What if LICQ does not hold?

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Suppose given a nonlinear optimization programming:



$min_x,y f(x,y)$ st



$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$



and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.










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  • There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
    – Eff
    Sep 7 at 10:48















up vote
0
down vote

favorite












Suppose given a nonlinear optimization programming:



$min_x,y f(x,y)$ st



$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$



and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.










share|cite|improve this question





















  • There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
    – Eff
    Sep 7 at 10:48













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Suppose given a nonlinear optimization programming:



$min_x,y f(x,y)$ st



$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$



and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.










share|cite|improve this question













Suppose given a nonlinear optimization programming:



$min_x,y f(x,y)$ st



$g_1(x,y)ge 0$ , $g_2(x,y)ge 0$ , $g_3(x,y)ge 0$



and suppose that at the solution $(x*, y*)$ the three constraints are active. This means that the constraints gradients will not be linearly dependent and the LICQ fails to hold. How could we handle this case since the LICQ is a necessary condition for optimality.







linear-algebra nonlinear-optimization






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asked Sep 7 at 9:52









yas are

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  • There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
    – Eff
    Sep 7 at 10:48

















  • There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
    – Eff
    Sep 7 at 10:48
















There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
– Eff
Sep 7 at 10:48





There are also other regularity conditions that can be used (where the KKT conditions are the same), LICQ is just among the most common. It simplifies things a great deal. See the wikipedia article.
– Eff
Sep 7 at 10:48











1 Answer
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oldest

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1
down vote













LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.



You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).



LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.






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  • Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
    – yas are
    Sep 7 at 10:20










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1 Answer
1






active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.



You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).



LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.






share|cite|improve this answer




















  • Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
    – yas are
    Sep 7 at 10:20














up vote
1
down vote













LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.



You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).



LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.






share|cite|improve this answer




















  • Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
    – yas are
    Sep 7 at 10:20












up vote
1
down vote










up vote
1
down vote









LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.



You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).



LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.






share|cite|improve this answer












LICQ is not a necessary condition for optimality. It is a prerequisite that the KKT conditions are necessary for optimality.



You might check other constraint qualifications (MFCQ, linearity, convexity + Slater point, etc).



LICQ fails trivially if $g_1=g_2=g_3$. Nevertheless, Lagrange multipliers might exist as other constraint qualifications might hold.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Sep 7 at 10:10









daw

22.2k1542




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  • Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
    – yas are
    Sep 7 at 10:20
















  • Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
    – yas are
    Sep 7 at 10:20















Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
– yas are
Sep 7 at 10:20




Thank you for you answer, I am referring to the book: "Numerical optimization, Nocedal" where the necessary conditions for optimality are stated. The theorem supposes that LICQ holds. Now what are the necessary conditions if the LICQ does not hold?
– yas are
Sep 7 at 10:20

















 

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