This paper focuses on coordinate update methods, which are useful for solving problems involving large or high-dimensional datasets. They decompose a problem into simple subproblems, where each updates one, or a small block of, variables while fixing others. These methods can deal with linear and nonlinear mappings, smooth and nonsmooth functions, as well as convex and nonconvex problems. In addition, they are easy to parallelize.

Recent years have witnessed the rapid development of block coordinate update (BCU) methods, which are particularly suitable for problems involving large-sized data and/or variables. In optimization, BCU first appears as the coordinate descent method that works well for smooth problems or those with separable nonsmooth terms and/or separable constraints. As nonseparable constraints exist, BCU can be applied under primal-dual settings. In the literature, it has been shown that for weakly convex problems with nonseparable linear constraints, BCU with fully Gauss--Seidel updating rule may fail to converge and that with fully Jacobian rule can converge sublinearly. However, empirically the method with Jacobian update is usually slower than that with Gauss--Seidel rule. To maintain their advantages, we propose a hybrid Jacobian and Gauss--Seidel BCU method for solving linearly constrained multiblock

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We discuss a variant of the multi-task <i>n</i>-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem requires all vehicles in a group to arrive at the same destination. Given <i>n</i> tasks with assigned consume-time and profit, it may also be viewed as a maximization of every processors average profit. Further, we propose a new kind of partition problem in fractional form and analyze its computational complexity. By regarding fractional partition as a special case, we prove that the average profit maximization problem is NP-hard when the number of processors is fixed and it is strongly NPhard in general. At last, a pseudo-polynomial time algorithm for the average profit maximization problem and the fractional partition problem is presented, using the idea of the pseudo-polynomial time algorithm for the classical partition problem.

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In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A x= Ax. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the PDA.

The support vector machine (SVM) has been used in a wide variety of classification problems. The original SVM uses the hinge loss function, which is non-differentiable and makes the problem difficult to solve in particular for regularized SVMs, such as with 1 -regularization. This paper considers the Huberized SVM (HSVM), which uses a differentiable approximation of the hinge loss function. We first explore the use of the proximal gradient (PG) method to solving binary-class HSVM (B-HSVM) and then generalize it to multi-class HSVM (M-HSVM). Under strong convexity assumptions, we show that our algorithm converges linearly. In addition, we give a finite convergence result about the support of the solution, based on which we further accelerate the algorithm by a two-stage method. We present extensive numerical experiments on both synthetic and real datasets which demonstrate the superiority of our

This paper introduces algorithms for the decentralized low-rank matrix completion problem. Assume a low-rank matrix W = [W ₁ ,W ₂ , ...,W _L ]. In a network, each agent observes some entries of W  . In order to recover the unobserved entries of W via decentralized computation, we factorize the unknown matrix W as the product of a public matrix X, common to all agents, and a private matrix Y = [Y ₁ ,Y ₂ , ...,Y _L ], where Y  is held by agent . Each agent alternatively updates Y  and its local estimate of X while communicating with its neighbors toward a consensus on the estimate. Once this consensus is (nearly) reached throughout the network, each agent recovers W  = XY  , and thus W is recovered. The communication cost is scalable to the number of agents, and W  and Y  are kept private to agent to a certain extent. The algorithm is accelerated by extrapolation and compares favorably to the centralized

We extend the RayleighRitz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for RayleighRitz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the RayleighRitz method with refinement. The numerical results illustrate that the refinement procedure produces excellent

In Chu (Syst Control Lett 56:303314, 2007), the pole assignment problem was considered for the control system \dot{x} = Ax + Bu with linear state-feedback \dot{x} = Ax + Bu An algorithm using the Schur form has been proposed, producing good suboptimal solutions which can be refined further using optimization. In this paper, the algorithm is improved, incorporating the minimization of the feedback gain \dot{x} = Ax + Bu It is also extended for the pole assignment of the descriptor system \dot{x} = Ax + Bu with linear state- and derivative-feedback \dot{x} = Ax + Bu Newton refinement for the solutions is discussed and several illustrative numerical examples are presented.