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In this paper, we consider a class of nonsmooth sum-of-ratios fractional optimization problems with block structure. This model class is ubiquitous and encompasses several important nonsmooth optimization problems in the literature. We first propose an inertial proximal block coordinate method for solving this class of problems by exploiting the underlying structure. The global convergence of our method is guaranteed under the Kurdyka-Łojasiewicz (KL) property and some mild assumptions. We then identify the explicit exponents of the KL property for three important structured fractional optimization problems. In particular, for the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraint, we show that the KL exponents are 1/2, and so the proposed method exhibits a linear convergence rate. Finally, we illustrate our theoretical results with both analytic and simulated numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, which encompass many important modern optimization problems arising from diverse areas such as the recently proposed scale-invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and that any one of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted fast iterative shrinking-threshold algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka–Łojasiewicz property. Our algorithm exhibits linear convergence for the scale-invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. When the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose another extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford algorithm is a classical and powerful splitting method for minimizing the sum of two convex functions and, more generally, finding a zero of the sum of two maximally monotone operators. Although this algorithm is well understood when the involved operators are monotone or strongly monotone, the convergence theory for weakly monotone settings is far from being complete. In this paper, we propose an adaptive Douglas-Rachford splitting algorithm for the sum of two operators, one of which is strongly monotone while the other one is weakly monotone. With appropriately chosen parameters, the algorithm converges globally to a fixed point from which we derive a solution of the problem. When one operator is Lipschitz continuous, we prove global linear convergence, which sharpens recent known results. [ABSTRACT FROM AUTHOR]

Projection algorithms are well known for their simplicity and flexibility in solving feasibility problems. They are particularly important in practice owing to minimal requirements for software implementation and maintenance. In this work, we study linear convergence of several projection algorithms for systems of finitely many closed sets. The results complement contemporary research on the same topic. [ABSTRACT FROM AUTHOR]