1,356 research outputs found
Global convergence of splitting methods for nonconvex composite optimization
We consider the problem of minimizing the sum of a smooth function with a
bounded Hessian, and a nonsmooth function. We assume that the latter function
is a composition of a proper closed function and a surjective linear map
, with the proximal mappings of , , simple to
compute. This problem is nonconvex in general and encompasses many important
applications in engineering and machine learning. In this paper, we examined
two types of splitting methods for solving this nonconvex optimization problem:
alternating direction method of multipliers and proximal gradient algorithm.
For the direct adaptation of the alternating direction method of multipliers,
we show that, if the penalty parameter is chosen sufficiently large and the
sequence generated has a cluster point, then it gives a stationary point of the
nonconvex problem. We also establish convergence of the whole sequence under an
additional assumption that the functions and are semi-algebraic.
Furthermore, we give simple sufficient conditions to guarantee boundedness of
the sequence generated. These conditions can be satisfied for a wide range of
applications including the least squares problem with the
regularization. Finally, when is the identity so that the proximal
gradient algorithm can be efficiently applied, we show that any cluster point
is stationary under a slightly more flexible constant step-size rule than what
is known in the literature for a nonconvex .Comment: To appear in SIOP
Calculus of the exponent of Kurdyka-{\L}ojasiewicz inequality and its applications to linear convergence of first-order methods
In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an
important quantity for analyzing the convergence rate of first-order methods.
Specifically, we develop various calculus rules to deduce the KL exponent of
new (possibly nonconvex and nonsmooth) functions formed from functions with
known KL exponents. In addition, we show that the well-studied Luo-Tseng error
bound together with a mild assumption on the separation of stationary values
implies that the KL exponent is . The Luo-Tseng error bound is known
to hold for a large class of concrete structured optimization problems, and
thus we deduce the KL exponent of a large class of functions whose exponents
were previously unknown. Building upon this and the calculus rules, we are then
able to show that for many convex or nonconvex optimization models for
applications such as sparse recovery, their objective function's KL exponent is
. This includes the least squares problem with smoothly clipped
absolute deviation (SCAD) regularization or minimax concave penalty (MCP)
regularization and the logistic regression problem with
regularization. Since many existing local convergence rate analysis for
first-order methods in the nonconvex scenario relies on the KL exponent, our
results enable us to obtain explicit convergence rate for various first-order
methods when they are applied to a large variety of practical optimization
models. Finally, we further illustrate how our results can be applied to
establishing local linear convergence of the proximal gradient algorithm and
the inertial proximal algorithm with constant step-sizes for some specific
models that arise in sparse recovery.Comment: The paper is accepted for publication in Foundations of Computational
Mathematics: https://link.springer.com/article/10.1007/s10208-017-9366-8. In
this update, we fill in more details to the proof of Theorem 4.1 concerning
the nonemptiness of the projection onto the set of stationary point
Peaceman-Rachford splitting for a class of nonconvex optimization problems
We study the applicability of the Peaceman-Rachford (PR) splitting method for
solving nonconvex optimization problems. When applied to minimizing the sum of
a strongly convex Lipschitz differentiable function and a proper closed
function, we show that if the strongly convex function has a large enough
strong convexity modulus and the step-size parameter is chosen below a
threshold that is computable, then any cluster point of the sequence generated,
if exists, will give a stationary point of the optimization problem. We also
give sufficient conditions guaranteeing boundedness of the sequence generated.
We then discuss one way to split the objective so that the proposed method can
be suitably applied to solving optimization problems with a coercive objective
that is the sum of a (not necessarily strongly) convex Lipschitz differentiable
function and a proper closed function; this setting covers a large class of
nonconvex feasibility problems and constrained least squares problems. Finally,
we illustrate the proposed algorithm numerically
Radiative corrections to Higgs couplings with weak gauge bosons in custodial multi-Higgs models
We calculate 1-loop radiative corrections to the and couplings in
models with next--to--simplest Higgs sectors satisfying the electroweak
parameter equal to 1 at tree level: the Higgs singlet model, the two-Higgs
doublet models, and the Georgi-Machacek model. Under theoretical and current
experimental constraints, the three models have different correlations between
the deviations in the and couplings from the standard model
predictions. In particular, we find for each model predictions with no overlap
with the other two models.Comment: 5 pages, 1 figur
Learners\u27 Stories: A Study of Hong Kong Post-Secondary Students\u27 English Learning Experiences and Identity Construction
This is a narrative study of Hong Kong post-secondary studentsā English learning experiences, focusing on: i) the meanings that the student participants attached to their English learning; and ii) their identity (re-)construction during the course of English learning. Theoretically informed by Nortonās (1997, 2000) work on identity and English learning, this study pays particular attention to how the interactions within the participantsā English classroom have shaped and informed their English learning experiences and their English learner identities. A multi-method approach was adopted in order to capture a more complete picture of post-secondary studentsā English learning. Data collection techniques included pre-interview questionnaires, interviews with student participants and post-interview classroom observations. The collected data were used to develop a narrative of each studentās English learning. The narratives were used to demonstrate the connections between the various stories told by each participant and how they made meaning of their English learning. A thematic analysis of the data was also conducted to highlight both common and idiosyncratic aspects of Hong Kong post-secondary studentsā English learning. Findings demonstrate that the participants constantly (re-)constructed their identities as situated and multiple in accordance with their immediate and imagined learning communities. Their investment in English learning was inextricably tied to their prior learning experiences, multiple identities, and hopes and desires for the future. Through documenting studentsā lived English learning experiences, this study helps raise post-secondary English language educatorsā awareness of studentsā prior English learning and multiple identities so as to provide students with the support and help they need
Epitaxial growth of highly mismatched III-V materials on (001) silicon for electronics and optoelectronics
Monolithic integration of III-V on silicon has been a scientifically appealing concept for decades. Notable progress has recently been made in this research area, fueled by significant interests of the electronics industry in high-mobility channel transistors and the booming development of silicon photonics technology. In this review article, we outline the fundamental roadblocks for the epitaxial growth of highly mismatched III-V materials, including arsenides, phosphides, and antimonides, on (001) oriented silicon substrates. Advances in hetero-epitaxy and selective-area hetero-epitaxy from micro to nano length scales are discussed. Opportunities in emerging electronics and integrated photonics are also presented
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