<font class="Apple-style-span" face="'courier new', monospace">Hello all,</font><div><font class="Apple-style-span" face="'courier new', monospace">Many thanks for reply.</font></div><div><font class="Apple-style-span" face="'courier new', monospace">it is a problem of minimum operating cost, in the context of electricity market. </font></div>
<div><font class="Apple-style-span" face="'courier new', monospace">- F(x,y) is the cost of power, and "mu" is the nodal price in the market (Lagrange multiplier of the equality constraints ), and y is the power produced in the market.</font></div>
<div><font class="Apple-style-span" face="'courier new', monospace">- "mu * y" represent the cost of power. </font></div><div><font class="Apple-style-span" face="'courier new', monospace"><br></font></div>
<div><font class="Apple-style-span" face="'courier new', monospace">the point is that "mu" is only available post solution, but here it appears in the objective function. </font></div><div><font class="Apple-style-span" face="'courier new', monospace">this problem seems to be non-convex because it entails the product of variables, and I am not sure if it can be converted into a convex form (if any).</font></div>
<div><font class="Apple-style-span" face="'courier new', monospace"><br></font></div><div><font class="Apple-style-span" face="'courier new', monospace">Writing the optimality conditions is a good way, but we should note that they are stationary points (saddle-points + local optima) </font></div>
<div><font class="Apple-style-span" face="'courier new', monospace"><br></font></div><div><font class="Apple-style-span" face="'courier new', monospace">Your comments are highly appreciated.</font></div><div>
<br></div><div><br></div><br><div class="gmail_quote">On Tue, Jul 26, 2011 at 9:36 PM, rony goldenthal <span dir="ltr"><<a href="mailto:ronygold@gmail.com">ronygold@gmail.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
Hello Hossein,<br>
<br>
The main challenge in implementing the function in IPOPT, as you<br>
describe it, is that the Lagrange multipliers are not easily available<br>
when evaluating the function to minimize.<br>
Maybe you can try something like adding another set of variables, 'z',<br>
and solve the following system instead:<br>
<br>
Min L (x,y,z)= F(x)+ z * y (a1)<br>
s.t.<br>
g(x,y) = 0 (a2)<br>
h(x,y)<= 0 (a3)<br>
z - mu = 0 (a4)<br>
Where mu are the Lagrange multipliers that correspond to g(x,y).<br>
<br>
I cannot say much about the convexity of either of the approaches.<br>
<br>
Good luck,<br>
Rony<br>
<div><div></div><div class="h5"><br>
On Mon, Jul 25, 2011 at 9:00 PM, Hossein Haghighat<br>
<<a href="mailto:hosein.haghighat@gmail.com">hosein.haghighat@gmail.com</a>> wrote:<br>
> Hello,<br>
> I was wondering if I can solve a problem in the following form by IPOPT:<br>
><br>
> Min L (x,y,mu)= F(x)+ mu * y (a1)<br>
><br>
> s.t.<br>
><br>
><br>
><br>
> g(x,y) = 0 : mu (a2)<br>
><br>
> h(x,y)<= 0 : lambda (a3)<br>
><br>
> where "mu" is the Lagrange multiplier of equality constraint (a2) appearing<br>
> in the objective function.<br>
> this problem is non-convex. is it true? and can it be solved by IPOPT?<br>
> --<br>
> regards,<br>
> Hossein.<br>
><br>
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</blockquote></div><br><br clear="all"><br>-- <br>regards,<br>Hossein.<br>