MINLPLib
A Library of Mixed-Integer and Continuous Nonlinear Programming Instances
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Instance smallinvDAXr3b050-055
Small Investor Mean-Variance Portfolio Optimization This is a natural extension of the standard Markovitz Mean-Variance-Optimization (MVO) model by constraints for small investors. The variables in the standard Markovitz model determine which fraction of the investment is made in the repective assets. When the investment is bounded within a range that is not very much bigger than the prices of the assets, one has to take into account that each asset has a minimum unit of which only integer multiples can be traded, e.g. 1, 0.1, or 0.01. Hence, the optimization problem for the small investor has integer variables in addition to the budget constraints that define lower and upper bounds on the investment.
| Formatsⓘ | ams gms lp mod nl osil pip py |
| Primal Bounds (infeas ≤ 1e-08)ⓘ | |
| Other points (infeas > 1e-08)ⓘ | |
| Dual Boundsⓘ | 9.79714345 (ALPHAECP) 9.79714344 (ANTIGONE) 9.79714344 (BARON) 9.79714345 (BONMIN) 9.79714345 (COUENNE) 9.79714326 (CPLEX) 9.79714345 (GUROBI) 9.79714345 (LINDO) 9.79714345 (SCIP) 9.79714345 (SHOT) |
| Sourceⓘ | POLIP instance portfolio/small-investor-MVO-DAX30-return_0.0030-budget_5000-5500 |
| Applicationⓘ | Portfolio Optimization |
| Added to libraryⓘ | 26 Feb 2014 |
| Problem typeⓘ | MIQCP |
| #Variablesⓘ | 31 |
| #Binary Variablesⓘ | 0 |
| #Integer Variablesⓘ | 30 |
| #Nonlinear Variablesⓘ | 30 |
| #Nonlinear Binary Variablesⓘ | 0 |
| #Nonlinear Integer Variablesⓘ | 30 |
| Objective Senseⓘ | min |
| Objective typeⓘ | linear |
| Objective curvatureⓘ | linear |
| #Nonzeros in Objectiveⓘ | 1 |
| #Nonlinear Nonzeros in Objectiveⓘ | 0 |
| #Constraintsⓘ | 4 |
| #Linear Constraintsⓘ | 3 |
| #Quadratic Constraintsⓘ | 1 |
| #Polynomial Constraintsⓘ | 0 |
| #Signomial Constraintsⓘ | 0 |
| #General Nonlinear Constraintsⓘ | 0 |
| Operands in Gen. Nonlin. Functionsⓘ | |
| Constraints curvatureⓘ | convex |
| #Nonzeros in Jacobianⓘ | 121 |
| #Nonlinear Nonzeros in Jacobianⓘ | 30 |
| #Nonzeros in (Upper-Left) Hessian of Lagrangianⓘ | 900 |
| #Nonzeros in Diagonal of Hessian of Lagrangianⓘ | 30 |
| #Blocks in Hessian of Lagrangianⓘ | 1 |
| Minimal blocksize in Hessian of Lagrangianⓘ | 30 |
| Maximal blocksize in Hessian of Lagrangianⓘ | 30 |
| Average blocksize in Hessian of Lagrangianⓘ | 30.0 |
| #Semicontinuitiesⓘ | 0 |
| #Nonlinear Semicontinuitiesⓘ | 0 |
| #SOS type 1ⓘ | 0 |
| #SOS type 2ⓘ | 0 |
| Minimal coefficientⓘ | 1.1388e-04 |
| Maximal coefficientⓘ | 1.0190e+02 |
| Infeasibility of initial pointⓘ | 5000 |
| Sparsity Jacobianⓘ |  |
| Sparsity Hessian of Lagrangianⓘ |  |
$offlisting
*
* Equation counts
* Total E G L N X C B
* 4 0 2 2 0 0 0 0
*
* Variable counts
* x b i s1s s2s sc si
* Total cont binary integer sos1 sos2 scont sint
* 31 1 0 30 0 0 0 0
* FX 0
*
* Nonzero counts
* Total const NL DLL
* 121 91 30 0
*
* Solve m using MINLP minimizing objvar;
Variables i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,i17,i18,i19
,i20,i21,i22,i23,i24,i25,i26,i27,i28,i29,i30,objvar;
Integer Variables i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,i17
,i18,i19,i20,i21,i22,i23,i24,i25,i26,i27,i28,i29,i30;
Equations e1,e2,e3,e4;
e1.. 0.00558391*sqr(i1) + 0.0103737*sqr(i2) + 0.0221075*sqr(i3) + 0.00399551*
sqr(i4) + 0.00267064*sqr(i5) + 0.00516451*sqr(i6) + 0.00421051*sqr(i7) +
0.00368008*sqr(i8) + 0.00372788*sqr(i9) + 0.00668969*sqr(i10) + 0.00613034
*sqr(i11) + 0.0129271*sqr(i12) + 0.00697595*sqr(i13) + 0.0104282*sqr(i14)
+ 0.00899168*sqr(i15) + 0.0206685*sqr(i16) + 0.0488863*sqr(i17) +
0.00894867*sqr(i18) + 0.0124118*sqr(i19) + 0.0122291*sqr(i20) + 0.0128973*
sqr(i21) + 0.00668043*sqr(i22) + 0.0153425*sqr(i23) + 0.0128605*sqr(i24)
+ 0.00718697*sqr(i25) + 0.0102616*sqr(i26) + 0.0123235*sqr(i27) +
0.00569465*sqr(i28) + 0.00818114*sqr(i29) + 0.00469456*sqr(i30) +
0.00901964*i1*i2 + 0.00860972*i1*i3 + 0.00248004*i1*i4 + 0.001821292*i1*i5
+ 0.00561078*i1*i6 + 0.0051287*i1*i7 + 0.000691196*i1*i8 + 0.000805982*i1
*i9 + 0.00531452*i1*i10 + 0.00556768*i1*i11 + 0.00745224*i1*i12 +
0.00478224*i1*i13 + 0.00610824*i1*i14 + 0.00577634*i1*i15 + 0.00613326*i1*
i16 + 0.01432596*i1*i17 + 0.007501*i1*i18 + 0.00716212*i1*i19 + 0.00512922
*i1*i20 + 0.0087283*i1*i21 + 0.00245846*i1*i22 + 0.0071572*i1*i23 +
0.00543966*i1*i24 + 0.00708258*i1*i25 + 0.00243422*i1*i26 + 0.00729094*i1*
i27 + 0.00386642*i1*i28 + 0.0061908*i1*i29 + 0.00366754*i1*i30 +
0.01583972*i2*i3 + 0.00394608*i2*i4 + 0.001773554*i2*i5 + 0.00861376*i2*i6
+ 0.00604454*i2*i7 + 0.00312866*i2*i8 + 0.00184686*i2*i9 + 0.00924638*i2*
i10 + 0.01131902*i2*i11 + 0.01253232*i2*i12 + 0.00675858*i2*i13 +
0.00804604*i2*i14 + 0.00869872*i2*i15 + 0.0094047*i2*i16 + 0.0251538*i2*
i17 + 0.01321532*i2*i18 + 0.01127964*i2*i19 + 0.0096635*i2*i20 + 0.0160783
*i2*i21 + 0.00271*i2*i22 + 0.01486022*i2*i23 + 0.01091018*i2*i24 +
0.01009426*i2*i25 + 0.00754144*i2*i26 + 0.01408844*i2*i27 + 0.00544162*i2*
i28 + 0.01096178*i2*i29 + 0.00574964*i2*i30 + 0.00299428*i3*i4 +
0.001239314*i3*i5 + 0.01256412*i3*i6 + 0.00899714*i3*i7 + 0.00444448*i3*i8
+ 0.00616612*i3*i9 + 0.0146019*i3*i10 + 0.01249836*i3*i11 + 0.0264968*i3*
i12 + 0.01266506*i3*i13 + 0.01358566*i3*i14 + 0.01419766*i3*i15 +
0.01033796*i3*i16 + 0.040104*i3*i17 + 0.01504214*i3*i18 + 0.0210518*i3*i19
+ 0.0169342*i3*i20 + 0.020394*i3*i21 + 0.006361*i3*i22 + 0.0173249*i3*i23
+ 0.01157254*i3*i24 + 0.01601196*i3*i25 + 0.01305808*i3*i26 + 0.018918*i3
*i27 + 0.0100768*i3*i28 + 0.01415258*i3*i29 + 0.00890208*i3*i30 +
0.00365082*i4*i5 + 0.0031533*i4*i6 + 0.001664882*i4*i7 + 0.000487746*i4*i8
+ 0.00074873*i4*i9 + 0.00279536*i4*i10 + 0.000948078*i4*i11 + 0.00218644*
i4*i12 + 0.001471884*i4*i13 + 0.001764448*i4*i14 + 0.001707856*i4*i15 +
0.00415534*i4*i16 + 0.00552118*i4*i17 + 0.00298928*i4*i18 + 0.000446818*i4
*i19 + 0.0042709*i4*i20 + 0.00437068*i4*i21 + 0.001584414*i4*i22 +
0.0028495*i4*i23 + 0.00550266*i4*i24 + 0.0019381*i4*i25 - 0.000779792*i4*
i26 + 0.00383714*i4*i27 + 0.00170793*i4*i28 + 0.00220852*i4*i29 +
0.001897386*i4*i30 + 0.00226608*i5*i6 + 0.001391572*i5*i7 + 0.001434726*i5
*i8 + 0.000718962*i5*i9 + 0.00117417*i5*i10 + 0.001240914*i5*i11 +
0.000587866*i5*i12 + 0.0020154*i5*i13 + 0.00126883*i5*i14 + 0.000645164*i5
*i15 + 0.0001425196*i5*i16 + 0.001199014*i5*i17 + 0.001896292*i5*i18 -
0.000289412*i5*i19 + 0.001457998*i5*i20 + 0.00199702*i5*i21 + 0.001266598*
i5*i22 + 0.000764624*i5*i23 + 0.001961312*i5*i24 + 0.001748826*i5*i25 -
0.00122625*i5*i26 + 0.000753266*i5*i27 + 0.00063941*i5*i28 + 0.001644068*
i5*i29 + 0.001587886*i5*i30 + 0.00454154*i6*i7 + 0.001157686*i6*i8 +
0.0032018*i6*i9 + 0.00727798*i6*i10 + 0.0064553*i6*i11 + 0.00791618*i6*i12
+ 0.00687526*i6*i13 + 0.00638032*i6*i14 + 0.00425538*i6*i15 + 0.00583332*
i6*i16 + 0.01491304*i6*i17 + 0.00876772*i6*i18 + 0.00814434*i6*i19 +
0.00549208*i6*i20 + 0.0103848*i6*i21 + 0.001352278*i6*i22 + 0.0063097*i6*
i23 + 0.0052012*i6*i24 + 0.00808494*i6*i25 + 0.00595234*i6*i26 +
0.00960786*i6*i27 + 0.0035648*i6*i28 + 0.00730486*i6*i29 + 0.0036145*i6*
i30 + 0.0027426*i7*i8 + 0.00224138*i7*i9 + 0.00558948*i7*i10 + 0.00489378*
i7*i11 + 0.0073565*i7*i12 + 0.0050794*i7*i13 + 0.00363244*i7*i14 +
0.00634576*i7*i15 + 0.001588982*i7*i16 + 0.00877926*i7*i17 + 0.00710862*i7
*i18 + 0.00675396*i7*i19 + 0.00621206*i7*i20 + 0.00746652*i7*i21 +
0.001927036*i7*i22 + 0.00410122*i7*i23 + 0.00344774*i7*i24 + 0.00594546*i7
*i25 + 0.00461784*i7*i26 + 0.00530234*i7*i27 + 0.00320122*i7*i28 +
0.00474356*i7*i29 + 0.00341222*i7*i30 + 0.00105347*i8*i9 + 0.001879822*i8*
i10 + 0.00290244*i8*i11 + 0.00353818*i8*i12 + 0.0035513*i8*i13 +
0.00294406*i8*i14 + 0.00389942*i8*i15 + 0.00286866*i8*i16 + 0.000920126*i8
*i17 + 0.00274282*i8*i18 + 0.0027675*i8*i19 + 0.00464592*i8*i20 +
0.001093444*i8*i21 + 0.000948594*i8*i22 + 0.00275316*i8*i23 + 0.001626794*
i8*i24 + 0.00209498*i8*i25 + 0.0031962*i8*i26 + 0.001767658*i8*i27 +
0.00109948*i8*i28 + 0.00292004*i8*i29 + 0.00215496*i8*i30 + 0.00329222*i9*
i10 + 0.00239978*i9*i11 + 0.00365066*i9*i12 + 0.00463422*i9*i13 +
0.00260888*i9*i14 + 0.00330432*i9*i15 + 0.000950274*i9*i16 + 0.00309664*i9
*i17 + 0.00325462*i9*i18 + 0.00494078*i9*i19 + 0.00339202*i9*i20 +
0.00283784*i9*i21 + 0.001862472*i9*i22 + 0.001457294*i9*i23 + 0.000292408*
i9*i24 + 0.00434258*i9*i25 + 0.0051917*i9*i26 + 0.00442724*i9*i27 +
0.00235362*i9*i28 + 0.0023207*i9*i29 + 0.00232972*i9*i30 + 0.00661128*i10*
i11 + 0.0099349*i10*i12 + 0.00670728*i10*i13 + 0.00688756*i10*i14 +
0.00814804*i10*i15 + 0.00387536*i10*i16 + 0.01709622*i10*i17 + 0.00921546*
i10*i18 + 0.01138012*i10*i19 + 0.0073598*i10*i20 + 0.012047*i10*i21 +
0.001953884*i10*i22 + 0.01110682*i10*i23 + 0.00744232*i10*i24 + 0.00846572
*i10*i25 + 0.00811902*i10*i26 + 0.01093528*i10*i27 + 0.00642736*i10*i28 +
0.00817838*i10*i29 + 0.00467066*i10*i30 + 0.01089978*i11*i12 + 0.00580646*
i11*i13 + 0.00479126*i11*i14 + 0.00655088*i11*i15 + 0.00784072*i11*i16 +
0.0171429*i11*i17 + 0.0099023*i11*i18 + 0.00881158*i11*i19 + 0.0065332*i11
*i20 + 0.01111462*i11*i21 + 0.00238226*i11*i22 + 0.00942038*i11*i23 +
0.00509366*i11*i24 + 0.0079177*i11*i25 + 0.00653764*i11*i26 + 0.00963386*
i11*i27 + 0.00518254*i11*i28 + 0.00839924*i11*i29 + 0.00396162*i11*i30 +
0.00812884*i12*i13 + 0.00932748*i12*i14 + 0.01172114*i12*i15 + 0.00937084*
i12*i16 + 0.033621*i12*i17 + 0.0125625*i12*i18 + 0.01635358*i12*i19 +
0.01460644*i12*i20 + 0.01374474*i12*i21 + 0.00526496*i12*i22 + 0.01402198*
i12*i23 + 0.00931776*i12*i24 + 0.01195866*i12*i25 + 0.00822682*i12*i26 +
0.01241788*i12*i27 + 0.00706034*i12*i28 + 0.01219462*i12*i29 + 0.00598988*
i12*i30 + 0.0068538*i13*i14 + 0.00620178*i13*i15 + 0.00379406*i13*i16 +
0.00889862*i13*i17 + 0.00816594*i13*i18 + 0.01033824*i13*i19 + 0.00577162*
i13*i20 + 0.00736548*i13*i21 + 0.00410776*i13*i22 + 0.00580558*i13*i23 +
0.00459074*i13*i24 + 0.0072167*i13*i25 + 0.00956086*i13*i26 + 0.00943468*
i13*i27 + 0.00587164*i13*i28 + 0.00902842*i13*i29 + 0.00550608*i13*i30 +
0.00635356*i14*i15 + 0.00709628*i14*i16 + 0.01555038*i14*i17 + 0.00826722*
i14*i18 + 0.00751614*i14*i19 + 0.00814342*i14*i20 + 0.00995652*i14*i21 +
0.00477798*i14*i22 + 0.0076843*i14*i23 + 0.00817698*i14*i24 + 0.00886056*
i14*i25 + 0.00579636*i14*i26 + 0.01128084*i14*i27 + 0.00483444*i14*i28 +
0.0068342*i14*i29 + 0.0077372*i14*i30 + 0.00973548*i15*i16 + 0.01556958*
i15*i17 + 0.00926266*i15*i18 + 0.01281188*i15*i19 + 0.00669072*i15*i20 +
0.00937684*i15*i21 + 0.00639856*i15*i22 + 0.00611934*i15*i23 + 0.00853942*
i15*i24 + 0.00964296*i15*i25 + 0.00704584*i15*i26 + 0.0119279*i15*i27 +
0.00648174*i15*i28 + 0.01050128*i15*i29 + 0.00502696*i15*i30 + 0.01809222*
i16*i17 + 0.00823288*i16*i18 + 0.01161214*i16*i19 + 0.00533676*i16*i20 +
0.01233794*i16*i21 + 0.00512778*i16*i22 + 0.00722276*i16*i23 + 0.01715638*
i16*i24 + 0.00677738*i16*i25 + 0.0069565*i16*i26 + 0.01691522*i16*i27 +
0.00246824*i16*i28 + 0.00934088*i16*i29 + 0.00393866*i16*i30 + 0.01858542*
i17*i18 + 0.0224912*i17*i19 + 0.01793624*i17*i20 + 0.0270204*i17*i21 +
0.01083832*i17*i22 + 0.0216678*i17*i23 + 0.0183347*i17*i24 + 0.01893*i17*
i25 + 0.01089098*i17*i26 + 0.0209142*i17*i27 + 0.01273162*i17*i28 +
0.0200902*i17*i29 + 0.00774366*i17*i30 + 0.01171594*i18*i19 + 0.00861454*
i18*i20 + 0.01414322*i18*i21 + 0.001961404*i18*i22 + 0.00910214*i18*i23 +
0.01003468*i18*i24 + 0.0094743*i18*i25 + 0.00825794*i18*i26 + 0.01336058*
i18*i27 + 0.00607998*i18*i28 + 0.01070732*i18*i29 + 0.00492858*i18*i30 +
0.0082848*i19*i20 + 0.0126004*i19*i21 + 0.00407366*i19*i22 + 0.01381284*
i19*i23 + 0.00838908*i19*i24 + 0.01198264*i19*i25 + 0.01583126*i19*i26 +
0.01664044*i19*i27 + 0.00924324*i19*i28 + 0.01214842*i19*i29 + 0.00592778*
i19*i30 + 0.01071434*i20*i21 + 0.00296964*i20*i22 + 0.00736528*i20*i23 +
0.00606396*i20*i24 + 0.00628822*i20*i25 + 0.00817696*i20*i26 + 0.00776894*
i20*i27 + 0.0026202*i20*i28 + 0.00717342*i20*i29 + 0.00579184*i20*i30 +
0.00469936*i21*i22 + 0.0138599*i21*i23 + 0.0125037*i21*i24 + 0.01211002*
i21*i25 + 0.00836436*i21*i26 + 0.016494*i21*i27 + 0.00602872*i21*i28 +
0.01180462*i21*i29 + 0.00570478*i21*i30 + 0.0032176*i22*i23 + 0.00379112*
i22*i24 + 0.00301976*i22*i25 + 0.00308424*i22*i26 + 0.00369962*i22*i27 +
0.00278784*i22*i28 + 0.00465846*i22*i29 + 0.00297212*i22*i30 + 0.01019176*
i23*i24 + 0.00779098*i23*i25 + 0.00577776*i23*i26 + 0.01267514*i23*i27 +
0.00735432*i23*i28 + 0.00786386*i23*i29 + 0.00559972*i23*i30 + 0.00725022*
i24*i25 + 0.00455648*i24*i26 + 0.0157223*i24*i27 + 0.00579512*i24*i28 +
0.00792398*i24*i29 + 0.0045755*i24*i30 + 0.00723442*i25*i26 + 0.01196012*
i25*i27 + 0.0063273*i25*i28 + 0.0099815*i25*i29 + 0.0041794*i25*i30 +
0.01139894*i26*i27 + 0.0080092*i26*i28 + 0.0080044*i26*i29 + 0.00493602*
i26*i30 + 0.00826208*i27*i28 + 0.01246152*i27*i29 + 0.0067556*i27*i30 +
0.00575648*i28*i29 + 0.0044929*i28*i30 + 0.00469952*i29*i30 - objvar =L= 0
;
e2.. 0.00418236*i1 + 0.00084825*i2 - 0.00595114*i3 + 0.013626*i4
+ 0.0104202*i5 + 0.00696825*i6 + 0.00454677*i7 - 0.00034318*i8
- 0.00570481*i9 - 0.00023195*i10 + 0.00048701*i11 - 0.00248068*i12
+ 0.00586372*i13 - 0.00923462*i14 - 0.00111219*i15 + 0.00557072*i16
- 0.00728342*i17 - 0.00068175*i18 - 0.00431856*i19 - 0.00011388*i20
+ 0.0152599*i21 + 0.00645572*i22 + 0.0087442*i23 + 0.0205808*i24
+ 0.00360132*i25 - 0.00424596*i26 + 0.00406*i27 + 0.00429918*i28
+ 0.00365023*i29 + 0.0055383*i30 =G= 0;
e3.. 39.19*i1 + 41.47*i2 + 5.71*i3 + 53.59*i4 + 43.65*i5 + 85.46*i6
+ 39.7*i7 + 44.91*i8 + 9.6*i9 + 11.26*i10 + 39.56*i11 + 46*i12
+ 45.25*i13 + 21.9*i14 + 11.85*i15 + 37.4*i16 + 4.75*i17 + 44.44*i18
+ 80.5*i19 + 49.46*i20 + 67.02*i21 + 59.25*i22 + 71.5*i23 + 48.8*i24
+ 73.22*i25 + 101.9*i26 + 20.06*i27 + 36.33*i28 + 41.31*i29 + 53.09*i30
=G= 5000;
e4.. 39.19*i1 + 41.47*i2 + 5.71*i3 + 53.59*i4 + 43.65*i5 + 85.46*i6
+ 39.7*i7 + 44.91*i8 + 9.6*i9 + 11.26*i10 + 39.56*i11 + 46*i12
+ 45.25*i13 + 21.9*i14 + 11.85*i15 + 37.4*i16 + 4.75*i17 + 44.44*i18
+ 80.5*i19 + 49.46*i20 + 67.02*i21 + 59.25*i22 + 71.5*i23 + 48.8*i24
+ 73.22*i25 + 101.9*i26 + 20.06*i27 + 36.33*i28 + 41.31*i29 + 53.09*i30
=L= 5500;
* set non-default bounds
Model m / all /;
m.limrow=0; m.limcol=0;
m.tolproj=0.0;
$if NOT '%gams.u1%' == '' $include '%gams.u1%'
$if not set MINLP $set MINLP MINLP
Solve m using %MINLP% minimizing objvar;
Last updated: 2024-08-26 Git hash: 6cc1607f