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Introduction Getting Started

Mixed Integer Programming Class LibraryIntroduction

N.N. [email protected]

Belarus State UniversityFaculty of Economics

Minsk - 2015

N.N. Pisaruk [email protected] MIPCL

Introduction Getting Started

Outline

1 IntroductionWhat is MIPCL?How good is MIPCL?

2 Getting StartedSolving InstancesSolving Problems

Introduction Getting Started What is MIPCL? How good is MIPCL?

Outline

What is MIPCL?

Mixed Integer Programming Class Library (MIPCL) is tool forsolving

Linear Programs (LP);Mixed Integer Programs (MIP).

MIPCL is implemented as library of C++ classes.It can be used

standalone as shared object (dynamic linked library) in any C++projects;together with modelling environment MIPshell;as application for solving MIPs given in MPS format.

What is MIPCL?

Jump to MIP

What is MIPCL?

MIPCL Features

1 MIPCL API makes it easy to integrate optimization into user’sapplications.

2 MIPCL is highly flexible through many virtual functions thatallow developers to implement problem specific

1 cut and column generators;2 primal heuristics to search for feasible solutions;3 node selectors to guide the search;4 branching rules to split the problem into subproblems;5 preprocessing to simplify solved problems;6 management of cuts and generated columns.

3 MIPCL comes with clear, comprehensive documentation andexamples to help you get started quickly.

MIPCL Features

Outline

Results for the MIPLIB-2010 Benchmark set:MIPCL on AMD FX8020 CPU (8 cores) vs SCIP on 2 Intel Xeon X5680 CPUs (6+6 cores)

Variables Time in sec.Name CtrsAll Int Bin MIPCL SCIP

30n20b8 576 18380 7344 11036 90 200.0%acc-tight5 3052 1339 1339 64 66aflow40b 1442 2728 1364 569 233

air04 823 8904 8904 14 70app1-2 53467 26871 13300 1080 490.3%

ash6gpia-3col 24748 3651 3651 710 94bab5 4964 21600 21600 332 0.9%

beasleyC3 1750 2500 1250 24 22.3%biella1 1203 7328 6110 94 723bienst2 576 505 35 246 128

bincar10_1 1026 2298 170 3 35bley_x11 175620 5831 5831 42 178bnatt350 4923 3150 3150 ∞% 344

core2536-691 2539 15293 15284 135 359cov1075 637 120 120 1473 1601

csched010 351 1758 1284 3792 1307danoint 664 521 56 2927 599

dfn-gwin-UUM 158 938 90 284 57eil32-2 32 4516 4516 100 163eilB101 100 2818 2818 184 445

enlight13 169 338 169 169 61 931enlight14 196 392 196 196 249 193

ex9 40962 10404 10404 4 87glass4 396 322 322 82.2% 570

gmu-35-40 424 1205 1200 1611 abortiis-100-0-cov 4803 345 345 385 718iis-bupa-cov 4803 345 345 2096 1350iis-pima-cov 7201 768 768 491 889

lectsched-4-obj 14163 7901 236 7665 72 121m100n500k4r1 100 500 500 4.2% 4.2%

macrophage 3164 2260 2260 1793 27.0%map18 328818 164547 146 0.98 3.03map20 328818 164547 146 464 1292

mcsched 2107 1747 14 1731 122 269mik-250-1-100-1 151 251 150 100 7 19

mine-166-5 8429 830 830 8 67mine-90-10 6270 900 900 633 320msc98-ip 15850 21143 53 20237 3205 57.0%mspp16 561657 29280 29280 5047 abortmzzv11 9499 10240 251 9989 37 850n3div36 4484 22120 22120 2766 11.0%n3seq24 6044 119856 119856 20.0% 20.0%

n4-3 1236 3596 174 43 441neos-1109824 28979 1520 1520 7 72neos-1337307 5687 2840 2840 0.0% 0.1%

neos-1396125 1494 1161 129 147 1278neos-1601936 3131 4446 3906 60 1223neos-476283 10015 11915 5588 1876 878neos-686190 3664 3660 60 3600 39 81neos-849702 1041 1737 1737 581 412neos-916792 1909 1474 717 ?% 15.7%neos-934278 11495 23123 19955 1258 4.0%

neos13 20852 1827 1815 170 655neos18 11402 3312 3312 34 65net12 14021 14115 1603 1129 44.9%

netdiversion 119589 129180 129180 1411 ∞%newdano 576 505 56 3463 924noswot 182 128 25 75 167 191

ns1208400 4289 2883 2880 212 ∞%ns1688347 4191 2685 2685 20 34.7%

ns1758913 624166 17956 17822 3107 ∞%ns1766074 182 100 90 53 2366ns1830653 2932 1629 1458 29 308opm2-z7-s2 31798 2023 2023 144 1572

pg5_34 225 2600 100 577 377pigeon-10 931 490 400 11.1% 11.1%

pw-myciel4 8164 1059 1 1058 537 1242qiu 1192 840 48 10 70

rail507 509 63019 6309 2078 3054ran16x16 288 512 256 36 54reblock67 2523 670 670 487 86

rmatr100-p10 7260 7359 100 31 397rmatr100-p5 8685 8784 100 35 1535

rmine6 7078 1096 1096 182 354rocII-4-11 21738 9234 9086 316 374err

rococoC10-001000 1293 3117 124 2993 1371 370roll3000 2295 1166 492 246 13 1440

satellites1-25 5996 9013 8509 616 240.0%sp98ic 825 10894 10894 1714 0.4%sp98ir 1531 1680 809 871 28 82

tanglegram1 2295 1166 492 246 740 0.0%tanglegram2 5996 9013 8509 14 2153

timtab1 825 10894 10894 92 174triptim1 1531 1680 809 871 953 3011unitcal_7 48939 25755 2856 498 0.1%vpphard 47280 51471 51471 ∞% ∞%

zib54-UUE 1809 5150 81 1079 811

Introduction Getting Started Solving Instances Solving Problems

Outline

Example

10x0 + 6.4x1→max50x0 + 31x1 ≤ 250,−3x0 + 2x1 ≤ 4,x0, x1 ≥ 0 and integer.

Simple MIPCL-program (Variant 1)

#include "cmip.h"

void main(){

CMIP prob("MIPCLtest");prob.openMatrix(2,2,4);prob.addColumn(0,VAR_INT,10.0,0.0,VAR_INF);prob.addColumn(1,VAR_INT,6.4,0.0,VAR_INF);prob.addRow(0,0,-INF,250);prob.addRow(1,0,-INF,4);prob.addEntry(50.0,0,0);prob.addEntry(31.0,0,1);prob.addEntry(-3.0,1,0);prob.addEntry( 2.0,1,1);prob.closeMatrix();prob.optimize();prob.printSolution("solution.txt");

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"

void main(){

#include "cmip.h"const int n=2, m=2, nz=4;double c[] = {10.0,6.4}, b[] = {250,4};double A[][n] = {{50,31},{-3,2}};int ind[] = {0,1};void main(){

CMIP prob("MIPCLtest");prob.openMatrix(m,n,nz);for (int j=0; j < n; ++j)

prob.addColumn(j,VAR_INT,c[j],0.0,VAR_INF);for (int i=0; i < m; ++i)

prob.addRow(i,0,-INF,b[i],n,A[i],ind);

prob.closeMatrix();prob.optimize();prob.printSolution("solution.txt");

}N.N. Pisaruk [email protected] MIPCL

Outline

Fixed Charge Flow Networks (FCNF)

Given:

G = (V,E) — digraph;

dv — demand at node v ∈ V (∑

v∈V dv = 0);

ue > 0 — capacity of arc e ∈ E;

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

— cost of shipping xe units of

flow along arc e ∈ E.

Find a flow x ∈ RE+ of minimum cost

∑e∈E c̄(xe) satisfying the

demand and capacity constraints.

Given:

v∈V dv = 0);

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

Given:

v∈V dv = 0);

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

Given:

v∈V dv = 0);

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

Given:

v∈V dv = 0);

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

Given:

v∈V dv = 0);

c̄(xe) =

{0, xe = 0,fe + cexe, xe > 0

FCNF MIP-formulation

Variables:xe — flow along arc e ∈ E,ye = 1 if xe > 0, and ye = 0 otherwise.

Formulation: ∑e∈E

(feye + cexe)→ min (2a)∑e∈E(V,v)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

where, for S,T ⊆ V , E(S,T)def= {(v,w) ∈ E : v ∈ S, w ∈ T}.

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

xe −∑

e∈E(v,V)

xe = dv, v ∈ V, (2b)

0 ≤ xe ≤ ueye, e ∈ E, (2c)

ye ∈ {0, 1}, e ∈ E, (2d)

Definition of Class CFCNF (file fcnf.h)

#include "cmip.h"

class CFCFN: public CMIP

{int n,m;int *d,*t,*h,*u,*f,*c;public:CFCFN(const char *name,int n,int m,int *d,

int *t,int *h,int *u,int *f,int *c);virtual ∼CFCFN() {};void printSolution(const char *FileName);private:void buildProb();

#include "cmip.h"

Jump to implementation

#include "cmip.h"

Implementation of CFCNF: Constructor

#include "fcnf.h"

CFCNF::CFCFN(const char *name,

int n0,int m0,int *d0,int *t0,int *h0,

int *u0,int *f0,int *c0): CMIP(name)

{n=n0; m=m0;d=d0; t=t0; h=h0; u=u0; f=f0; c=c0;buildProb();

Return to definition

#include "fcnf.h"

Implementation of CFCNF: function buildProb()

void CFCFN::buildProb(){

double w, val[]={-1.0,1.0,1.0};int r=n,row[3];openMatrix(m+n,2*m,6*m);for (int i=0; i < n; ++i)

addRow(i,0,d[i],d[i]);for (int e=0; e < m; ++e)

addRow(r++,0,-INF,0.0);for (int e=0; e < m; ++e) {

row[0]=t[e]; row[1]=h[e]; row[2]=n+e;addColumn(e,0,-c[e],0,VAR_INF,3,val,row);w=-u[e]; r=n+e;addColumn(e+m,VAR_BIN,-f[e],0,1,1,&w,&r);

}closeMatrix();

Implementation of CFCNF: function printSolution()

void CFCNF::printSolution(const char* name){

double *x;int m2=2*m,*hd;std::ofstream fout(name);if (isSolution()) {

getSolution(x,hd);fout << "Nonzero flows:\n";for (int i=0; i < m2; ++i)

if (hd[i] < m && x[i] > 0.5))fout << "flow(" << t[hd[i]] << ","

<< h[hd[i]] << ")="<< x[i] << std::endl;

}else fout << "Problem has no solution!\n";fout.close();

<< h[hd[i]] << ")="<< x[i] << std::endl;

Using CFCNF: function main()

int main(int argc, char *argv[])

{int n,m,*d,*t,*h,*u,*f,*c;readNet(argv[1],n,m,d,t,h,u,f,c);CFCFN prob("FCFN",n,m,d,t,h,u,f,c,d);prob.optimize();char name[128];strcpy(name,argv[1]);strcat(name,"_sol.txt");prob.printSolution(name);delete[] d;return 0;

Appendix Miscellaneous

3 AppendixMiscellaneous

Outline

3 AppendixMiscellaneous

Mixed Integer Programming

Mixed Integer Program (MIP):

cTx→ max

b1 ≤ Ax ≤ b2,

d1 ≤ x ≤ d2,

xj ∈ Z, for j ∈ S,

b1, b2 ∈ Rm,

c, d1, d2 ∈ Rn,

A — real m× n-matrix,

x — n-vector of variables (unknowns),

S ⊆ {1, . . . , n}— set of integer variables.Return to "What Is MIPCL?"

Using CFCNF: function readNet()

#include <fstream>#include "fcnf.h"void readNet(const char *fileName,

int &n,int &m,int* &d,int* &t,int* &h,int* &u,int* &f,int* &c)

{std::ifstream fin(fileName);fin >> n >> m;d = new int[n+5*m];c=(f=(u=(h=(t=d+n)+m)+m)+m)+m;for (int i=0; i < n; ++i)

fin >> d[i];for (int e=0; e < m; ++e)