this theorem relates to the existence of minimization solution for an optimization problem.

Definition of min solution

Another expression

infimum n. 下确界

Example of nonexistence of a minimizing solution

Proof the existence of minimum solution with nonempty, compact set and continuous map

definition of the cone of tangents

examples of cone of tangents

Step size scheme for self-regulated MSA

scheme and guarantee the convergence because this scheme satisfies the conditions.

MSA in a whole solution procedure

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MSA is a step size adjust strategy that serves the whole decent direct based solution algorithm
It is also known that for more details, refer to Sheffi (1985)

MSA is a step-size adjust scheme in convex combination method.
So, the SUE solution problems consist of two steps: direction and step size. for the stepsize, the simplest way is MSA. for the direction, like dial's method the solution follow the logit model.
Critique of Logit-Based Network Loading, IID property
Dial's method still reflect logit model.

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it can see, that algorithm is, of course, correct theoretically.
Next step is to debug in programs.
Anthony Chen 2009 self-adaptive method, said stop criteria is UE, so, it is inferred that it also exists SUE stop criteria. But, I am not sure.