# An efficient sieving based secant method for sparse optimization problems with least-squares constraints

arXiv (Cornell University)（2023）

摘要

In this paper, we propose an efficient sieving based secant method to address
the computational challenges of solving sparse optimization problems with
least-squares constraints. A level-set method has been introduced in [X. Li,
D.F. Sun, and K.-C. Toh, SIAM J. Optim., 28 (2018), pp. 1842–1866] that solves
these problems by using the bisection method to find a root of a univariate
nonsmooth equation φ(λ) = ϱ for some ϱ > 0, where
φ(·) is the value function computed by a solution of the
corresponding regularized least-squares optimization problem. When the
objective function in the constrained problem is a polyhedral gauge function,
we prove that (i) for any positive integer k, φ(·) is piecewise
C^k in an open interval containing the solution λ^* to the equation
φ(λ) = ϱ; (ii) the Clarke Jacobian of φ(·) is
always positive. These results allow us to establish the essential ingredients
of the fast convergence rates of the secant method. Moreover, an adaptive
sieving technique is incorporated into the secant method to effectively reduce
the dimension of the level-set subproblems for computing the value of
φ(·). The high efficiency of the proposed algorithm is demonstrated
by extensive numerical results.

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关键词

sparse optimization,secant method,efficient sieving,constraints,least-squares

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