Optimal proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems

The augmented Lagrangian method (ALM) is fundamental in solving convex programming problems with linear constraints. The proximal version of ALM, which regularizes ALM’s subproblem over the primal variable at each iteration by an additional positive-definite quadratic proximal term, has been well studied in the literature. In this paper we show that it is not necessary to employ a positive-definite quadratic proximal term for the proximal ALM and the convergence can be still ensured if the positive definiteness is relaxed to indefiniteness by reducing the proximal parameter. An indefinite proximal version of the ALM is thus proposed for the generic setting of convex programming problems with linear constraints. We show that our relaxation is optimal in the sense that the proximal parameter cannot be further reduced. The consideration of indefinite proximal regularization is particularly meaningful for generating larger step sizes in solving ALM’s primal subproblems. When the model under discussion is separable in the sense that its objective function consists of finitely many additive function components without coupled variables, it is desired to decompose each ALM’s subproblem over the primal variable in Jacobian manner, replacing the original one by a sequence of easier and smaller decomposed subproblems, so that parallel computation can be applied. This full Jacobian splitting version of the ALM is known to be not necessarily convergent, and it has been studied in the literature that its convergence can be ensured if all the decomposed subproblems are further regularized by sufficiently large proximal terms. But how small the proximal parameter could be is still open. The other purpose of this paper is to show the smallest proximal parameter for the full Jacobian splitting version of ALM for solving multi-block separable convex minimization models.

Touching tacit knowledge: handwork as ethnographic method in a glassblowing studio

This article draws from an enacted ethnography conducted over four years in a glassblowing studio, where I immersed myself in the learning process to become a glassblower. Specifically, it uses the visceral ethnographic experience of handwork in glassblowing to unpack the micro-meanings of hand coordination and examine Michael Polanyi’s theory of tacit knowledge ‘from the body’ (Ingold, 2000; Pink, 2009; Wacquant, 2015: 5). Methodologically, handwork is the ‘point of production’ by which to reflect upon Polanyi’s analytical concepts (Wacquant, 2015: 5). Broadly engaging anthropology’s study of the relation of gesture and form both within and outside of glassblowing studios and the sociology of skill, this analysis brings the body’s embedded experience and constitutive power to bear on analyses of tacit knowledge to reveal how handwork is itself constitutive of form and meaning (Atkinson, 2013b; Harper, 1987; Keller and Keller, 1996; Malafourius, 2008; Marchand, 2010, 2009, 2008, 2001; Sudnow, 1978). It also grounds a reinterpretation of the proximal term in Polanyi’s theory of tacit knowledge.

A Line-Search-Based Partial Proximal Alternating Directions Method for Separable Convex Optimization

We propose an appealing line-search-based partial proximal alternating directions (LSPPAD) method for solving a class of separable convex optimization problems. These problems under consideration are common in practice. The proposed method solves two subproblems at each iteration: one is solved by a proximal point method, while the proximal term is absent from the other. Both subproblems admit inexact solutions. A line search technique is used to guarantee the convergence. The convergence of the LSPPAD method is established under some suitable conditions. The advantage of the proposed method is that it provides the tractability of the subproblem in which the proximal term is absent. Numerical tests show that the LSPPAD method has better performance compared with the existing alternating projection based prediction-correction (APBPC) method if both are employed to solve the described problem.

Convergence of a Proximal Point Algorithm for Solving Minimization Problems

We introduce and consider a proximal point algorithm for solving minimization problems using the technique of Güler. This proximal point algorithm is obtained by substituting the usual quadratic proximal term by a class of convex nonquadratic distance-like functions. It can be seen as an extragradient iterative scheme. We prove the convergence rate of this new proximal point method under mild assumptions. Furthermore, it is shown that this estimate rate is better than the available ones.