Keywords: Unit commitment, stochastic optimization, multi-horizon scenario trees, non-convex hydro-production function, regularized benders decomposition, bundle methods.

Abstract: For an electric power mix subject to uncertainty, the stochastic unit-commitment problem finds short-term optimal generation schedules that satisfy several system-wide constraints. In regulated electricity markets, this very practical and important problem is used by the system operator to decide when each unit is to be started or stopped, and to define how to generate enough energy to meet the load. For hydro-dominated systems, an accurate description of the hydro-production function involves non-convex relations. This feature, combined with the fine time discretization needed to represent uncertainty of renewable generation, yields a large-scale mathematical optimization model that is nonlinear and has mixed-integer variables. To make the problem tractable, a novel solution strategy, based on multi-horizon scenario trees, is proposed. The approach deals in a first level with the integer decision variables representing whether units are on or off. Once units are committed, the expected operational cost is minimized by solving a continuous second-level problem, which is separable by scenarios. The coordination between the two decision levels is done by means of a bundle-like variant of Benders decomposition that proves very efficient for the considered setting. To assess the quality of the optimal commitment on out-of-sample scenarios, a new simulation technique, based on certain sustainable pseudo-distance is proposed. For the numerical experiments, a mix of hydro, thermal, and wind power plants extracted from the Brazilian power system is considered. The results confirm the interest of the approach, particularly regarding a more efficient management of hydro-plants, because non-convex operational regions are considered by the model.

Keywords: Packing, ellipsoids, nonlinear programming, algorithms, matheuristic.

Abstract: The problem of packing ellipsoids in the three-dimensional space is considered in the present work. The proposed approach combines heuristic techniques with the resolution of recently introduced nonlinear programming models in order to construct solutions with a large number of ellipsoids. The introduced approach is able to pack identical and non-identical ellipsoids within a variety of containers. Moreover, it allows the inclusion of additional positioning constraints. This fact makes the proposed approach suitable for constructing large-scale solutions with specific positioning constraints in which density may not be the main issue. Numerical experiments illustrate that the introduced approach delivers good quality solutions with a computational cost that scales linearly with the number of ellipsoids; and solutions with more than a million ellipsoids are exhibited.

Keywords: Cutting and packing problems, constrained two-dimensional guillotine cuts, integer programming models, branch-and-Benders-cut algorithm.

Abstract: In this paper, we address the constrained two-dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangularitems from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can beproduced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinearprogramming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulationfrom it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of theinteger linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, wederive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consideronly two-staged or one-group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP,we apply Benders decomposition to the proposed ILP formulation and develop a branch-and-Benders-cutalgorithm. All proposed approaches are evaluated through computational experiments using benchmarkinstances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.

Keywords: Cutting and packing ellipsoids, optimization, nonlinear programming, models, numerical experiments.

Abstract: The problem of packing ellipsoids is considered in the present work. Usually, the computational effort associated with numerical optimization methods devoted to packing ellipsoids grows quadratically with respect to the number of ellipsoids being packed. The reason is that the number of variables and constraints of ellipsoids' packing models is associated with the requirement that every pair of ellipsoids must not overlap. As a consequence, it is hard to solve the problem when the number of ellipsoids is large. In this paper, we present a nonlinear programming model for packing ellipsoids that contains a linear number of variables and constraints. The proposed model finds its basis in a transformation-based non-overlapping model recently introduced by Birgin, Lobato, and Martínez [Journal of Global Optimization (2015), DOI: 10.1007/s10898-015-0395-z]. Numerical experiments show the efficiency and effectiveness of the proposed model.

Keywords: Cutting and packing ellipsoids, nonlinear programming, models, numerical experiments.

Abstract: In this paper, continuous and differentiable nonlinear programming models and algorithms for packing ellipsoids in the $n$-dimensional space are introduced. Two different models for the non-overlapping and models for the inclusion of ellipsoids within half-spaces and ellipsoids are presented. By applying a simple multi-start strategy combined with a clever choice of starting guesses and a nonlinear programming local solver, illustrative numerical experiments are presented.

Keywords: Inexact restoration, mixed-integer nonlinear programming (MINLP), projected gradients.

Abstract: Inexact restoration (IR) is a well established technique for continuous minimization problems with constraints that can be applied to constrained optimization problems with specific structures. When some variables are restricted to be integer, an IR strategy seems to be appropriate. The IR strategy employs a restoration procedure in which one solves a standard nonlinear programming problem and an optimization procedure in which the constraints are linearized and techniques for mixed-integer (linear or quadratic) programming can be employed.

Keywords: Cutting and packing, two-dimensional non-guillotine cutting pattern, dynamic programming, recursive approach, distributor's pallet loading problem.

Abstract: In this study, a dynamic programming approach to deal with the unconstrained two-dimensional non-guillotine cutting problem is presented. The method extends the recently introduced recursive partitioning approach for the manufacturer's pallet loading problem. The approach involves two phases and uses bounds based on unconstrained two-staged and non-staged guillotine cutting. Since a counterexample for which the approach fails to find known optimal solutions was not found, it is conjectured that it always finds an optimal unconstrained non-guillotine cutting. The method is able to find the optimal cutting pattern of a large number of problem instances of moderate sizes known in the literature. For the instances that the required computer runtime is excessive, the approach is combined with simple heuristics to reduce its running time. Detailed numerical experiments show the reliability of the method.

Keywords: Packing and cutting of rectangles, orthogonal packing, isotropic convex regions, feasibility problems, nonlinear programming, models.

Abstract: A mixed integer continuous nonlinear model and a solution method for the problem of orthogonally packing identical rectangles within an arbitrary convex region are introduced in the present work. The convex region is assumed to be made of an isotropic material in such a way that arbitrary rotations of the items, preserving the orthogonality constraint, are allowed. The solution method is based on a combination of branch and bound and active-set strategies for bound-constrained minimization of smooth functions. Numerical results show the reliability of the presented approach.

Keywords: Cutting and packing, manufacturer's pallet loading problem, woodpulp stowage problem, non-guillotine cutting pattern, dynamic programming, raster points.

Abstract: In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an $L$-approach for packing rectangles into larger rectangles and $L$-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50000 instances). It is also effective for solving the instances of problem set Cover III (almost 100000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.