Convex optimization monotone inclusion

  • What is a monotone inclusion problem?

    .

    1. In general, a monotone inclusion problem refers to finding a zero of a set-valued maximally monotone operator.
      2. When there is no confusion, we use monotone inclusion problems to refer to the constrained single-valued monotone inclusion problems.Aug 10, 2022

  • Monotone operator theory is a fertile area of nonlinear analysis which emerged in 1960 in in- dependent papers by Kacurovski˘ı, Minty, and Zarantonello.
    Let D be a nonempty subset of H, let.
    A: D → H, and let B: D → H.
    Extending the ordering of functions on the real line which results.
This formulation yields an equivalent inclusion problem for an operator which can be decomposed as the sum of two maximally monotone operators amenable to 

Can a nicely behaved fixed-point equation solve monotone inclusions with three operators?

The hard part of this conversion, then, is to design nicely behaved fixed-point equations

In this paper, we design a new, and thus far, the only nicely behaved fixed-point equation for solving monotone inclusions with three operators; the equation employs resolvent and forward operators, one at a time, in succession

Can monotone inclusions be derived from optimization concepts?

In principle, however, even though monotone inclusions provide a more synthetic and natural framework, it is possible (at least a posteriori) to derive their convergence in the scenario of Problem 4

2 from optimization concepts only, without invoking monotone operator theory

What is the interplay between monotone operator theory and convex optimization?

Several aspects of the interplay between monotone operator theory and convex optimization are presented

The crucial role played by monotone operators in the analysis and the numerical solution of convex minimization problems is emphasized


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