Epigraph

Abstract: Vibration signals always contain noise and irregularities which makes spectrum analysis difficult to extract high-level features. Recently, graph theory has been applied to spectrum analysis to improve the performance of feature extraction. By converting the raw data into graphs, hidden structural and topological information can be obtained. In this paper, a spatial-temporal graph-based feature extraction, called SuperGraph, for rotating machinery fault diagnosis is proposed. Specifically, graph theory-based spectrum analysis is used to construct the spatial-temporal graph. Then, the Laplacian matrix-based feature vector is extracted from the constructed spatial-temporal graph. By this means, the spatial-temporal graph is converted into the one-dimensional vector for further constructing SuperGraph, where each node of the SuperGraph represents a spatial-temporal graph and the SuperGraph is composed of many local graphs. In the local graph, only the same type of nodes are connected to form a fully connected graph. Thus, the task of graph classification can be transformed into classifying the nodes in the SuperGraph. After graph convolutional network is established for learning and obtaining deep features, the label of nodes is identified from a softmax model. Experiments are conducted on two benchmarking datasets and a practical experimental platform to verify effectiveness of the proposed fau

Abstract: Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist (C;.), or the local uniqueness of the projection mapping, but also because in the case where C is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C 2 property) of the function. In this paper we provide an adapted geometrical concept, called sub smoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C 1 property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

Abstract: This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objective functions, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast body of literature on distributed optimization, most of them require the graph to be balanced, which is quite restrictive and not necessary. To solve it, this work proposes a novel idea of using the epigraph form of the constrained optimization, which can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.

Abstract: The paper deals with deterministic optimal control problem with state constraints and non-linear dynamics. It is known for such a problem that the value function is in general discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumption, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypass the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.