Combining Exact and Heuristic Approaches for the Multidimensional 0-1 Knapsack Problem

Saïd HANAFI

Said.hanafi@univ-valenciennes.fr

SYM-OP-IS 2009

2

Motivation

LowerBound

v(P)

UpperBoundv(P)

Heuristic &Metaheuristic

ExactMethod

OptimalValuev(P)

Maximization

Small sizeLarge size Large size

Relaxation &Duality

P : Hard Optimization Problem

3

Outline

Variants of Multidimensional Knapsack problem (MKP)

Tabu Search & Dynamic Programming Relaxations of MKP Iterative Relaxation-based Heuristics Inequalities and Target Objectives

Conclusions

4

Multidimensional Knapsack Problem

MKP is NP-Hard Dantzig 1957, Lorie & Savage 1955, Markowitz & Manne 1957 Martello & Toth, 1990; Kellerer & Pferschy & Pisinger 2004 Fréville & Hanafi 2005, Wilbaut & Hanafi & Salhi 2008; Hanafi & Wilbaut 2009 Boussier & Vasquez & Vimont & Hanafi & Michelon 2009 Hanafi & Lazic & Mladenovic & Wilbaut 2009

nx

bAts

c

MKP

1;0

x ..

x max

A(m,n),b(m),c(n) 0

5

Applications Resource allocation problems Production scheduling problems Transportation management problems Adaptive multimedia system with QoS Telecommunication networks problems Wireless switch design Service selection for web services with QoS constraints Polyedral approach, (1,k)-configuration

Sub-problem : Lagrangean / Surrogate relaxation Benchmark

6

Variants of Knapsack Problem

- Unidimensional Knapsack Problem m = 1

- Bi-Dimensional Knapsack Problem m = 2

- Quadratic- Hyperbolic- Min-Max- Multi-Objectives- Bi-Level

- Cardinality- Multiple-choice- Precedence - Demand- Quadratic

ObjectiveObjective ConstraintsConstraints

7

Multidemand Multidimensional Knapsack Problem

Knapsack & Covering Problem

8

,...,gj,...,gi,x

,...,gix

,...,mkbxats

xc

MMKP

iij

g

jij

kij

g

i

g

j

k

g

i

g

jijij

i

i

ij

i

1 ,11 0

11

1..

max

1

1 1

1 1

Multi-Choice Multidimensional Knapsack Problem

• g : Number of groups• gi : Number of items in group i• cij : Profit associated to item j of group i• : Consuming resource k by item j of group i• bk : Capacity of resource k

kija

9

,...,gj,...,gi,x

,...,gix

,...,mkbxats

xc

GUBMKP

iij

g

jij

kij

g

i

g

j

k

g

i

g

jijij

i

i

ij

i

1 ,11 0

11

1..

max

1

1 1

1 1

Multidimensional Knapsack Problem with Generalized Upper Bound constraints

10

Knapsack Constrained Maximum Spanning Tree Problem

KCMST problem arises in practice in situations where the aim is to design a mobile communication network under a strict limit on total costs.

Aa,x

x

bxwts

xc

KCMST

a

Aaaa

Aaaa

1 0

G of treecovering

..

max

Aggarwal et al (1982)

11

Generalized Knapsack Problem

nj,yx

nj ybxya

cxts

xc

GKP

jj

jjjjj

n

jj

n

jjj

,...,11 0;0

,...,1

..

max

1

1

Suzuki (1978)

12

Disjunctively Constrained Knapsack Problem

nj,x

Ii,jxx

bxats

xc

DCKP

j

ji

n

jjj

n

jjj

,...,11 0

)( 1

..

max

1

1

Yamada et al (2002)

13

Max Min Multidimensional Knapsack

Problem

n

k

xbAx

ts

pkxcMinMax

MMMKP

1,0

..

,...,1:

Introduction to minimax, Demyanov Molozemov 1974

Minimax and applications, Du, Pardalos 1995

Robust optimization applications, Yu 1996

A virtual pegging approach, Taniguchi, Yamada, Kataoka 2009

Iterative Heuristics, Mansi, Hanafi, Wilbaut 2009

14

Nj,x

liNNNN

bxats

mixc

KSP

j

li

m

ii

Njjj

Njjj

i

1 0

,

..

,...,1:minmax

1

Knapsack Sharing ProblemBrown (1979)Brown (1979)

15

Moore, Bard 1990 => B2 = 0Dempe, Richter 2000Hanafi, Mansi, Brotcorne 2009

15

yxMax

, f 1 ( x , y ) = d 1 x + d 2 y

s.t. B1 x + B2 y ≤ b1 x 1nZ

y

Max f 2 ( y ) = c y

s.t. A1 x + A2 y ≤ b2

y 2n1,0

Leader objective function

Follower objective function

Follower problem

(BKP)

Bilevel Knapsack Problem

16

Tabu Search & Dynamic Programming

Metaheuristic: Tabu Search (TS) Very powerful method on many optimization

problems Based on the exploration of a neighbourhood

Exact method: Dynamic Programming (DP) Efficient for small-sized problem Can be used for solving a series of "similar" problem Can be enhanced by adding reduction rules

17

Tabu Search & Dynamic Programming

|N 1| small and depends on the instance and the machine. How defining N 1 and N 2 ?

N1

N2

Dynamic Programmingwith Fixation on N1

Tabu Search on N2

Partition

Recurrence

|N| = 8

18

Dynamic Programming (Bellman, 1957)

Recurrence relation for the MKP

In our case all the subproblems concern the same variables and only g varies

19

Reduction Rules

Let x0 be a feasible solution of the MKP. Order the variables such that:

Proposition 1 (Optimality):

Let ,

Proposition 2 (Fixation):

Balev, Fréville, Yanev and Andonov, 2001.

20

Dynamic Programming as a Heuristic

P Initial solution: x0

Order the variables:uj upper bound of P(e-x0,{j})

Partition the set of variables: N 1 and N 2

Dynamic Programming

if cx1 > cx0

or if |F| > 0

N 1

N 2

n, m

the machine

List L (with all the optimal values)Fixation Fx1

21

Global Intensification (GI) Mechanism

Dynamic Programming

List L (with all the optimal values)Fixation Fx1

Tabu Search:performed on N 2 and uses L

x*

If cx* > cx1 then restart the algorithm with x* as the initial solution

22

? 0 1 ? ? 1 011 0 1 0 1 1 00

Tabu Search

TS on N2 (|N2| = n2) Neighbourhood of the current solution x2 on N2:

Evaluation of a solution over N from a solution over N2

N 1

N 2

Scan in L

23

Computational Environment

Algorithm coded in C language The results have been obtained on a Solaris Ultra

Sparc workstation 300MHz n1 15% of n OR-Library : 270 correlated instances

n = 100, 250, 500 and m = 5, 10, 30 30 instances for each (n,m)

24

A Summary of the Results

Each value = average over 30 instances

%: average gap wrt CPLEX (with the same computational time)

CPU: in seconds

TS alone: Tabu Search algorithm without global intensification mechanism

25

RelaxationDefinition : Let

(P) max{ f(x) : x X}

(R) max{ g(x) : x Y}

Problem R is a relaxation of P if

1) X Y

2) x X : f(x) ≤ g(x).Properties

- v(P) ≤ v(R)

- Let x* an optimal solution of R, if x* is feasible for P and f(x*)=g(x*) then x* is an optimal solution of P

26

LP-RelaxationDantzig (1957)

max max cxcx

(LP)(LP) s.t.s.t. Ax ≤ bAx ≤ b

x x [0, 1] [0, 1]nn

Properties :

- v(MKP) ≤ v(LP)

- Let x* an optimal solution of LP,

if x* {0,1}n then x* is an optimal solution of MKP

27

MIP Relaxation

Let x0 {0, 1}n and J N

Remarks: MIP(x0,) = LP(P), MIP(x0,N) = P v(MIP(x0,J)) v(MIP(x0,J’) if J J’

Stronger bound : v(P) v(MIP(P,J)) v(LP(P))

Hanafi & Wilbaut (06), Glover (06)

28

Lagrangean Relaxation

Lagrangean Relaxation : 0

LR() max{cx + (b – Ax) : x X} Lagrangean Dual :

(L) min{v(LR()) : 0}

Properties :

- v(MKP) ≤ v(L) ≤ v(LR())

- Let x* be an optimal solution of LR(*), if *(b – Ax*) = 0 and x* is feasible for P then x* is an optimal solution of P

Held & Krap (71), Geoffrion (74)

29

Geometric Interpretation : v(L) = max{cx : Ax ≤ b, x co(X)}

eDxx

Conv nZxeDxx ,

bAxx

cx

LP

LR

P

nZxeDxxX ,

30

Lagrangean Decomposition

nZx

eDx

bAx

cxZ

max

n

n

Zy

Zx

yx

eDy

bAx

cxZ

max

n

n

LD

Zy

Zx

eDy

bAx

xyvcxvZ

)(max)(

Guignard & Kim (87)

31

Lagrangean Decomposition

)()(min 210

vZvZZ LDLDv

LD

n

n

LD

Zy

Zx

eDy

bAx

xyvcxvZ

)(max)(

n

LD

Zx

bAx

xvcvZ

)(max)(1

n

LD

Zy

eDy

vyvZ

max)(2

Sub-problem 1 Sub-problem 2

32

P

Geometric InterpretationSub Pb 1

eDxx bAxx

cx

LP

LR

Conv nZxeDxx ,

Conv nZxbxAx ,

LD

Sub Pb 2

33

Theorem

)()()()( LPvLRvLDvPv

34

Surrogate Relaxation

Relaxation surrogate : 0SR()max{cx : Ax ≤ b, x X}

Dual surrogate corresponding : (S) min{v(SR()) : 0}.

Glover (65)

Propriétés :

- v(MKP) ≤ v(S) ≤ v(SR())

- Let x* be an optimal solution of SR(),

if x* is feasible for MKP then x* is an optimal solution of MKP

- (S) min{v(SR()) : U = { 0 : |||| = 1}}

35

Composite Relaxation Composite Relaxation : , ≥ 0

CR(, ) max{cx + (b – Ax) : Ax ≤ b and x X}

Composite Dual :

(C) min{v(CR(, )) : , ≥ 0}

Greenberg & Pierskalla (70)

Remarks : - v(LR()) = v(CR(, 0)) - v(SR()) = v(CR(0, ))

36

Comparisons

v(P) v(C) v(S) v(L) v(LP)

37

Properties of Relaxation Functions

Lagrangean function v(LR()) is convex and piecewise-linear if X is finite

Surrogate function surrogate v(SR()) is quasi-convex and piecewise-linear if X is finite

Composite function v(CR(, )) is non convex, non quasi-convex

38

Methods for Dual

(Quasi-)Sub-gradient : ’ = + t gg : sub-gradient of f( )=v(R())

t : Stepwise

x*() an optimal solution of R() g() = b – Ax*() a sub-gradient of f()

Bundle method, linearization, etc.

39

Intersection Algorithmvv(LR((LR())))

11 2200 maxmax**

Fisher (75), Yu (90), (90), Hanafi (98)

40

Extended Dichotomy Algorithmfor Bi-dimensional Knapsack

AA11xx≤≤bb11

AA22xx≤≤bb22

x*(u)x*(u)

uAxuAx≤u≤ubb

Fréville & Plateau (93), Fréville & Plateau (93), Hanafi (93)Hanafi (93)

41

Relaxation-based Heuristics

u = u°, UB = g(u), LB = f(x0) Repeat

let x*(u) an optimal of the current relaxation R(u) let x(u) the projection of x*(u) an the feasible set of P UB = min(UB, v(R(u))) LB = max(LB, f(x(u))) Update the current multiplier u

42

Relaxation-based Heuristics

Step 1: Solve one or more relaxations of the current problem P to generate one or more constraints.

Step 2: Solve one or more reduced problems induced by the optimal solution(s) of the previous relaxation(s) to obtain one or more feasible solution(s) of the initial problem.

Step 3: If a stopping criteria is satisfied then return the best lower bound. Else add a generated constraint(s) to the problem P and return to step 1.

Hanafi & Wilbaut (2006)

43

Space Search

Binary Solution : x {0, 1}n

Improved Heuristic Continuous Solution : x [0, 1]n

LP-Relaxation Partial Solution : x {0, 1, *}n where * = free

Branch&Bound, Dynamic Programming, Heuristic Feasible and Infeasible Solutions

Oscillation, Resolution Search, Dynamic B&B

44

Reduced Problem

Let x0 {0, 1}n and J N

P(x0, J) = (P| xJ = x0J)

Remarks : P(x0,) = P, v(P(x0,J)) v(P(x0,J’) if J J’ xJ = x0

J (eJ – 2 x0J)xJ + eJ x0

J = 0

|J| = 1 If v(P(e - x0,{j})) ≤ cx0 then x*j = x0

j , x* an optimal solution

45

LP-based Heuristic

Step 1: Solve the LP-relaxation of P to obtain an optimal solution xLP

Step 2: Solve the reduced problem P(xLP,J(xLP)) to generate an optimal solution x0

where J (x) = {j N : xj {0, 1}}

Remarks: cx0 ≤ v(P) ≤ cxLP

Relaxation induced neighbourhoods search, Danna et al., 2003P((xLP+x*)/2,J((xLP+x*))/2) with cx > cx*

where x* the best feasible solution

46

Pseudo-Cut

Let y be a vector in {0, 1}n, the following inequality (*) cuts off the solution y without cutting off any other solution in {0, 1}n

(*)

where J0(x) = {j N : xj = 0} J1(x) = {j N : xj = 1}.

Example:

(*) is called canonical cut in Balas and Jeroslow, 1972.

47

Partial Pseudo cuts

Let x’ be a partial solution in {0, 1, *}n, the inequality

dJ(x, x’) = (eJ – 2 x’J)xJ + eJ x’J 1 (*)cuts off x’ and all partial solution dominated by x’.

J = {j N: x’j {0,1}}, xJ = (xj : j J)Remarks : (*) is called

Pseudo Cut by Glover, 2005. Canonical Cut by Balas & Jeroslow, 1972, if J = N

dN (x, x’) = Hamming distance over {0, 1}n

48

Iterative LP-based heuristic (ILPH)

Let x* a feasible solution of Pv* = cx*; Q = P ; stop = False;while (stop = False) do Let xLP be an optimal solution of LP(Q); J = J(xLP);

Let x0 be an optimal solution of (Q|(eJ – 2 xLPJ)xJ k – eJ xLP

J) if cx0 > v* then x* = x0; v* = cx0

Q = (Q | (eJ – 2 xLPJ)xJ k + 1 – eJ xLP

J) if cxLP-v* < 1 then stop = True end while

49

Theorem : Finite Convergence

The algorithm ILPH converges to an optimal solution of the problem in a finite number of iterations ( 3n).

Proof : v(P) = max(v(P|dx d0), v(P|dx d0 + 1))

d = (eJ – 2 xLPJ, 0) and d0 = – eJ xLP

J

Number of partial solutions = |{0, 1, *}n|=3n

50

Convergence of IPLH

CPU 400sec (Cplex 10sec)

Replace the optimal condition by a total number of iterations

F(x): set of fractional variables of solution x

51

Remark and Dominance

1 001 1 0 *01 * 01

1 001 1 0 *01 * * *

1 001 1 0 *01 * 01

1 001 1 0 *01 * * *x1

x2

J1 J0 F

Reduced problem included

52

Improvements: Dominance Properties

Definition: Let x1 and x2 be two solutions in [0,1]n,x1 dominates x2 if F(x2) F(x1), J 1(x1) J 1(x2) and J 0(x1) J 0(x2)

Property 1: If x1 dominates x2 then v(P(x1,J(x1))) v(P(x2,J(x2)))

Property 2: The solution x defined bydominates x1 and x2, and we have :

v(P(x,J(x))) max { v(P(x1,J(x1))) , v(P(x2,J(x2))) }

Reduction of the number of reduced problems to solve

53

Illustration of Dominance

ILPH_D: ILPH with property 1

ILPH_D*(t) : ILPH with properties 1 and 2

Reduction between 30 and 50%

54

Iterative independent relaxations-based heuristic (IIRH)

xk optimal solution of MIP(Qk,F(xk-1))

k = k + 1

Pk+1 = (Pk | {f kx ≤ | J1(yk)| - 1}

yk optimal solution of LP(Pk)

wk optimal solution of P(yk,J(yk)) zk optimal solution of Q(xk,J(xk))

v*= max(v*,cwk,czk) u = min(cyk,cxk)

Q k+1 = (Q k | {f kx ≤ | J1(xk)| - 1}

Q = P, x0 optimal solution of LP(P); k = 1

Stop No Yes

If u–v*< 1

55

Iterative Relaxations-based Heuristic (IRH)

k = k + 1

yk optimal solution of LP(Pk)

Pk+1 = (Pk | {f kx ≤ | J1(yk)| - 1 and/or f kx ≤ | J1(xk)| - 1}

xk optimal solution of MIP(Pk,F(yk))

wk optimal solution of P(yk,J(yk)) zk optimal solution of P(xk,J(xk))

v*= max(v*,cwk,czk) u = min(cxk,cyk)Stop

No

Yes

If u–v*< 1

56

Comparison on a small instance

n = 30; m = 10

v* = 376

57

Results for n = 500

1 line = average / 10 instances

Average computational time 70 hours 2 hours

58

Summary (n = 500)

1 line = results for 30 instances

59

Some Related Works

Canonical Cuts on the Unit Hypercube, Balas & Jeroslow, 1972 0-1 IP with many variables and few constraints, Soyster, Lev,

Slivka, 1978 A hybrid approach to discrete mathematical programming,

Marsten, Morin, 1978 A new hybrid method combining exact solution and local

search, Mautor, Michelon, 1997 Parametric Branch and Bound, Glover, 1978 Large neighborhood search, Shaw, 1998 Local branching, Fischetti, Lodi, 2002

60

Some Related Works Short Hot Starts for B&B, Spielberg & Guignard, 2003 Relaxation induced neighbourhood search, Danna, Rothberg, Pape, 2003 Adaptive memory projection method, Glover, 2005 Feasibility Pump, Fischetti, Glover, Lodi, 2005 Relaxation guided VNS, Puchinger, Raidl, 2005 Global intensification using dynamic programming, Wilbaut, Hanafi,

Fréville, Balev, 2006 Convergent Heuristics for 0-1 MIP, Wilbaut, Hanafi, 2006 Variable Neighbourhood Decomposition Search, Lazic, Hanafi,

Mladenovic, Urosevic, 2009 Inequalities and Target Objectives in Metaheuristics for MIP, Glover,

Hanafi, 2009

61

Inequalities and Target Objectives in Metaheuristics for MIP

Recent adaptive memory and evolutionary metaheuristics for MIP have included proposals for introducing inequalities and target objectives to guide the search.

Two types of search strategies Fix subsets of variables to particular values within

approaches for exploiting strongly determined and consistent variables,

Use of solution targeting procedures.

Glover & Hanafi (09)

62

Inequalities and Target Objectives

Let a Target Solution x′, solveLP(x′, c′, M) min{Mfx + c′x : x X}

M is a large positive number. The coefficients c′j of the Target Objective seek to induce assignments xj = x′j for different variables with varying degrees of emphasis.An alternative to solve LP(LP(x′, c′, M) in two stages: 1- Solve LP(x′, c′, 0) = min{c′x : x X}2- Solve the problem min{fx : x X, cx′ = v(LP(x′, c′, 0))}Remark : The residual problem can be significantly smaller than the first stage problem, allowing it to be solved very efficiently

Glover & Hanafi (09)

63

Strong Pseudo Cut

x {0, 1}n x X {x : (x, x) max{1, v(LP(x))}} = X

LP(x) min{(x, x) : x X}

We call x the target solution for LP(x’) Remarks : x X v(LP(x)) = 0 (x, x ) = v(LP(x)) fractional x X Parametric branch and bound, Glover (1978) Feasibility pump, Fischetti, Glover and Lodi (2005)

Nj

jjjj xxxxxx )1(')'1(),'(

64

Illustration

MartinGrötsche

l

64

x

x

(x, x) = min{(x, x) : x X}d(x, x) d(x, x)

65

Partial Pseudo Cuts

For all y s.t. (J, x, y) = 0, we have

y X {x : (J, x, x) max{1, v(LP(J, x))}}

LP(J, x) min{(J, x, x) : x X}

Remarks : Partial Hamming distance over {0, 1, *}n

Jj

jjjj xxxxxxJ )1(')'1(),',(

Let x [0,1]n, J N(x) = {j : xj {0, 1}}

66

Theorem

Let x [0,1]n, N(x) = {j N: xj {0, 1}}, k an integer 0 k n - |N(x)|

The canonical hyperplane associated to x :H(x, k) = { x [0,1]n : (N(x), x, x) = k}We have :

Co(H(x, k) {0,1}n) = H(x, k)

where Co(X) is the convex hull of the set X.

Balas & Jeroslow 1972, Glover & Hanafi 2008

67

Weighted Partial Pseudo Cuts

Nj

jjjjj xxxxcxxc )1(')'1(),',(Let x [0,1]n, c C(x) = {c INn : cjxj (1 – xj) = 0}

x {0, 1}n y : (c, x, y) = 0 y X {x : (c, x, x) max{1, v(LP(c, x))}}

LP(c, x) min{(c, x, x) : x X}Remarks : (e, x, x) = || x – x||1 = || x – x||2 (J, x, x) = (c, x, x) : cj = 1 if j J else cj = 0.

68

Additional and Stronger Inequalities from Basic LP Solutions

Let x″ a basic feasible solution for

LP(c, x) min{(c, x, x) : x X}

Let d = c - rc where rc is reduced cost to x″, consider the pseudo cut

(d, x, x) v(LP(d, x)) (*)

LP(d, x) min{(d, x, x) : x X}

The inequality (*) is satisfied by all binary vectors x X, and excludes the solution x = x″ when (c, x,x″) is fractional.

If the basic solution x″ for LP(x, c) is optimal then the inequality (*) dominates

(c, x, x) v(LP(c, x))

69

Intensification Procedure

Pp

PpsXx

xxdsxxdts

sws

p

pppp

ppPp

pp

,

,0,

)",',(),',(..

min 0

Pp

Xx

xxdxxdsts

sppppp

,),',()",',(..

min

0

0

Min-Sum

Min-Max

70

Diversification Procedure

Pp

PpsXx

xxdsxxdts

sws

p

pppp

ppPp

pp

,

,0,

)",',(),',(..

max 0

Pp

Xx

xxdxxdsts

sppppp

,),',()",',(..

max

0

0

Max-Sum

Max-Min

71

Conclusions

Many MIP problems resist to be solved by the best current B&B and B&C methods.

Metaheuristics can likewise profit from generating inequalities to supplement their basic functions.

This framework can be exploited in generating target objectives, employing both adaptive memory ideas and newer ones proposed here.

72

Some Prospects

Extension to other problems Improve the implementation of dominance rules Integrate intensification and diversification Strengthen the use of memory Parallel versions