152 research outputs found
Structured Sparsity Promoting Functions: Theory and Applications
Motivated by the minimax concave penalty based variable selection in high-dimensional linear regression, we introduce a simple scheme to construct structured semiconvex sparsity promoting functions from convex sparsity promoting functions and their Moreau envelopes. Properties of these functions are developed by leveraging their structure. In particular, we show that the behavior of the constructed function can be easily controlled by assumptions on the original convex function. We provide sparsity guarantees for the general family of functions via the proximity operator. Results related to the Fenchel Conjugate and Łojasiewicz exponent of these functions are also provided. We further study the behavior of the proximity operators of several special functions including indicator functions of closed convex sets, piecewise quadratic functions, and linear combinations of the two. To demonstrate these properties, several concrete examples are presented and existing instances are featured as special cases. We explore the effect of these functions on the penalized least squares problem and discuss several algorithms for solving this problem which rely on the particular structure of our functions. We then apply these methods to the total variation denoising problem from signal processing
Local convergence of a sequential quadratic programming method for a class of nonsmooth nonconvex objectives
A sequential quadratic programming (SQP) algorithm is designed for nonsmooth
optimization problems with upper-C^2 objective functions. Upper-C^2 functions
are locally equivalent to difference-of-convex (DC) functions with smooth
convex parts. They arise naturally in many applications such as certain classes
of solutions to parametric optimization problems, e.g., recourse of stochastic
programming, and projection onto closed sets. The proposed algorithm conducts
line search and adopts an exact penalty merit function. The potential
inconsistency due to the linearization of constraints are addressed through
relaxation, similar to that of Sl_1QP. We show that the algorithm is globally
convergent under reasonable assumptions. Moreover, we study the local
convergence behavior of the algorithm under additional assumptions of
Kurdyka-{\L}ojasiewicz (KL) properties, which have been applied to many
nonsmooth optimization problems. Due to the nonconvex nature of the problems, a
special potential function is used to analyze local convergence. We show that
under acceptable assumptions, upper bounds on local convergence can be proven.
Additionally, we show that for a large number of optimization problems with
upper-C^2 objectives, their corresponding potential functions are indeed KL
functions. Numerical experiment is performed with a power grid optimization
problem that is consistent with the assumptions and analysis in this paper
Inexact proximal methods for weakly convex functions
This paper proposes and develops inexact proximal methods for finding
stationary points of the sum of a smooth function and a nonsmooth weakly convex
one, where an error is present in the calculation of the proximal mapping of
the nonsmooth term. A general framework for finding zeros of a continuous
mapping is derived from our previous paper on this subject to establish
convergence properties of the inexact proximal point method when the smooth
term is vanished and of the inexact proximal gradient method when the smooth
term satisfies a descent condition. The inexact proximal point method achieves
global convergence with constructive convergence rates when the Moreau envelope
of the objective function satisfies the Kurdyka-Lojasiewicz (KL) property.
Meanwhile, when the smooth term is twice continuously differentiable with a
Lipschitz continuous gradient and a differentiable approximation of the
objective function satisfies the KL property, the inexact proximal gradient
method achieves the global convergence of iterates with constructive
convergence rates.Comment: 26 pages, 3 table
Splitting Method for Support Vector Machine in Reproducing Kernel Banach Space with Lower Semi-continuous Loss Function
In this paper, we use the splitting method to solve support vector machine in
reproducing kernel Banach space with lower semi-continuous loss function. We
equivalently transfer support vector machines in reproducing kernel Banach
space with lower semi-continuous loss function to a finite-dimensional tensor
Optimization and propose the splitting method based on alternating direction
method of multipliers. By Kurdyka-Lojasiewicz inequality, the iterative
sequence obtained by this splitting method is globally convergent to a
stationary point if the loss function is lower semi-continuous and subanalytic.
Finally, several numerical performances demonstrate the effectiveness.Comment: arXiv admin note: text overlap with arXiv:2208.1252
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