HEURISTICS IN MIXED INTEGER depending on the main ingredient (building PROGRAMMING block) they use, namely LP-based heuristics in the section titled ‘‘LP-Based Heuristics’’ MATTEO FISCHETTI and MILP-based approaches in the section DEI, UniversitadiPadova,Padova,´ titled ‘‘MILP-Based Heuristics.’’ Italy With a little abuse of terminology, in what ANDREA LODI follows, we will say that a point x is integer ∈ I DEIS, Universita´ di Bologna, if xj is integer for all j (no matter what Bologna, Italy the value of the other components), whereas the rounding x˜ of a given x will be defined as x˜j := [xj]ifj ∈ I and x˜j := xj otherwise ([·] representing scalar rounding to the nearest INTRODUCTION integer). We end the introduction by noting that, We consider a generic mixed integer linear although Achterberg [1] has shown that the program (MILP) in the form impact of heuristics is not dramatic in terms of ability of an MILP solver to prove optimal- min cTx (1) ity in a (much) quicker way, the psychological impact for the user who sees a high quality Ax ≥ b (2) feasible solution provided very early is huge. xj ∈{0, 1}∀j ∈ B,(3)For this reason, the availability of very effec- tive general-purpose heuristics for MILP is x integer ∀ j ∈ G,(4) j among the most crucial improvements over xj continuous, ∀ j ∈ C,(5)the last 10 years. where A is an m × n input matrix, and b and c are input vectors of dimension m and LP-BASED HEURISTICS n, respectively. Here, the variable index set N :={1, ..., n} is partitioned into (B, G, C), In this section we address some basic heuris- where B is the index set of the 0–1 variables tics that only assume the availability of a (if any), while the sets G and C index the ‘‘black-box’’ LP solver. general integer and the continuous vari- Folklore ables, respectively. Bound on the variables, including the 0–1 bounds on the binary The so-called rounding as well as diving ones, are assumed to be part of system heuristics belong to folklore. (2). Removing the integrality requirement Rounding methods solve the LP relaxation on variables indexed by I := B ∪ G leads to to get an optimal point x∗, whose fractional { T ∈ } ∗ ∈ I the LP relaxation min c x : x P where components xj with j are rounded, for P :={x ∈ Rn : Ax ≥ b}. example, to their nearest integer value. Con- MILP heuristics aim at finding a feasible tinuous variables, if any, can optimally be (and hopefully good) solution of the prob- recomputed once the integer variables have lem above, which is an NP-hard problem by been fixed by solving an LP. itself. We next present the main ideas under- Diving (also known as Relax-and-Fix) lying some of the heuristics proposed in the methods, instead, mimic an enumerative literature. In this article, in particular, we scheme by sequentially fixing some integer- focus on those algorithms developed with the valued variables that assume a fractional aim of being tightly integrated within MILP value in the solution of the current LP- solvers. We group and discuss the algorithms relaxation. The sequence can be viewed Wiley Encyclopedia of Operations Research and Management Science, edited by James J. Cochran Copyright © 2010 John Wiley & Sons, Inc. 1 2 HEURISTICS IN MIXED INTEGER PROGRAMMING as ‘‘diving’’ on a branch-and-bound tree In the mid-1980s Balas and Martin [4] until a feasible (and hopefully good) MILP introduced a generalization of P&C to MILPs, solution is found. A limited backtracking called pivot and shift, where general integer is often allowed to investigate alternative variables are treated as 0–1 variables ‘‘cen- fixing patterns. Various variable-fixing (i.e., tered’’ on the current LP solution, and some branching) policies have been used by dif- new types of ‘‘nonstandard’’ pivots are intro- ferent authors, all meant to hopefully reach duced. An elaboration with the same name of deep branching nodes while maintaining a this idea was later proposed by Balas et al. good LP relaxation lower bound value. [5]. Different extensions called pivot, cut, and dive and pivot and Gomory cut were proposed by Nediak and Eckstein [6] and by Gosh and Pivoting Methods Hayward [7], respectively. These methods originate from the seminal work of Balas and Martin [2] in the late OCTANE and Line Search Methods 1970s. The basic heuristic, called Pivot and Line search methods are based on the fol- Complement (P&C), applies to pure 0–1 inte- lowing construction. One starts from an LP G = C =∅ ger linear programs (ILPs) with .It relaxation optimal vertex x∗ and draws a is based on the observation that a feasible line toward a second point, for example, y, solution is just a basic solution of the LP to be chosen according to a certain criterion. relaxation where all 0–1 variables are non- Thus, moving in a discretized way from x∗ basic (either at their lower or upper bound), to y traces a discrete sequence of (say) K that is, only the slack (continuous) variables k ∗ ∗ points x := x + αk(y − x )for0= α1 < ···< on Equation (2) are allowed to be basic. Given k αK = 1. Each x can therefore be associated an optimal basis of the LP relaxation, P&C (e.g., by rounding) with an integer point x˜k then performs a systematic series of ‘‘non- which is then checked for feasibility and pos- standard’’ pivots in the attempt of driving as sibly used to update a current-best MILP many 0–1 variables as possible out of the solution. Local shifts of the rounded vari- basis, while preserving primal feasibility and ables are typically allowed in the attempt of worsening the objective function as little as improving the feasibility and/or quality of the possible. rounded points. More specifically, the method prescribes Hillier [8] introduced this scheme by defin- the use of three types of pivots: (i) Type ing y so as to go inside P along a so-called 1 pivots that maintain primal feasibility interior path, the rational being supported and increase the number of nonbasic 0–1 by the geometrical intuition that (mainly variables; (ii) Type 2 pivots that also main- for general integer MILPs) going inside P tain primal feasibility but leave the number increases the chances the rounded point be of nonbasic 0–1 variables unchanged, feasible. A similar scheme was later adopted while reducing the infeasibility degree of by Ibaraki et al. [9] and by Faaland and ∗ the current LP solution x ,computedas Hiller [10], where the sampling line segment { ∗ − ∗} ∗ j∈B min xj ,1 xj ; (iii) Type 3 pivots [x , y] is replaced by a sequence of linear that increase the number of nonbasic 0–1 segments [x∗, y1], [y1, y2], ... approximating variables at the price of producing a primal- a curved search trajectory. Very recently, infeasible point—a situation to be repaired Naoum-Sawaya and Elhedhli [11] investi- at a later step of the procedure. gated the effect of replacing the LP solver Nonbasic 0–1 variables can also be com- by an analytic-center method. plemented with the aim of improving the OCTANE (for Octahedral Neighborhood solution quality or reducing its primal infea- Enumeration) is a line search method pro- sibility, and reduced costs are exploited to posed by Balas et al.[12]for0–1ILPs.It fix variables when mathematically correct. also starts with an LP optimal vertex x∗ The P&C heuristic was later improved by and moves it along a certain search direction Løkketangen et al. [3] by adding tabu search y − x∗. However, the integer (possibly infea- to the basic mechanisms. sible) points discovered along this direction HEURISTICS IN MIXED INTEGER PROGRAMMING 3 ∗ are not defined through rounding, but exploit FP scheme above may stall in case (x , x˜) the following construction. Consider the cen- is not reduced at some iteration, hence a ter x0 = (1/2, ...,1/2) of the unit hypercube. number of diversification mechanisms are For any vertex x˜ of the unit hypercube (not introduced. necessarily in P), define the hyperplane H(x˜) As stated, FP is meant to only produce passing through x˜ and orthogonal to x˜ − a feasible MILP solution, as the objective x0.1 Starting from x∗ and going toward y function is only taken into account implicitly ∗ one crosses, in sequence, certain hyperplanes in the definition of the very first x .In H(x˜1), H(x˜2), ...whose generating 0–1 points order to produce good quality solutions, the x˜1, x˜2, ... are checked for feasibility and used original FP approach [13] uses a ‘‘sliding to possibly update the current-best feasible window’’ method that introduces the cut T solution. OCTANE prescribes to visit only a c x ≤ UB into the LP model and updates limited number of hyperplanes, and exploits UB, on the fly, by taking an intermediate an effective method for their enumeration. value between the optimal LP-relaxation and the UB values. (Note that the value UB can also be a ‘‘guess’’ of a feasible Feasibility Pump solution value and in such a case the The Feasibility Pump (FP) is a method orig- cut can be possibly invalid.) A different inally proposed by Fischetti et al. [13] for approach, called objective FP,wasdevel- 0–1 MILPs (G =∅),andthenextendedby oped by Achterberg and Berthold [15], Bertacco et al. [14] to the general case. It who modified the objective function of is based on the observation that a feasible the auxiliary LP to (x∗, x˜) + α cTx,where MILP solution is a point x of P that coincides α is a dynamically updated parameter.