<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"><head><meta http-equiv=Content-Type content="text/html; charset=us-ascii"><meta name=Generator content="Microsoft Word 12 (filtered medium)"><style><!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
@font-face
{font-family:Tahoma;
panose-1:2 11 6 4 3 5 4 4 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
margin-bottom:.0001pt;
font-size:12.0pt;
font-family:"Times New Roman","serif";}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
span.EstiloCorreo17
{mso-style-type:personal-reply;
font-family:"Calibri","sans-serif";
color:#1F497D;}
.MsoChpDefault
{mso-style-type:export-only;}
@page WordSection1
{size:612.0pt 792.0pt;
margin:70.85pt 3.0cm 70.85pt 3.0cm;}
div.WordSection1
{page:WordSection1;}
--></style><!--[if gte mso 9]><xml>
<o:shapedefaults v:ext="edit" spidmax="1026" />
</xml><![endif]--><!--[if gte mso 9]><xml>
<o:shapelayout v:ext="edit">
<o:idmap v:ext="edit" data="1" />
</o:shapelayout></xml><![endif]--></head><body lang=EN-US link=blue vlink=purple><div class=WordSection1><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>If the model is nonconvex, the only way to assure that is trying to solve the optimization problem with different initial points (better if these are logical with the constraints). On the other hand, and more if you have a nonconvex model, you are lucky if you can find a local minimum/maximum. <o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm'><p class=MsoNormal><b><span lang=ES style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>De:</span></b><span lang=ES style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> ipopt-bounces@list.coin-or.org [mailto:ipopt-bounces@list.coin-or.org] <b>En nombre de </b>Hossein Haghighat<br><b>Enviado el:</b> martes, 18 de octubre de 2011 07:19 a.m.<br><b>Para:</b> ipopt@list.coin-or.org; ipopt-request@list.coin-or.org<br><b>Asunto:</b> [Ipopt] optimality of solution<o:p></o:p></span></p></div><p class=MsoNormal><o:p> </o:p></p><p class=MsoNormal>Hello,<o:p></o:p></p><div><p class=MsoNormal>I have solved a nonconvex EPEC (equilibrium model with equilibrium constraints) model with Ipopt, using two modeling approaches involving:<o:p></o:p></p></div><div><p class=MsoNormal>1- an NLP approach, where the problem is reformulated as a set of nonlinear equations, <o:p></o:p></p></div><div><p class=MsoNormal>2- a diagonalization approach where the problem is reformulated as a set of MPECs (mathematical problem with equilibrium constraints) <o:p></o:p></p></div><div><p class=MsoNormal><o:p> </o:p></p></div><div><p class=MsoNormal>I get the same solution from these different approaches which supports the optimality of the solution. <o:p></o:p></p></div><div><p class=MsoNormal>Is there any other way to ensure that the solution is indeed optimal not a saddle point or a local maximum? (e.g by using IPOPT options, or initial point manipulations)<o:p></o:p></p></div><div><p class=MsoNormal><o:p> </o:p></p></div><div><p class=MsoNormal>thank you in advance,<o:p></o:p></p></div></div></body></html>