NLP. KKT Practice and Second Order Conditions from Nash and Sofer. Unconstrained. First Order Necessary Condition Second Order Necessary Second Order Sufficient. Easiest Problem. Linear equality constraints. KKT Conditions. Note for equality – multipliers are unconstrained
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NLP
KKT Practice and Second Order Conditions from Nash and Sofer
Note for equality – multipliers are unconstrained
Complementarity not an issue
{x : x*+p pN(A)}
where N(A) is null space of A
See Section 3.2 of Nash and Sofer for example
You can convert any linear equality constrained optimization problem to an equivalent unconstrained problem
becomes
becomes
Gradient is not in Null(A), thus it must be in Range(A’)
then x* is a strict local minimizer
then x* is a strict local minimizer
SOSC
so SOSC satisfied, and x* is a strict local minimum
Objective is convex, so KKT conditions are sufficient.
Constraints form a polyhedron
x*
Equality FONC:
a2x = b
a2x = b
-a2
Polyhedron Ax>=b
a3x = b
a4x = b
-a1
a1x = b
contour set of function
unconstrained minimum
Which i are 0? What is the sign of I?
x*
Equality FONC:
a2x = b
a2x = b
-a2
Polyhedron Ax>=b
a3x = b
a4x = b
-a1
a1x = b
Which i are 0? What is the sign of I?
x*
Inequality FONC:
a2x = b
a2x = b
-a2
Polyhedron Ax>=b
a3x = b
a4x = b
-a1
a1x = b
Nonnegative Multipliers imply gradient points to the less than
Side of the constraint.
where Z+ is a basis matrix for Null(A +) and A + corresponds to nondegenerate active constraints)
i.e.
Lagrangian function
and Jacobian matrix
were each row is a gradient of a constraint
where Z+ is a basis matrix for Null(A +) and A + corresponds to Jacobian of nondegenerate active constraints)
i.e.
should have linearly independent rows.
is regular and find KKT point
KKT point might not exist.
X* is global min
Convex f
Convex constraints
X* is local min
SOSC
CQ
KKT Satisfied