maxmin.gms : Max Min Location of Points in Unit Square

**Description**

This test problem locates points in the unit square such that the distance between any two points is maximized. For certain number of points we know optimal arrangements. This knowledge is also used to find a lower bound on the objective by looking for perfect square arrangements (suggested by S Dirkse). Several formulations are presented which serve as good examples to investigate the performance of different solution approaches. The problem was originally proposed by Dick van Hertog and has been implemented by Janos Pinter and used extensively by LGO with 13 and 20 points.

**References**

- Pinter, J D, LGO - A Model Development System for Continuous Global Optimization, User's Guide, Revised edition, Pinter Consulting Services, Halifax, NS, Canada, 2003.
- Stinstra, E, den Hertog, D, Stehouwer, P, and Vestjens, A, Constrained Maximin Designs for Computer Experiments. Technometrics 45, 4 (2003), 340-346.

**Small Model of Type :** DNLP

**Category :** GAMS Model library

**Main file :** maxmin.gms

```
$title Max Min Location of Points in Unit Square (MAXMIN,seq=263)
* This test problem locates points in the unit square such that the
* distance between any two points is maximized. For certain number
* of points we know optimal arrangements. This knowledge is also
* used to find a lower bound on the objective by looking for perfect
* square arrangements (suggested by S Dirkse).
*
* Several formulations are presented which serve as good examples to
* investigate the performance of different solution approaches. The
* problem was originally proposed by Dick van Hertog and has been implemented
* by Janos Pinter and used extensively by LGO with 13 and 20 points.
*
*
* E. Stinstra, D. den Hertog, H.P. Stehouwer, A. Vestjens,
* Constrained Maximin Designs for Computer Experiments,
* Technometrics, 2002. (under revision)
*
* Janos Pinter, LGO - Users Guide, Pinter Consulting Services, Halifax,
* Canada, 2003.
*
$eolcom //
$if NOT set points $set points 13
sets d dimension of space / x, y /
n number of points / p1 * p%points% /
low(n,n) lower triangle
alias (n,nn);
low(n,nn) = ord(n) > ord(nn);
variable point(n,d) coordinates of points
dist(n,n) distance between all points
mindist
equations defdist(n,n) distance definitions
mindist1(n,n) minimum distance formulation 1
mindist1a(n,n) minimum distance formulation 1 without dist
mindist2 minimum distance formulation 2
mindist2a minimum distance formulation 2 without dist ;
defdist(low(n,nn)).. dist(low) =e= sqrt(sum(d, sqr(point(n,d)-point(nn,d))));
mindist1 (low) .. mindist =l= dist(low);
mindist1a(low(n,nn)).. mindist =l= sqrt(sum(d, sqr(point(n,d)-point(nn,d))));
mindist2 .. mindist =e= smin(low, dist(low));
mindist2a.. mindist =e= smin(low(n,nn), sqrt(sum(d, sqr(point(n,d)-point(nn,d)))));
model maxmin1 / defdist, mindist1 /
maxmin2 / defdist, mindist2 /
maxmin1a / mindist1a /
maxmin2a / mindist2a /;
scalar p; p = 0; // Pinter's
loop((n,d), // original
p = round(mod(p,10)) + 1; // nominal
point.l(n,d) = p/10 ); // point 0.1,.2, ... 1.0, 0.1, ...
point.lo(n,d) = 0;
point.up(n,d) = 1;
point.l (n,d) = uniform(0,1);
dist.l(low(n,nn)) = sqrt(sqr(point.l(n,'x')-point.l(nn,'x')) + sqr(point.l(n,'y')-point.l(nn,'y')));
point.fx('p1',d) = 0; // fix one point
parameter bnd lower bound on objective;
bnd = 1/max(ceil(sqrt(card(n)))-1,1); display bnd;
option limcol=0,limrow=0; if(card(n) > 9, option solprint=off;);
* for experimentation we will combine different model version
* with different bounds and starting points
*
* dist.lo(low) = -inf;
* dist.lo(low) = 0;
* dist.lo(low) = 0.01;
* dist.lo(low) = bnd/2;
* dist.lo(low) = bnd;
*
* solve maxmin1 max mindist us nlp;
* solve maxmin1a max mindist us nlp;
* solve maxmin2 max mindist us dnlp;
* solve maxmin2a max mindist us dnlp;
* maxmin2 and maxmin2a without bounds are well suited for LGO
* maxmin1a with bounds is well suited for conopt3 (bounds 200 point is ok)
solve maxmin1a max mindist us dnlp;
display bnd,mindist.l, point.l;
```