Implementations of the simplex method differ mostly in specific aspects such as the pivot rule. Similarly, most relaxation methods for mixed-integer program- ming differ mostly in the type of cuts and the exploration of the search tree. We provide a scripting mechanism to easily implement and experiment with primal and dual pivot rules for the simplex method, by building upon COIN-OR’s open-source linear programming package CLP, without explicitly interacting with the underlying C++ layers of CLP. In the same manner, users can customize the solution process of mixed-integer linear programs using the CBC and CGL COIN-OR packages by cod- ing branch-and-cut strategies and cut generators in Python. The Cython programming language ensures communication between Python and C++ libraries and activates user-defined customizations as callbacks. Our goal is to emphasize the ease of devel- opment in Python while maintaining acceptable performance. The resulting software, named CyLP, has become a part of COIN-OR and is available under open-source terms. For illustration, we provide an implementation of the positive edge rule—a recently proposed rule that is particularly efficient on degenerate problems—and demonstrate how to customize branch-and-cut node selection in the solution of a mixed-integer program.

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Werevisitanalgorithm[calledEdgePushing(EP)]forcomputingHessians using Automatic Differentiation (AD) recently proposed by Gower and Mello (Optim Methods Softw 27(2): 233–249, 2012). Here we give a new, simpler derivation for the EP algorithm based on the notion of live variables from data-flow analysis in compiler theory and redesign the algorithm with close attention to general applicability and performance. We call this algorithm Livarh and develop an extension of Livarh that incorporates preaccumulation to further reduce execution time—the resulting algo- rithm is called Livarhacc. We engineer robust implementations for both algorithms Livarh and Livarhacc within ADOL-C, a widely-used operator overloading based AD software tool. Rigorous complexity analyses for the algorithms are provided, and the performance of the algorithms is evaluated using a mesh optimization application and several kinds of synthetic functions as testbeds. The results show that the new algo- rithms outperform state-of-the-art sparse methods (based on sparsity pattern detection, coloring, compressed matrix evaluation, and recovery) in some cases by orders of mag- nitude. We have made our implementation available online as open-source software and it will be included in a future release of ADOL-C.

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We present a quasi-Newton sequential quadratic programming (SQP) algorithm for nonlinear programs in which the Hessian of the Lagrangian function is block-diagonal. Problems with this characteristic frequently arise in the context of optimal control; for example, when a direct multiple shooting parametrization is used. In this article, we describe an implementation of a filter line-search SQP method that computes search directions using an active-set quadratic programming (QP) solver. To take advantage of the block-diagonal structure of the Hessian matrix, each block is approximated separately by quasi-Newton updates. For nonconvex instances, that arise, for example, in optimum experimental design control problems, these blocks are often found to be indefinite. In that case, the block-BFGS quasi-Newton update can lead to poor convergence. The novel aspect in this work is the use of SR1 updates in place of BFGS approximations whenever possible. The resulting indefinite QPs neces- sitate an inertia control mechanism within the sparse Schur-complement factorization that is carried out by the active-set QP solver. This permits an adaptive selection of the Hessian approximation that guarantees sufficient progress towards a stationary point of the problem. Numerical results demonstrate that the proposed approach reduces the number of SQP iterations and CPU time required for the solution of a set of optimal control problems.

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The Steiner tree problem with revenues, budget and hop-constraints (STPRBH) is a variant of the classical Steiner tree problem. The goal is to find a tree maximizing the collected revenue, which is associated with nodes, subject to a given budget for the edge cost of the tree and a hop-limit for the distance between the given root node and any other node in that tree. In this work, we introduce a novel generic way to model hop-constrained tree problems as integer linear programs and apply it to the STPRBH. Our approach is based on the concept of layered graphs that gained widespread attention in the recent years, due to their computational advantage when compared to previous formulations for modeling hop-constraints. Contrary to previous MIP formulations based on layered graphs (that are arc-based models), our model is node-based. Thus it contains much less variables and allows to tackle large- scale instances and/or instances with large hop-limits, for which the size of arc-based layered graph models may become prohibitive. The aim of our model is to provide a good compromise between quality of root relaxation bounds and the size of the under- lying MIP formulation. We implemented a branch-and-cut algorithm for the STPRBH based on our new model. Most of the instances available for the DIMACS challenge, including 78 (out of 86) previously unsolved ones, can be solved to proven optimality within a time limit of 1000 s, most of them being solved within a few seconds only. These instances contain up to 500 nodes and 12,500 edges, with hop-limit up to 25.

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