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A Conceptual Conjugate Epi-Projection Algorithm of Convex Optimization: Superlinear, Quadratic and Finite Convergence
This paper considers a conceptual version of a convex optimization algorithm
whic is based on replacing a convex optimization problem with the root-finding
problem for the approximate sub-differential mapping which is solved by
repeated projection onto the epigraph of conjugate function. Whilst the
projection problem is not exactly solvable in finite space-time it can be
approximately solved up to arbitrary precision by simple iterative methods,
which use linear support functions of the epigraph. It seems therefore useful
to study computational characteristics of the idealized version of this
algorithm when projection on the epigraph is computed precisely to estimate the
potential benefits for such development. The key results of this study are that
the conceptual algorithm attains super-linear rate of convergence in general
convex case, the rate of convergence becomes quadratic for objective functions
forming super-set of strongly convex functions, and convergence is finite when
objective function has sharp minimum. In all cases convergence is global and
does not require differentiability of the objective.
Keywords: convex optimization, conjugate function, approximate
sub-differential, super-linear convergence, quadratic convergence, finite
convergence, projection, epigrap