Area of Hexagon Test Problem

TomSym implementation of GAMS Example (HIMMEL16,SEQ=36)

The physical problem is to maximize the area of a hexagon in which the diameter must be less than or equal to one. The formulation in Himmelblau is different from the one given here, because certain fixed variables have been eliminated. However, the formulation given here is more natural and easier to understand. It makes use of the fact that one vertex is fixed at the origin and uses the vector scalar product to calculate the areas of all triangles originating from the origin. All terms in x(1) and y(1) thus vanish when the algebraic expression is simplified.

The problem appears many other places, e.g. as example 108 in W. Hock and K. Schittkowski: Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems, 187, Springer Verlag, 1981, and as the test example in P. E. Gill, W. Murray, M. A. Saunders, and M. Wright: User's Guide for SOL/NPSOL: A FORTRAN Package for Nonlinear Programming, Tech. Rep. 83-12, Dept. of Operation Research, Stanford University.

Himmelblau, D M, Problem Number 16. In Applied Nonlinear Programming. Mc Graw Hill, New York, 1972.

i: indices for the 6 points (1-6);

% toms 6x1 x y area. This would confuse Matlab, because "area" is the name
% of a function.

% x-coordinates of the points
x    = tom('x',6,1);

% y-coordinates of the points
y    = tom('y',6,1);

% area of the i'th triangle (circular)
area = tom('area',6,1);

% Maximal distance between i and j
eq1 = {};
for i=1:6
    for j=1:6
        if i<j
            eq1 = {eq1; (x(i)-x(j))^2+(y(i)-y(j))^2 <= 1};
        end
    end
end

% Area definition for triangle i
eq2 = {area == 0.5*(x.*y([2:end,1],1)-y.*x([2:end,1],1))};

% Total area
totarea = sum(area);

% Initial conditions
cbnd = {x(1) == 0; y(1) == 0; y(2) == 0; area >= 0};

% Starting point
x0 = {x == [0;0.5;0.5;0.5;0;0]
    y == [0;0;0.4;0.8;0.8;0.4]
    area == 0.5*(x.*y([2:end,1],1)-y(i)*x([2:end,1],1))};

solution = ezsolve(-totarea,{cbnd, eq1, eq2},x0);

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Problem 1                      f_k      -0.674981446903530680
                                       sum(|constr|)      0.000000052086637559
                              f(x_k) + sum(|constr|)     -0.674981394816893120
                                              f(x_0)     -0.300000000000000040

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   31 ConJacEv   31 Iter   26 MinorIter   41
CPU time: 0.062500 sec. Elapsed time: 0.062000 sec.