Coordinate descent methods with high-order regularized models for box-constrained minimization are introduced. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order ε-stationarity with respect to the variables of each coordinate-descent block is O(ε −(p+1)/p ) whereas the computer work for getting first-order ε-stationarity with respect to all the variables simultaneously is O(ε −(p+1) ). Numerical examples involving multidimensional scaling problems are presented. The numerical performance of the methods is enhanced by means of coordinate-descent strategies for choosing initial points.

At each iteration of a Block Coordinate Descent method one minimizes an approximation of the objective function with respect to a generally small set of variables subject to constraints in which these variables are involved. The unconstrained case and the case in which the constraints are simple were analyzed in the recent literature. In this paper we address the problem in which block constraints are not simple and, moreover, the case in which they are not defined by global sets of equations and inequations. A general algorithm that minimizes quadratic models with quadratric regularization over blocks of variables is defined and convergence and complexity are proved. In particular, given tolerances δ > 0 and ε > 0 for feasibility/complementarity and optimality, respectively, it is shown that a measure of (δ, 0)-criticality tends to zero; and the the number of iterations and functional evaluations required to achieve (δ, ε)-criticality is O(ε 2 ). Numerical experiments in which the proposed method is used to solve a continuous version of the traveling salesman problem are presented.

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This paper addresses an integrated lot sizing and scheduling problem in the industry of consumer goods for personal care, a very competitive market in which good customer service and cost management are crucial in the competition for clients. In this research, a complex operational environment composed of unrelated parallel machines with limited production capacity and sequence-dependent setup times and costs is studied. There is also a limitation in the total storage capacity for finished goods, a characteristic not found in the literature. Backordering is allowed, but it is extremely undesirable. The problem is described through a mixed integer linear programming formulation. Since the problem is NP-hard, relax-and-fix heuristics with hybrid partitioning strategies are investigated. Computational experiments with randomly generated and real-world instances are presented. The results show the efficacy and efficiency of the proposed approaches. Compared to the current solutions used by the company, the best proposed strategies yield results with substantially lower costs, primarily from the reduction in inventory levels and better allocation of production batches on the machines.

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