MINLPLib

A Library of Mixed-Integer and Continuous Nonlinear Programming Instances

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Instance ball_mk4_10

A simple MINLP with a feasible set described by a ball.
The basic model over which these variations are made is:
min  sum_i=1^n x_i
s.t. sum_i=1^n (x_i - 0.5)^2 <= (n-1)/4
x integer between -1 and 1.
Obvisouly, this problem is infeasible and has no solution.
It can be shown that any outer-approximation based method will need 2^n linear inequalities to show infeasibility, see reference.
In this instance, the ball is an empty ellipse and the quadratic form is not diagonal.

Formatsⓘ	ams gms lp mod nl osil pip py
Primal Bounds (infeas ≤ 1e-08)ⓘ
Other points (infeas > 1e-08)ⓘ
Dual Boundsⓘ	-6.65024410 (ALPHAECP) 6.99327009 (ANTIGONE) 71.00000000 (BARON) 124.16373590 (BONMIN) -14240.00000000 (COUENNE) inf (CPLEX) 6.00000000 (GUROBI) 0.00000000 (LINDO) inf (SCIP) inf (SHOT)
Referencesⓘ	Hijazi, Hassan, Bonami, Pierre, and Ouorou, Adam, An Outer-Inner Approximation for Separable Mixed-Integer Nonlinear Programs, INFORMS Journal on Computing, 26:1, 2014, 31-44.
Sourceⓘ	Pierre Bonami
Applicationⓘ	Geometry
Added to libraryⓘ	11 Sep 2017
Problem typeⓘ	IQCP
#Variablesⓘ	20
#Binary Variablesⓘ	0
#Integer Variablesⓘ	20
#Nonlinear Variablesⓘ	20
#Nonlinear Binary Variablesⓘ	0
#Nonlinear Integer Variablesⓘ	20
Objective Senseⓘ	min
Objective typeⓘ	linear
Objective curvatureⓘ	linear
#Nonzeros in Objectiveⓘ	20
#Nonlinear Nonzeros in Objectiveⓘ	0
#Constraintsⓘ	1
#Linear Constraintsⓘ	0
#Quadratic Constraintsⓘ	1
#Polynomial Constraintsⓘ	0
#Signomial Constraintsⓘ	0
#General Nonlinear Constraintsⓘ	0
Operands in Gen. Nonlin. Functionsⓘ
Constraints curvatureⓘ	convex
#Nonzeros in Jacobianⓘ	20
#Nonlinear Nonzeros in Jacobianⓘ	20
#Nonzeros in (Upper-Left) Hessian of Lagrangianⓘ	40
#Nonzeros in Diagonal of Hessian of Lagrangianⓘ	20
#Blocks in Hessian of Lagrangianⓘ	10
Minimal blocksize in Hessian of Lagrangianⓘ	2
Maximal blocksize in Hessian of Lagrangianⓘ	2
Average blocksize in Hessian of Lagrangianⓘ	2.0
#Semicontinuitiesⓘ	0
#Nonlinear Semicontinuitiesⓘ	0
#SOS type 1ⓘ	0
#SOS type 2ⓘ	0
Minimal coefficientⓘ	1.0000e+00
Maximal coefficientⓘ	1.0000e+02
Infeasibility of initial pointⓘ	1
Sparsity Jacobianⓘ
Sparsity Hessian of Lagrangianⓘ

$offlisting
*  
*  Equation counts
*      Total        E        G        L        N        X        C        B
*          2        1        0        1        0        0        0        0
*  
*  Variable counts
*                   x        b        i      s1s      s2s       sc       si
*      Total     cont   binary  integer     sos1     sos2    scont     sint
*         21        1        0       20        0        0        0        0
*  FX      0
*  
*  Nonzero counts
*      Total    const       NL      DLL
*         41       21       20        0
*
*  Solve m using MINLP minimizing objvar;


Variables  objvar,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,i17,i18
          ,i19,i20,i21;

Integer Variables  i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15,i16,i17,i18
          ,i19,i20,i21;

Equations  e1,e2;


e1..    objvar - 19*i2 - 18*i3 - 17*i4 - 16*i5 - 15*i6 - 14*i7 - 13*i8 - 12*i9
      - 11*i10 - 10*i11 - 9*i12 - 8*i13 - 7*i14 - 6*i15 - 5*i16 - 4*i17 - 3*i18
      - 2*i19 - i20 - 20*i21 =E= 0;

e2.. 100*sqr(i20) - 98*i20 + 100*sqr(i19) - 98*i19 + 100*sqr(i18) - 98*i18 + 
     100*sqr(i17) - 98*i17 + 100*sqr(i16) - 98*i16 + 100*sqr(i15) - 98*i15 + 
     100*sqr(i14) - 98*i14 + 100*sqr(i13) - 98*i13 + 100*sqr(i12) - 98*i12 + 
     100*sqr(i11) - 98*i11 + 100*sqr(i10) - 98*i10 + 100*sqr(i9) - 98*i9 + 100*
     sqr(i8) - 98*i8 + 100*sqr(i7) - 98*i7 + 100*sqr(i6) - 98*i6 + 100*sqr(i5)
      - 98*i5 + 100*sqr(i4) - 98*i4 + 100*sqr(i3) - 98*i3 + 100*sqr(i2) - 98*i2
      + 100*sqr(i21) - 98*i21 - 2*i20*i19 - 2*i20*i19 - 2*i18*i17 - 2*i18*i17
      - 2*i16*i15 - 2*i16*i15 - 2*i14*i13 - 2*i14*i13 - 2*i12*i11 - 2*i12*i11
      - 2*i10*i9 - 2*i10*i9 - 2*i8*i7 - 2*i8*i7 - 2*i6*i5 - 2*i6*i5 - 2*i4*i3
      - 2*i4*i3 - 2*i2*i21 - 2*i2*i21 =L= -1;

* set non-default bounds
i2.lo = -100;
i3.lo = -100;
i4.lo = -100;
i5.lo = -100;
i6.lo = -100;
i7.lo = -100;
i8.lo = -100;
i9.lo = -100;
i10.lo = -100;
i11.lo = -100;
i12.lo = -100;
i13.lo = -100;
i14.lo = -100;
i15.lo = -100;
i16.lo = -100;
i17.lo = -100;
i18.lo = -100;
i19.lo = -100;
i20.lo = -100;
i21.lo = -100;

Model m / all /;

m.limrow=0; m.limcol=0;
m.tolproj=0.0;
$if gamsversion 242 option intvarup = 0;

$if NOT '%gams.u1%' == '' $include '%gams.u1%'

$if not set MINLP $set MINLP MINLP
Solve m using %MINLP% minimizing objvar;