FunctionalQuadraticPerturb.convex_conj¶

property FunctionalQuadraticPerturb.convex_conj¶

Convex conjugate functional of the functional.

Notes

Given a functional $f$ , the convex conjugate of a linearly perturbed version $f(x) + <y, x>$ is given by a translation of the convex conjugate of $f$ :

$(f + \langle y, \cdot \rangle)^* (x^*) = f^*(x^* - y).$

For reference on the identity used, see [KP2015]. Moreover, the convex conjugate of $f + c$ is by definition

$(f + c)^* (x^*) = f^*(x^*) - c.$

References

[KP2015] Komodakis, N, and Pesquet, J-C. Playing with Duality: An overview of recent primal-dual approaches for solving large-scale optimization problems. IEEE Signal Processing Magazine, 32.6 (2015), pp 31--54.