We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x,y)=f(x)+Q(x,y)+g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L.We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to “metrically regular” problems.Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with “regular” intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted ℓ1 algorithm for compressive sensing and an application to rank reduction problems.

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This paper offers an explanation of the fact that some foreign firms are favored at the expense of others, and characterizes the distribution of favors in terms of the cost parameters of firms. We present a model where favors must be bought: they come from competing contributions. This model is compared with a benchmark model with a benevolent government. We show how the distribution of favors in the favor-seeking model deviates from the distribution that would be obtained if the government were really benevolent. Copyright © 2006 The Authors; Journal compilation © 2006 Blackwell Publishing Ltd.

We develop a new framework for the analysis of the impact of trade liberalization on the wage structure and on welfare. Our model focuses on the decision of workers to accumulate firm-specific skills, by "on-the-job" training, knowing that this means their future wages will have to be negotiated, and that the outcome of negotiation will depend on the profitability prospect of firms operating in a new trading environment. We show that trade liberalization may reduce the welfare of a developing country because of its adverse effect on skill accumulation. We also explore the effects of trade liberalization on the wage gap between skilled and unskilled workers. Copyright (c) 2007 The Authors.