Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
149 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Convergence rate analysis of a Dykstra-type projection algorithm (2301.03026v2)

Published 8 Jan 2023 in math.OC

Abstract: Given closed convex sets $C_i$, $i=1,\ldots,\ell$, and some nonzero linear maps $A_i$, $i = 1,\ldots,\ell$, of suitable dimensions, the multi-set split feasibility problem aims at finding a point in $\bigcap_{i=1}\ell A_i{-1}C_i$ based on computing projections onto $C_i$ and multiplications by $A_i$ and $A_iT$. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto $\bigcap_{i=1}\ell A_i{-1}C_i$; we refer to this problem as the best approximation problem in multi-set split feasibility settings (BA-MSF). We adapt the Dykstra's projection algorithm, which is classical for solving the BA-MSF in the special case when all $A_i = I$, to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto $C_i$ and multiplications by $A_i$ and $A_iT$ in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-Lojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each $C_i$ is $C{1,\alpha}$-cone reducible for some $\alpha\in (0,1]$: this class of sets covers the class of $C2$-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions.

Summary

We haven't generated a summary for this paper yet.