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Constrained Global Optimization

[Graphics:../Images/index_gr_124.gif] [Graphics:../Images/index_gr_125.gif]
[Graphics:../Images/index_gr_126.gif] [Graphics:../Images/index_gr_127.gif]

Figure 4. Functions for constrained global optimization.

This gives the square of the minimal distance of a solution of [Graphics:../Images/index_gr_128.gif] from the origin.

[Graphics:../Images/index_gr_129.gif]
[Graphics:../Images/index_gr_130.gif]

Here is a graphical representation of the problem. The set of solutions is colored blue, and the red dot represents the point where the minimum is attained.

[Graphics:../Images/index_gr_131.gif]

[Graphics:../Images/index_gr_132.gif]

We can improve the computation time by noticing that the minimum has to be attained on the boundary of the solution set. The cylindrical algebraic decomposition algorithm can then use a simpler projection operator making use of the equation constraint.

[Graphics:../Images/index_gr_133.gif]
[Graphics:../Images/index_gr_134.gif]

If all we want is the minimal value, and if we have a reason to believe that [Graphics:../Images/index_gr_135.gif] is the closure of [Graphics:../Images/index_gr_136.gif], we can compute the answer even faster, using an even simpler version of the algorithm. This version is used if the constraints are all strong inequalities and if the input contains only rational functions with rational number coefficients.

[Graphics:../Images/index_gr_137.gif]
[Graphics:../Images/index_gr_138.gif]

Converted by Mathematica      April 24, 2000

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