resolvent and cayley operator - github pages€¦ · as a mnemonic (unverified), in a composite...
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def: monotone operators definitions and related
Lipschitz mapping, (e.g., nonexpansive and contraction mapping
proof: (\lambda>=0, F monotone) =
thm: for positive coefficient and monotone operator resolvent is a nonexpansive function
Resolvent_and_Cayley...
Resolvent and Cayley Operator9:37 PM
EE 364b Convex Optimization II Page 1
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statement: overloaded sum operator for relations has additivity
As a mnemonic (unverified), in a composite relation, the constituent relation gulor majhe jara function, tader jonno ukto equ ation e functional operation chalano jabe
Old proof , I suggest skip it.
EE 364b Convex Optimization II Page 2
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(\lambda>=0, F monotone) => C [*nonexpansive function*]
thm: proximal operator is the resolvent of subdifferential operator
normal cone operator
thm: proximal operator is the resolvent of subdifferential operator
def: multiplier to residual mapping
def: multiplier to residual mapping
thm: for positive coefficient and monotone operator resolvent is anonexpansive function
[# Note, we want to find out z=R(y), but to find out z here we also need to know x*, so this is what weare going to do, solve for x* first, and then set the value in z=y-\lambda (b- Ax*) #]
# resolvent for multiplier to residual mapping
# Resolvent of normal cone operator
EE 364b Convex Optimization II Page 3
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[# Note, we want to find out z=R(y), but to find out z here we also need to know x*, so this is what weare going to do, solve for x* first, and then set the value in z=y-\lambda (b- Ax*) #]
def: maximal monotone
As a mnemonic (unverified), in a composite relation, the constituent relation gulor majhe jara function, tader jonno ukto equation e functional operation chalano jabe
thm: for positive coefficient and monotone operator resolvent is anonexpansive function
Why find fixed point of Cayley and resolvent of some operator?•
why maximality is needed?
Compact form: Resolvent of the multiplier to residual mapping
EE 364b Convex Optimization II Page 4
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