete quantities, and thus BOMIP is an important class of problems. Exact algorithms for BOMIP are proposed in [11, 16]; the former is an extension of the standard branch-and-bound (BB) algorithm for MIPs whereas the latter is a search method in the (f1, f2)-space that recursively solves MIPs.
In this work we introduce a branch-and-bound framework for general BOMIP. Our main body of work is in developing new node processing techniques that account for the challenges of biobjective problems. We are also first to discuss the extension of a variety of dual presolve techniques to the multiobjective setting and we provide empirical evidence of the utility of these methods. Additionally, we show that one of these presolve techniques, probing on integer variables, can also be used alongside branching in order to develop tighter bounds and reduce the computational effort needed for BB. Through the use of locally valid cutting planes and the solution of single objective MIPs, we are able to propose methods for developing tighter dual bounds than have previously been proposed. We also introduce a new, challenging set of test instances which we develop from classical single objective instances available in the MIPlib 2010 library. As many of these instances are quite challenging, we propose the first technique for measuring a “duality gap” for multiobjective BB which relies on the computation of an approximated version of the well-known Hausdorff distance between two subsets of R2.
3.2
Preliminaries
3.2.1
Definitions and Notation
The idea of optimality for single objective optimization is replaced with the idea of efficiency in multiobjective problems. Given distinct x0, x00∈ XI, we say that y0= f (x0) dominates y00= f (x00)
if f (x0) ≤ f (x00). We denote this relationship as y0 y00. We then say that x ∈ X
I is efficient if
there is no x0 ∈ X
I such that f (x0) f (x). The set of efficient solutions is denoted by XE. Let
YI = {y = f (x) ∈ Rp: x ∈ XI}. Then y ∈ YI is called Pareto optimal if its preimage is efficient. YN
denotes the set of Pareto optimal points.
For each k ∈ {1, 2} we also define the set
YIk:= {y ∈ YI : yk= min x∈XI
{fk(x) : fi(x) ≤ fi(ˆx) for all i 6= k, ˆx ∈ XI}} (3.1)
and let ykI denote an arbitrary element of YIk. Note that for either XI, YI, or YIk, if we drop the “I”
subscript we are indicating the continuous relaxation of the given set. Also, if we add a subscript “s”, then it means that the set is associated with node s of the BB tree. We use OS to denote
the objective space, the smallest rectangle in R2 containing Y . Given S ⊆ OS ⊆ R2, the ideal
point of S, denoted Sideal
, is the point y ∈ R2 for which y
k = miny∈S{yk} for each k ∈ {1, 2}.
Additionally, we denote the nondominated subset of S as Snd. Throughout this work we also utilize
the set R2
≥= {y ∈ R2: y1≥ 0, y2≥ 0}.
We assume background in branch-and-cut algorithms for single objective problems; see [60] for a survey. One of the key differences and challenging aspects of BOMIP versus MIP is the concept of primal and dual bound sets, which we explain next.
3.2.2
Bound sets for BOMIP
In biobjective BB, unlike the single objective case, primal and dual bounds are not scalars but rather subsets of R2. These bound sets were introduced by [28]. We treat the dual bound set as
a single polyhedron in R2
and the primal bound set as a finite union of polyhedra in R2. Note that
this deviates from the traditional view of bound sets which defines them in terms of the boundary of these polyhedra. However it is straightforward to see that equivalent fathoming rules exist for each definition.
Consider any arbitrary node s of the BB tree. We use Ls to denote the locally valid dual
bound set generated from Pareto solutions of the BOLP relaxation at this node; this dual bound can be calculated as Ls= Ys+ R2≥. Let Nsdenote the current nondominated set of solutions in OS
that correspond to some feasible integer solutions in XI; this set Nsis the nondominated subset of
∪s0∈N
s(Ys0)I, where Ns is the set of nodes processed prior to s. The globally valid primal bound
generated from the solutions in Ns is Us := Ns+ R2≥. Using these sets, the most basic idea of
fathoming for BOMIP is: node s can be fathomed if Ls⊆ Us. Figure 3.1 shows an example of these
bound sets. Notice that node s2can be fathomed but we cannot say anything about fathoming node
s1 since Ls1 * Us.
Recognize that, similar to the single objective case, correct fathoming rules are essential for any BB algorithm to solve BOMIP to Pareto optimality. However, as can be imagined from Figure 3.1, fathoming is even more crucial and computationally intensive for BOMIPs since it involves checking inclusion and intersection of polyhedral sets as opposed to comparing scalar values in the MIP case. Thus, the majority of the computational effort in multiobjective BB is spent processing a node s of the BB tree, in particular checking various fathoming rules.
U
sL
s1L
s2N
sFigure 3.1: Biobjective bound sets
y1 s yλ s y 2 s LP ideal points LP ideal segment MIP ideal pts. (y1 s)I (yλ s)I (ys2)I MIP ideal seg.
Figure 3.2: Ideal points and segment
3.3
Node processing
Processing a node consists of three basic steps: (i) Generate a valid dual bound; (ii) Check a fathoming rule to determine whether or not s can be eliminated from the search tree; (iii) Optionally, if s is not fathomed in (ii), generate a tighter dual bound and repeat (ii). Figure 3.2 provides a visual example of how one might carry out these three steps. Most of the fathoming rules for biobjective BB are designed to check whether or not Usdominates (Ys)I by exploiting the transitivity of dominance.
First, a set
T
is generated such thatT
(Ys)I. Then if UsT
, Us (Ys)I and s can be fathomed.Otherwise, a tighter bound on (Ys)I is needed. The first bound we use is a set of two ideal points
which we obtain by solving three single objective LPs; one for each fk and an one with a weighted
sum objective fλin which the weights, denoted λs, are given by the normal vector of the line segment
Hspassing through ys1and y2s. We begin with these points because it is straightforward to determine
whether or not Us dominates a singleton. In Figure 3.2 these points are labelled “LP ideal points.”
Notice that they are not dominated. Consider the intersection of (Ys)ideal+ R2≥ and the line with
normal vector λspassing through yλ
s. Recognize that this intersection, which we denote Hsλ, is also
a valid dual bound. In Figure 3.2 the resulting line segment is labelled “LP ideal segment,” but is not dominated. A tighter bound can next be found by explicitly generating Ls. In Figure 3.2 this is
the set indicated by the red points, which is again not dominated. After generating Ls, one cannot
hope to find a tighter bound on (Ys)I resulting from LP solutions. Instead, one can solve single
objective MIPs to generate elements of (Ys)I and use these elements to form a valid dual bound. We
In Figure 3.2 these points are labelled “MIP ideal points.” Yet again they are not dominated. We can then consider the intersection of ((Ys)I)ideal+ R2≥ and the line with normal vector λs passing
through (yλ
s)I, which we denote ˜Hsλ. This intersection forms another valid dual bound. In Figure
3.2 the resulting line segment is labelled “MIP ideal segment” and is dominated. Hence, s can be fathomed in this example.
We now formally introduce a proposition outlining the fathoming rules we employ in this work. For use in this proposition, we introduce some additional notation. For each k ∈ {1, 2}, define
Pk s := ∪i6=ky˜si ∪ ˜y λ s. (3.2) Then let Ps:= (Ps1) ideal ∪ (Ps2) ideal. (3.3)
Additionally, given any I ⊂ {1, 2, λ}, define
DIs := ∪2k=1 Pk
s \ {∪i∈Iyis} ∪ {∪i∈I\{k}(yis)I} ideal
. (3.4)
Recognize that Ps represents the sets of ideal points obtained from LP solutions, while and DsI
represents a set of ideal points obtained from a mixture of LP and MIP solutions. Proposition 3.1 (Fathoming Rules). Node s can be fathomed if:
0. Ls⊂ (Ys)I 1a. (Us Ps) 2a. Us Hsλ 1b. (Us DsI for some I ⊂ {1, 2, λ}) 2b. (Us ˜Hsλ). 3. Ls⊆ Us Proof.
Rule 0: The result is trivial since Lsis integer feasible.
Rule 1a: The result is trivial since by construction Ps Lsand so Us Ls.
Rule 2a: The result is trivial since by construction ˜Hλ
Rule 1b: By construction, for any I ⊂ {1, 2, λ}, DIs dominates every (ys)I ∈ (Ys)I and thus DIs is
a valid dual bound at node s. The desired result follows. Rule 2b: By construction Hλ
s dominates every (ys)I ∈ (Ys)I and thus Hsλ is a valid dual bound at
node s. The desired result follows. Rule 3: The result is trivial.
Proposition 3.1 outlines five fathoming rules. Rule 0 expresses the idea of fathoming due to optimality, while the remainder of the rules indicate situations in which s can be fathomed due to bound dominance.
Before we outline the process we use for processing a node s, we briefly discuss another important task that ought to be carried out while processing node s: Updating Ns. We do this in
two ways: (i) add each integer-feasible line segment discovered while checking Fathoming Rule 0 to Ns, and (ii) for each discovered x∗∈ XI, generate the nondominated subset of
Y(x∗) := {y = f (x) : x ∈ X, xi= x∗i for all i ∈ {m + 1, . . . , m + n}} (3.5)
and add each defining line segment of this set to Ns. Consider the latter of these strategies. Observe
that the feasible set of Y(x∗) can be interpreted as a leaf node of the BB tree, which we denote s(x∗). Hence, the Y(x∗)+R2≥= Ls(x∗). This leads to a need for generating the nondominated subset
of Ls, i.e. Lnds . Typical techniques for generating Lnds include the multiobjective simplex method
and the parametric simplex algorithm (PSA) [27]. However, the multiobjective simplex method is far more robust than is necessary for biobjective problems. Also, we found in practice that using the PSA often resulted in many basis changes yielding the same extreme point of Lsin OS. Since
much work is done during the PSA to determine the entering and exiting variables, we found that generating Lnd
s using the PSA required a significant amount of computational effort. We decided to
use an alternative method for generating Lnd
s which relies on sensitivity analysis. We first solve the
single objective LP using objective f2to obtain ys2. Next we create the LP
Ps(α) := min{f1(x) + αf2(x) : x ∈ Xs} (3.6)
and then carry out the procedure outlined in Algorithm 3.1.
In lines 3 and 4 of Algorithm 3.1 we compute the south-east and north-west most extreme points of Lnd
Algorithm 3.1 Generate Lnds
Input: Node s.
Output: A set B containing all defining line segments of Lnds .
1: function GenerateDualBd(s) 2: Set B = ∅.
3: Solve the LP min{f2(x) : x ∈ Xs} to obtain ys2.
4: SolvePs(0) to obtain solution x∗ and set y = f (x∗).
5: while y 6= y2 s do
6: Use sensitivity analysis to obtain an interval [α0, α00] such that x∗ is optimal toPs(α) for
all α ∈ [α0, α00].
7: Set x∗= argmin{Ps(α00+ )} for sufficiently small > 0.
8: if f (x∗) 6= y then
9: Add the line segment connecting f (x∗) and y to B. Update y to be f (x∗).
10: Return B.
adjacent extreme points of Lnds in a west to east pattern, until the south-east most extreme point is
rediscovered. Each line segment joining a pair of adjacent extreme points of Lnds is stored and the
set of all computed segments is returned at the end of the procedure.
Recognize from Proposition 3.1 that Fathoming Rules 0 and 3 each impose a condition on Ls and therefore require knowledge of Lnds in order to be employed. We note, however, that for
each of these rules it is often unnecessary to generate Lnd
s entirely. In particular, the generation
of Lnd
s should cease if: (i) one is checking Fathoming Rule 0 and a defining line segment of Lnds is
generated that is not integer feasible, or (ii) one is checking Fathoming Rule 3 and a defining line segment of Lnds is generated that is not contained in Us. Hence, the procedures in Algorithm 3.1 can
be modified in order to develop strategies for checking Fathoming Rules 0 and 3. These strategies are outlined in Algorithms 3.2 and 3.3, respectively.
Algorithm 3.2 follows almost the same procedure as Algorithm 3.1, except it terminates prematurely on line 10 if a line segment is computed that is not integer feasible. Algorithm 3.3 also follows almost the same procedure as Algorithm 3.1. However, this procedure terminates prematurely on line 5 or 12 if a point or line segment is computed that is not dominated by Us. We have now
built the tools necessary to present our proposed procedure for processing a node s. We do so in Algorithm 3.4.
Line 2 of Algorithm 3.4 is an optional procedure in which we can generate locally valid cutting planes to strengthen the representation of Xs if so desired. We then compute y1s and y
2 s
on line 3. We then check to see if either of these solutions are integer feasible, and if they are, we generate the dual bound associated with the integer solution in order to update N . Furthermore,
Algorithm 3.2 Fathoming Rule 0 Input: Node s and solutions y1
s and ys2.
Output: 1 if node s should be fathomed, 0 otherwise.
1: function FR 0(s, y1s, y2s)
2: ys1is the solution toPs(0). Let x∗ represent the preimage of ys1. Set y = y1s.
3: if y = y2s then return 1
4: else
5: while y 6= y2 s do
6: Use sensitivity analysis to obtain an interval [α0, α00] such that x∗is optimal toPs(α)
for all α ∈ [α0, α00].
7: Update x∗ to be the solution ofPs(α00+ ) for an arbitrarily small > 0.
8: if f (x∗) 6= y then
9: Let S represent the line segment connecting f (x∗) and y.
10: if S 6⊂ (Ys)I then return 0
11: else Update y to be f (x∗).
12: return 1
Algorithm 3.3 Fathoming Rule 3 Input: Node s and solutions y1s and ys2.
Output: 1 if node s should be fathomed, 0 otherwise.
1: function FR 3(s, y1s, y2 s)
2: y1
s is the solution toPs(0). Let x∗ represent the preimage of ys1. Set y = y1s.
3: if y = y2 s then 4: if Us y then return 1 5: else return 0 6: else 7: while y 6= y2s do
8: Use sensitivity analysis to obtain an interval [α0, α00] such that x∗is optimal toPs(α)
for all α ∈ [α0, α00].
9: Update x∗ to be the solution ofPs(α00+ ) for an arbitrarily small > 0.
10: if f (x∗) 6= y then
11: Let S represent the line segment connecting f (x∗) and y. 12: if Us6 S then return 0
13: else Update y to be f (x∗).
14: return 1
if both solutions are integer feasible, we check Fathoming Rule 0 on line 6. On line 7 we compute the value λs, the value of the weights on the objectives so that the level curves of fλ have the same
slope as the line segment joining y1
s and ys2. We then solve the LP associated with fλ. If the solution
is integer feasible, we again update Nsas before. On line 9 we check whether or not ys1, ys2 and yλs
are dominated by Us. If they are, we proceed to check Fathoming Rules 1a, 2a, and 3. Otherwise,
we solve the MIP associated with fλ and fk for each k ∈ {1, 2} such that the ideal point (Psk)ideal
is not dominated by Us. On lines 21 and 22 we utilize the solutions of each MIP to (optionally) add
Algorithm 3.4 Process node s
1: function ProcessNode(s)
2: Compute valid cutting planes for (Xs)I and add them to the description of Xs.
3: for k ∈ {1, 2} do Solve min{fk(x) : x ∈ Xs} to find optimal solution ¯xk and generate
yks ∈ Ysk.
4: if ysk∈ (Ys)I then let N = GenerateDualBd(s(¯xk)) and set Ns= (Ns∪ N )nd.
5: if y1
s, ys2∈ (Ys)I then
6: if FR 0(s, ys1, y2
s) = 1 then Fathom s, STOP! (Fathoming Rule 0)
7: Calculate Hsand λsusing y1sand ys2. Solve min{fλ(x) : x ∈ Xs} to find optimal solution ¯xλ
and generate yλs ∈ Yλ s.
8: if yλs ∈ (Ys)I then let N = GenerateDualBd(s(¯xλ)) and set Ns= (Ns∪ N )nd.
9: if Us ys1, Us y2s and Us ysλthen
10: if Us Psthen Fathom s, STOP! (Fathoming Rule 1a)
11: else
12: Calculate ˜Hλ s.
13: if Us ˜Hsλthen Fathom s, STOP! (Fathoming Rule 2a)
14: else
15: if FR 3(s, ys1, y2
s) = 1 then Fathom s, STOP! (Fathoming Rule 3)
16: else
17: Define the set I = ∅.
18: for k ∈ {1, 2} do
19: if Us6 (Psk)ideal then add ({1, 2} \ {k}) ∪ {λ} to I
20: for each k ∈ I do solve the MIP min{fk(x) : x ∈ (Xs)I} to find optimal solution ˆxk and
obtain (yk
s)I ∈ (Ysk)I.
21: Add a local cut to Xs which lies on the level curve of fk associated with the best
found dual solution.
22: Let N = GenerateDualBd(s(ˆxk)) and set N
s= (Ns∪ N )nd.
23: if Us DIs then Fathom s, STOP! (Fathoming Rule 1b)
24: else if λ ∈ I then 25: Calculate Hλ
s.
26: if Us Hsλthen Fathom s, STOP! (Fathoming Rule 2b)
We now proceed to Section 3.4 in which we discuss the extension of the remaining major aspects of single objective BB to the biobjective setting.
3.4
Biobjective BB
In this section we discuss the specifics of how the different components of single objective BB – presolve/preprocessing, node processing, and branching, can each be extended to the biobjective setting. We then briefly discuss optional additions to our basic biobjective BB procedure.
3.4.1
Presolve/Preprocessing
It has been shown in a variety of works that examining the structure of an instance of single objective MIP prior to solving it, and utilizing information found during this examination to simplify the structure of the instance often has a significant impact on the time and effort needed to solve that instance. It has also been shown that knowledge of feasible solutions for an instance of MIP can have quite an impact on solution time. Hence, it makes sense to extend the techniques used in these procedures to the biobjective case. For the discussion that follows we distinguish the idea of simplifying an instance of BOMIP based on its problem structure from the idea of determining a set of initial integer-feasible solutions. We refer to the first as presolve and the latter as preprocessing. We propose a procedure which is carried out in three phases: (i) Presolve phase 1, (ii) Preprocessing and (iii) Probing on variables.
3.4.1.1 Presolve Phase 1
When presolve techniques are utilized for single objective MIP, both primal and dual in- formation is used. Fortunately, the primal information of a BOMIP instance is no different than