We study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term. We introduce an auxiliary variable to separate the objective function components and utilize the Moreau envelope of the regularization term to derive the proximal augmented Lagrangian -a continuously differentiable function obtained by constraining the augmented Lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term. The continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers (MM) to compute optimal primaldual pairs by solving a sequence of differentiable problems. The MM algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step-size update rules than the alternating direction method of multipliers. These features make it an attractive option for solving structured optimal control problems. We also develop an algorithm based on the primal-descent dual-ascent gradient method and prove global (exponential) asymptotic stability when the differentiable component of the objective function is (strongly) convex and the regularization term is convex. Finally, we identify classes of problems for which the primal-dual gradient flow dynamics are convenient for distributed implementation and compare/contrast our framework to the existing approaches. I. INTRODUCTIONWe study a class of composite optimization problems in which the objective function is a sum of a differentiable but possibly nonconvex component and a convex nondifferentiable component. Problems of this form are encountered in diverse fields including compressive sensing [1], machine learning [2], statistics [3], image processing [4], and control [5]. In feedback synthesis, they typically arise when a traditional performance metric (such as the H2 or H∞ norm) is augmented with a regularization function to promote certain structural properties in the optimal controller. For example, the 1 norm and the nuclear norm are commonly used nonsmooth convex regularizers that encourage sparse and low-rank optimal solutions, respectively.The lack of a differentiable objective function precludes the use of standard descent methods for smooth optimization. Proximal gradient methods [6] and their accelerated variants [7] generalize gradient descent, but typically require the nonsmooth term to be separable over the optimization variable. Furthermore, standard acceleration techniques are not well-suited for problems with constraint sets that do not admit an easy projection (e.g., closed-loop stability).An alternative approach is to split the smooth and nonsmooth components in the objective function over separate variables which are coupled via an equality constraint. Such a reformulation facilitates the use of the alternating direction method of multi...

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A generalization of Vinnicombe's ν-gap metric and corresponding robust feedback stability results are proposed for a class of linear time-varying (LTV) systems in "Robust stability analysis of time-varying linear systems,"

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