Ride sharing has important implications in terms of environmental, social and individual goals by reducing carbon footprints, fostering social interactions and economizing commuter costs. The ride sharing systems that are commonly available lack adaptive and scalable techniques that can simultaneously learn from the large scale data and predict in real-time dynamic fashion. In this paper, we study such a problem towards a smart city initiative, where a generic ride sharing system is conceived capable of making predictions about ride share opportunities based on the historically recorded data while satisfying real-time ride requests. Underpinning the system is an application of a powerful machine learning convex optimization framework called Network Lasso that uses the Alternate Direction Method of Multipliers (ADMM) optimization for learning and dynamic prediction. We propose an application of a robust and scalable unified optimization framework within the ride sharing case-study. The application of Network Lasso framework is capable of jointly optimizing and clustering different rides based on their spatial and model similarity. The prediction from the framework clusters new ride requests, making accurate price prediction based on the clusters, detecting hidden correlations in the data and allowing fast convergence due to the network topology. We provide an empirical evaluation of the application of ADMM network Lasso on real trip record and simulated data, proving their effectiveness since the mean squared error of the algorithm's prediction is minimized on the test rides.

Different versions of this problem have received is coping with uncertainties arising from limited a-considerable attention from several research communities, e.g., priori knowledge of the environment. Acquiring necessary as a "pursuit-evasion game" in game theory [13], [14], as information and achieving the overall goal are complementary a "cow-path problem" in computer science [15] or as a subtasks that require adapting the motion of a robot during "coverage problem" in control [16], [17], but its solution for mission execution, typically accompanied by minimizing a a general probability distribution or a general geometry of the performance criterion. In this work we address an Optimal region is, to a large extent, still an open question. Effective Control Problem (OCP) for a robot with fourth-order dynamics approaches for the related persistent monitoring problem based that has to find, collect and move a finite number of on estimation [18], linear programming [19] or parametric objects to a designated spot in minimum time. The objects optimization [20] have been also been proposed. OCPs with with a-priori known masses are located in a bounded twodimensional uncertainties have also been addressed by certainty equivalent space, where the robot is capable of localizing event-triggered [21], minimax [22] and sampling-based [23] itself using a state-of-the-art simultaneous localization and optimization schemes.

In this paper, we propose a clustering-based iterative algorithm to solve certain optimization problems in machine learning when data size is large and thus it becomes impractical to use out-of-the-box algorithms. We rely on the principle of data aggregation and then subsequent disaggregations. While it is standard practice to aggregate the data and then calibrate the machine learning algorithm on aggregated data, we embed this into an iterative framework where initial aggregations are gradually disaggregated to the extent that even an optimal solution is obtainable. Early studies in data aggregation consider transportation problems [1, 10], where either demand or supply nodes are aggregated. Zipkin [31] studied data aggregation for linear programming (LP) and derived error bounds of the approximate solution.

We propose an adaptive smoothing algorithm based on Nesterov's smoothing technique in \cite{Nesterov2005c} for solving "fully" nonsmooth composite convex optimization problems. Our method combines both Nesterov's accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the $\mathcal{O}\left(\frac{1}{\varepsilon}\right)$-worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov's method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms.

This paper considers global optimization with a black-box unknown objective function that can be non-convex and non-differentiable. Such a difficult optimization problem arises in many real-world applications, such as parameter tuning in machine learning, engineering design problem, and planning with a complex physics simulator. This paper proposes a new global optimization algorithm, called Locally Oriented Global Optimization (LOGO), to aim for both fast convergence in practice and finite-time error bound in theory. The advantage and usage of the new algorithm are illustrated via theoretical analysis and an experiment conducted with 11 benchmark test functions. Further, we modify the LOGO algorithm to specifically solve a planning problem via policy search with continuous state/action space and long time horizon while maintaining its finite-time error bound.

We consider the problem of approximate joint triangularization of a set of noisy jointly diagonalizable real matrices. Approximate joint triangularizers are commonly used in the estimation of the joint eigenstructure of a set of matrices, with applications in signal processing, linear algebra, and tensor decomposition. By assuming the input matrices to be perturbations of noise-free, simultaneously diagonalizable ground-truth matrices, the approximate joint triangularizers are expected to be perturbations of the exact joint triangularizers of the ground-truth matrices. We provide a priori and a posteriori perturbation bounds on the `distance' between an approximate joint triangularizer and its exact counterpart. The a priori bounds are theoretical inequalities that involve functions of the ground-truth matrices and noise matrices, whereas the a posteriori bounds are given in terms of observable quantities that can be computed from the input matrices. From a practical perspective, the problem of finding the best approximate joint triangularizer of a set of noisy matrices amounts to solving a nonconvex optimization problem. We show that, under a condition on the noise level of the input matrices, it is possible to find a good initial triangularizer such that the solution obtained by any local descent-type algorithm has certain global guarantees. Finally, we discuss the application of approximate joint matrix triangularization to canonical tensor decomposition and we derive novel estimation error bounds.

We give a simple proof that the Frank-Wolfe algorithm obtains a stationary point at a rate of $O(1/\sqrt{t})$ on non-convex objectives with a Lipschitz continuous gradient. Our analysis is affine invariant and is the first, to the best of our knowledge, giving a similar rate to what was already proven for projected gradient methods (though on slightly different measures of stationarity).

In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous applications in machine learning. The proximal damped Newton (PDN) methods have been well studied in the literature for solving this problem that enjoy a nice iteration complexity. Given that at each iteration these methods typically require evaluating or accessing the Hessian of $f$ and also need to solve a proximal Newton subproblem, the cost per iteration can be prohibitively high when applied to large-scale problems. Inspired by the recent success of block coordinate descent methods, we propose a randomized block proximal damped Newton (RBPDN) method for solving the CSC minimization. Compared to the PDN methods, the computational cost per iteration of RBPDN is usually significantly lower. The computational experiment on a class of regularized logistic regression problems demonstrate that RBPDN is indeed promising in solving large-scale CSC minimization problems. The convergence of RBPDN is also analyzed in the paper. In particular, we show that RBPDN is globally convergent when $g$ is Lipschitz continuous. It is also shown that RBPDN enjoys a local linear convergence. Moreover, we show that for a class of $g$ including the case where $g$ is Lipschitz differentiable, RBPDN enjoys a global linear convergence. As a striking consequence, it shows that the classical damped Newton methods [22,40] and the PDN [31] for such $g$ are globally linearly convergent, which was previously unknown in the literature. Moreover, this result can be used to sharpen the existing iteration complexity of these methods.

Enforcing local consistencies in cost function networks is performed by applying so-called Equivalent Preserving Transformations (EPTs) to the cost functions. As EPTs transform the cost functions, they may break the property that was making local consistency enforcement tractable on a global cost function. A global cost function is called tractable projection-safe when applying an EPT to it is tractable and does not break the tractability property. In this paper, we prove that depending on the size r of the smallest scopes used for performing EPTs, the tractability of global cost functions can be preserved (r = 0) or destroyed (r > 1). When r = 1, the answer is indefinite. We show that on a large family of cost functions, EPTs can be computed via dynamic programming-based algorithms, leading to tractable projection-safety. We also show that when a global cost function can be decomposed into a Berge acyclic network of bounded arity cost functions, soft local consistencies such as soft Directed or Virtual Arc Consistency can directly emulate dynamic programming. These different approaches to decomposable cost functions are then embedded in a solver for extensive experiments that confirm the feasibility and efficiency of our proposal.

Motivated by big data applications, first-order methods have been extremely popular in recent years. However, naive gradient methods generally converge slowly. Hence, much efforts have been made to accelerate various first-order methods. This paper proposes two accelerated methods towards solving structured linearly constrained convex programming, for which we assume composite convex objective. The first method is the accelerated linearized augmented Lagrangian method (LALM). At each update to the primal variable, it allows linearization to the differentiable function and also the augmented term, and thus it enables easy subproblems. Assuming merely weak convexity, we show that LALM owns $O(1/t)$ convergence if parameters are kept fixed during all the iterations and can be accelerated to $O(1/t^2)$ if the parameters are adapted, where $t$ is the number of total iterations. The second method is the accelerated linearized alternating direction method of multipliers (LADMM). In addition to the composite convexity, it further assumes two-block structure on the objective. Different from classic ADMM, our method allows linearization to the objective and also augmented term to make the update simple. Assuming strong convexity on one block variable, we show that LADMM also enjoys $O(1/t^2)$ convergence with adaptive parameters. This result is a significant improvement over that in [Goldstein et. al, SIIMS'14], which requires strong convexity on both block variables and no linearization to the objective or augmented term. Numerical experiments are performed on quadratic programming, image denoising, and support vector machine. The proposed accelerated methods are compared to nonaccelerated ones and also existing accelerated methods. The results demonstrate the validness of acceleration and superior performance of the proposed methods over existing ones.