Global convergence of splitting methods for nonconvex composite optimization

We consider the problem of minimizing the sum of a smooth function $h$ with a bounded Hessian, and a nonsmooth function. We assume that the latter function is a composition of a proper closed function $P$ and a surjective linear map $\cal M$, with the proximal mappings of $\tau P$, $\tau > 0$, 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 $h$ and $P$ 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 $\ell_{1/2}$ regularization. Finally, when $\cal M$ 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 $h$.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 $\frac12$. 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 $\frac12$. 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 $\ell_1$ 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 $hZZ$ and $hWW$ couplings in models with next--to--simplest Higgs sectors satisfying the electroweak $\rho$ 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 $hZZ$ and $hWW$ 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 students’ 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 Norton’s (1997, 2000) work on identity and English learning, this study pays particular attention to how the interactions within the participants’ 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 students’ 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 student’s 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 students’ 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 students’ lived English learning experiences, this study helps raise post-secondary English language educators’ awareness of students’ 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