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Moreau envelope

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The Moreau envelope (or the Moreau-Yosida regularization) M f {\displaystyle M_{f}} of a proper lower semi-continuous convex function f {\displaystyle f} is a smoothed version of f {\displaystyle f} . It was proposed by Jean-Jacques Moreau in 1965.

The Moreau envelope has important applications in mathematical optimization: minimizing over M f {\displaystyle M_{f}} and minimizing over f {\displaystyle f} are equivalent problems in the sense that the sets of minimizers of f {\displaystyle f} and M f {\displaystyle M_{f}} are the same. However, first-order optimization algorithms can be directly applied to M f {\displaystyle M_{f}} , since f {\displaystyle f} may be non-differentiable while M f {\displaystyle M_{f}} is always continuously differentiable. Indeed, many proximal gradient methods can be interpreted as a gradient descent method over M f {\displaystyle M_{f}} .

Definition

The Moreau envelope of a proper lower semi-continuous convex function f {\displaystyle f} from a Hilbert space X {\displaystyle {\mathcal {X}}} to ( , + ] {\displaystyle (-\infty ,+\infty ]} is defined as

M λ f ( v ) = inf x X ( f ( x ) + 1 2 λ x v 2 2 ) . {\displaystyle M_{\lambda f}(v)=\inf _{x\in {\mathcal {X}}}\left(f(x)+{\frac {1}{2\lambda }}\|x-v\|_{2}^{2}\right).}

Given a parameter λ R {\displaystyle \lambda \in \mathbb {R} } , the Moreau envelope of λ f {\displaystyle \lambda f} is also called as the Moreau envelope of f {\displaystyle f} with parameter λ {\displaystyle \lambda } .

Properties

  • The Moreau envelope can also be seen as the infimal convolution between f {\displaystyle f} and ( 1 / 2 ) 2 2 {\displaystyle (1/2)\|\cdot \|_{2}^{2}} .
  • The proximal operator of a function is related to the gradient of the Moreau envelope by the following identity:

M λ f ( x ) = 1 λ ( x p r o x λ f ( x ) ) {\displaystyle \nabla M_{\lambda f}(x)={\frac {1}{\lambda }}(x-\mathrm {prox} _{\lambda f}(x))} . By defining the sequence x k + 1 = p r o x λ f ( x k ) {\displaystyle x_{k+1}=\mathrm {prox} _{\lambda f}(x_{k})} and using the above identity, we can interpret the proximal operator as a gradient descent algorithm over the Moreau envelope.

M λ f ( v ) = max p X ( p , v λ 2 p 2 f ( p ) ) , {\displaystyle M_{\lambda f}(v)=\max _{p\in {\mathcal {X}}}\left(\langle p,v\rangle -{\frac {\lambda }{2}}\|p\|^{2}-f^{*}(p)\right),} where f {\displaystyle f^{*}} denotes the convex conjugate of f {\displaystyle f} . Since the subdifferential of a proper, convex, lower semicontinuous function on a Hilbert space is inverse to the subdifferential of its convex conjugate, we can conclude that if p 0 X {\displaystyle p_{0}\in {\mathcal {X}}} is the maximizer of the above expression, then x 0 := v λ p 0 {\displaystyle x_{0}:=v-\lambda p_{0}} is the minimizer in the primal formulation and vice versa.

See also

References

  1. Moreau, J. J. (1965). "Proximité et dualité dans un espace hilbertien". Bulletin de la Société Mathématique de France. 93: 273–299. doi:10.24033/bsmf.1625. ISSN 0037-9484.
  2. ^ Neal Parikh and Stephen Boyd (2013). "Proximal Algorithms" (PDF). Foundations and Trends in Optimization. 1 (3): 123–231. Retrieved 2019-01-29.
  3. Heaton, Howard; Fung, Samy Wu; Osher, Stanley (2022-10-09). "Global Solutions to Nonconvex Problems by Evolution of Hamilton–Jacobi PDEs". arXiv:2202.11014 .
  4. Osher, Stanley; Heaton, Howard; Fung, Samy Wu (2022-11-23). "A Hamilton–Jacobi-based Proximal Operator". arXiv:2211.12997 .

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