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In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

We consider the problem of estimating multiple related Gaussian graphical models from a high dimensional data set with observations belonging to distinct classes. We propose the joint graphical lasso, which borrows strength across the classes to estimate multiple graphical models that share certain characteristics, such as the locations or weights of non‐zero edges. Our approach is based on maximizing a penalized log‐likelihood. We employ generalized fused lasso or group lasso penalties and implement a fast alternating directions method of multipliers algorithm to solve the corresponding convex optimization problems. The performance of the method proposed is illustrated through simulated and real data examples.

We analyze the convergence rate of the alternating direction method of multipliers (ADMM) for minimizing the sum of two or more nonsmooth convex separable functions subject to linear constraints. Previous analysis of the ADMM typically assumes that the objective function is the sum of only two convex functions defined on two separable blocks of variables even though the algorithm works well in numerical experiments for three or more blocks. Moreover, there has been no rate of convergence analysis for the ADMM without strong convexity in the objective function. In this paper we establish the global R-linear convergence of the ADMM for minimizing the sum of any number of convex separable functions, assuming that a certain error bound condition holds true and the dual stepsize is sufficiently small. Such an error bound condition is satisfied for example when the feasible set is a compact polyhedron and the objective function consists of a smooth strictly convex function composed with a linear mapping, and a nonsmooth ℓ 1 regularizer. This result implies the linear convergence of the ADMM for contemporary applications such as LASSO without assuming strong convexity of the objective function.

This paper is devoted to the existence of non-trivial bound states of prescribed mass for the mass-supercritical nonlinear Schrödinger equation on compact metric graphs. The investigation is based upon a min-max principle for some constrained functionals which combines the monotonicity trick and second-order information on the Palais–Smale sequences, and upon the blow-up analysis of bound states with prescribed mass and bounded Morse index.

This paper presents the bibliometrics of a Keynesian and neoclassical discussion about the multiplier-accelerator effect. Having its oldest roots in the 1930s, there was a special emphasis in the 1960s and 1970s on discussions regarding the dependence of current investment on economic growth (the accelerator effect). Through a bibliometric analysis, we also consider the Hicks-Samuelson contribution, also known as the multiplier-accelerator model. We identified, among other things, the most relevant authors on the topics, the economic areas that have been contributed to the most through keyword analysis, and the most notable contributions through citation analysis. We concluded that several areas in economics have taken advantage of the discussion around the multiplier-accelerator effect, especially the discussion on the business cycle, structural dynamics, and public finance.

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend the ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of more than two separable convex functions. However, the convergence of this extension has been missing for a long time—neither an affirmative convergence proof nor an example showing its divergence is known in the literature. In this paper we give a negative answer to this long-standing open question: The direct extension of ADMM is not necessarily convergent. We present a sufficient condition to ensure the convergence of the direct extension of ADMM, and give an example to show its divergence.

This article considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There is a variety of economic applications where solving this problem allows to carry out inference on causal and structural parameters; a notable example is the market structure model of Ciliberto and Tamer (2009) where p = 2m+1 with m being the number of firms that could possibly enter the market. We consider the test statistic given by the maximum of p Studentized (or t-type) inequality-specific statistics, and analyse various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (1) the union bound combined with a moderate deviation inequality for self-normalized sums, (2) the multiplier and empirical bootstraps, and (3) two-step and three-step variants of (1) and (2) by incorporating the selection of uninformative inequalities that are far from being binding and a novel selection of weakly informative inequalities that are potentially binding but do not provide first-order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with the error in size decreasing polynomially in n while allowing for p being much larger than n; indeed p can be of order exp(nc ) for some c > 0. Importantly, all these results hold without any restriction on the correlation structure between p Studentized statistics, and also hold uniformly with respect to suitably large classes of underlying distributions. Moreover, in the online supplement, we show validity of a test based on the block multiplier bootstrap in the case of dependent data under some general mixing conditions.

Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for different models. Moreover, its theoretical studies such as rate of convergence and extensions to nonconvex problems also achieve much progress. In this paper, we give a survey on some recent developments of ADMM and its variants.

How large economic stimuli generate individual and aggregate responses is a central question in economics, but has not been studied experimentally. We provided one-time cash transfers of about USD 1000 to over 10,500 poor households across 653 randomized villages in rural Kenya. The implied fiscal shock was over 15 percent of local GDP. We find large impacts on consumption and assets for recipients. Importantly, we document large positive spillovers on non-recipient households and firms, and minimal price inflation. We estimate a local transfer multiplier of 2.5. We interpret welfare implications through the lens of a simple household optimization framework.

In this article we study consumption network effects. Does the consumption of our peers affect our own consumption? How large is such effect? What are the economic mechanisms behind it? We use administrative panel data on Danish households to construct a measure of consumption based on tax records on income and assets. We combine tax record data with matched employer–employee data to identify peer groups based on workplace, which gives us a much tighter and credible definition of networks than used in previous literature. We use the non-overlapping network structure of one’s peers group, as well as firm-level shocks, to build valid instruments for peer consumption. We estimate non-negligible and statistically significant network effects, capable of generating sizable multiplier effect at the macro-level. We also investigate what mechanisms generate such effects, distinguishing between intertemporal and intratemporal consumption effects as well as a more traditional risk sharing view.