Answers to: Convert NLP constraint into a linear onehttp://www.or-exchange.com/questions/10550/convert-nlp-constraint-into-a-linear-one<p>Hy,
I have a problem with a non linear constraint. In an Optimization problem i have this constraint</p>
<pre><code>X*Y >= K
with X,Y continuous variables and bounded (X€[Xm,XM] Y€[Ym,YM]) and K constant.
</code></pre>
<p>There is some way to convert this non linear constraint into a linear one (i mean a set of linear).</p>
<p>Thanks </p>enMon, 27 Oct 2014 14:33:33 -0400Answer by fbahrhttp://www.or-exchange.com/questions/10550/convert-nlp-constraint-into-a-linear-one/10552<p>With \(Z_1 = \frac{1}{2}(X + Y)\) and \(Z_2 = \frac{1}{2}(X - Y)\)<br>
\(\Rightarrow\) \(X * Y = Z_1^2 - Z_2^2\), and – hence –<br>
\(X * Y \ge K \Leftrightarrow Z_1^2 - Z_2^2 \ge K\)<br>
...which allows for <a href="http://www.aimms.com/aimms/download/manuals/aimms3om_integerprogrammingtricks.pdf#page=10">piecewise linear <i>approximation</i></a><br>
[using λ-formulation, SOS2 variables, and/or other fancy stuff your modeling language + solver support].</p>fbahrMon, 27 Oct 2014 14:33:33 -0400http://www.or-exchange.com/questions/10550/convert-nlp-constraint-into-a-linear-one/10552