This paper aims to propose an efficient numerical method for the most challenging problem known as the robust Euclidean embedding (REE) in the family of multi-dimensional scaling (MDS). The problem is notoriously known to be nonsmooth, nonconvex and its objective is non-Lipschitzian. We first explain that the semidefinite programming (SDP) relaxations and Euclidean distance matrix (EDM) approach, popular for other types of problems in the MDS family, failed to provide a viable method for this problem. We then propose a penalized REE (PREE), which can be economically majorized. We show that the majorized problem is convex provided that the penalty parameter is above certain threshold. Moreover, it has a closed-form solution, resulting in an efficient algorithm dubbed as PREEEDM (for Penalized REE via EDM optimization). We prove among others that PREEEDM converges to a stationary point of PREE, which is also an approximate critical point of REE. Finally, the efficiency of PREEEDM is compared with several state-of-the-art methods including SDP and EDM solvers on a large number of test problems from sensor network localization and molecular conformation.

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This paper addresses the problem of generating synthetic test cases for experimentation in linear programming. We propose a method which maps instance generation and instance space search to an alternative encoded space. This allows us to develop a generator for feasible bounded linear programming instances with controllable properties. We show that this method is capable of generating any feasible bounded linear program, and that parameterised generators and search algorithms using this approach generate only feasible bounded instances. Our results demonstrate that controlled generation and instance space search using this method achieves feature diversity more effectively than using a direct representation.

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In this paper, we describe an algorithm for the personalized nurse scheduling problem. We focus on the deterministic counterpart of the specific problem that has been described in the second international nurse rostering competition. One specificity of this version is that most constraints are soft, meaning that they can be violated at the price of a penalty. We model the problem as an integer program (IP) that we solve using a branch-and-price procedure. This model is, to the best of our knowledge, comparable to no other from the literature, since each column of the IP corresponds to a rotation, i.e., a sequence of consecutive worked days for a nurse. In contrast, classical models involve individual nurse schedules over the complete horizon. We tackle instances involving up to 120 nurses and 4 shifts over an 8-weeks horizon by embedding the branch-and-price in a large-neighborhood-search framework. Initial solutions of the large-neighborhood search are found by a rolling-horizon algorithm well-suited to the rotation model.

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We propose a new self-adaptive and double-loop smoothing algorithm to solve composite,
nonsmooth, and constrained convex optimization problems. Our algorithm is based on
Nesterov’s smoothing technique via general Bregman distance functions. It self-adaptively
selects the number of iterations in the inner loop to achieve a desired complexity bound
without requiring to set the accuracy a priori as in variants of augmented Lagrangian
methods (ALM). We prove O(1/k)-convergence rate on the last iterate of the outer sequence
for both unconstrained and constrained settings in contrast to ergodic rates which are c
ommon in ALM as well as alternating direction method-of-multipliers literature.
Compared to existing inexact ALM or quadratic penalty methods, our analysis does not rely
on the worst-case bounds of the subproblem solved by the inner loop. Therefore, our
algorithm can be viewed as a restarting technique applied to the ASGARD method in
Tran-Dinh et al. (SIAM J Optim 28(1):96–134, 2018) but with rigorous theoretical
guarantees or as an inexact ALM with explicit inner loop termination rules and
adaptive parameters. Our algorithm only requires to initialize the parameters once,
and automatically updates them during the iteration process without tuning.
We illustrate the superiority of our methods via several examples as compared
to the state-of-the-art.

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In the analysis of networks, one often searches for tightly knit clusters. One property of a “good” cluster is a small diameter (say, bounded by k), which leads to the concept of a k-club. In this paper, we propose new path-like and cut-like integer programming formulations for detecting these low-diameter subgraphs. They simplify, generalize, and/or dominate several previously existing formulations. Our best-performing formulation uses only node variables (quite unlike previous formulations) and imposes the diameter-at-most-k constraints via an exponentially large class of cut-like inequalities. A relatively simple implementation of the cut-like formulation easily outperforms previous approaches, solving dozens of instances of the maximum k-club problem in a second or two that would take hours by other formulations. Moreover, the cut-like formulation is more general in the sense that it applies even when distances are not measured in terms of hops. While we consider only the k-club problem in this paper, the proposed techniques may also be useful in other applications where compact solutions are key (e.g., political districting and wildlife reserve design).

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