The SCIP Optimization Suite 9.0

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in the SCIP Optimization Suite 9.0. The updates in SCIP 9.0 include improved symmetry handling, additions and improvements of nonlinear handlers and primal heuristics, a new cut generator and two new cut selection schemes, a new branching rule, a new LP interface, and several bug fixes. The SCIP Optimization Suite 9.0 also features new Rust and C++ interfaces for SCIP, new Python interface for SoPlex, along with enhancements to existing interfaces. The SCIP Optimization Suite 9.0 also includes new and improved features in the LP solver SoPlex, the presolving library PaPILO, the parallel framework UG, the decomposition framework GCG, and the SCIP extension SCIP-SDP. These additions and enhancements have resulted in an overall performance improvement of SCIP in terms of solving time, number of nodes in the branch-and-bound tree, as well as the reliability of the solver.Comment: The release report of the SCIP Optimization Suite version 9.

On the Computational Efficiency of Subgradient Methods: a Case Study with Lagrangian Bounds

Subgradient methods (SM) have long been the preferred way to solve the large-scale Nondifferentiable Optimization problems arising from the solution of Lagrangian Duals (LD) of Integer Programs (IP). Although other methods can have better convergence rate in practice, SM have certain advantages that may make them competitive under the right conditions. Furthermore, SM have significantly progressed in recent years, and new versions have been proposed with better theoretical and practical performances in some applications. We computationally evaluate a large class of SM in order to assess if these improvements carry over to the IP setting. For this we build a unified scheme that covers many of the SM proposed in the literature, comprised some often overlooked features like projection and dynamic generation of variables. We fine-tune the many algorithmic parameters of the resulting large class of SM, and we test them on two different Lagrangian duals of the Fixed-Charge Multicommodity Capacitated Network Design problem, in order to assess the impact of the characteristics of the problem on the optimal algorithmic choices. Our results show that, if extensive tuning is performed, SM can be competitive with more sophisticated approaches when the tolerance required for solution is not too tight, which is the case when solving LDs of IPs

Branch and Cut based on the volume algorithm: Steiner trees in graphs and Max-cut

We present a Branch-and-Cut algorithm where the volume algorithm is applied instead of the traditionally used dual simplex algorithm to the linear programming relaxations in the root node of the search tree. This means that we use fast approximate solutions to these linear programs instead of exact but slower solutions. We present computational results with the Steiner tree and Max-Cut problems. We show evidence that one can solve these problems much faster with the volume algorithm based Branch-and-Cut code than with a dual simplex based one. We discuss when the volume based approach might be more efficient than the simplex based approach

Simultaneous column-and-row generation for solving large-scale linear programs with column-dependent-rows

In this thesis, we handle a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within a new solution method. The proposed approach in this thesis is called simultaneous column-and-row generation. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now. We then introduce, in detail, a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. Additionally, this generic algorithm is combined with Lagrangian relaxation approach, which provides a different angle to deal with simultaneous column-and-row generation. This observation then leads to another method to solve problems with column-dependent-rows. Throughout the thesis, the proposed solution methods are applied to solve different problems, namely, the multi-stage cutting stock problem, the time-constrained routing problem and the quadratic set covering problem. We also conduct computational experiments to evaluate the performance of the proposed approaches

Exact rotamer optimization for computational protein design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (leaves 235-244).The search for the global minimum energy conformation (GMEC) of protein side chains is an important computational challenge in protein structure prediction and design. Using rotamer models, the problem is formulated as a NP-hard optimization problem. Dead-end elimination (DEE) methods combined with systematic A* search (DEE/A*) have proven useful, but may not be strong enough as we attempt to solve protein design problems where a large number of similar rotamers is eligible and the network of interactions between residues is dense. In this thesis, we present an exact solution method, named BroMAP (branch-and-bound rotamer optimization using MAP estimation), for such protein design problems. The design goal of BroMAP is to be able to expand smaller search trees than conventional branch-and-bound methods while performing only a moderate amount of computation in each node, thereby reducing the total running time. To achieve that, BroMAP attempts reduction of the problem size within each node through DEE and elimination by energy lower bounds from approximate maximurn-a-posteriori (MAP) estimation. The lower bounds are also exploited in branching and subproblem selection for fast discovery of strong upper bounds. Our computational results show that BroMAP tends to be faster than DEE/A* for large protein design cases. BroMAP also solved cases that were not solvable by DEE/A* within the maximum allowed time, and did not incur significant disadvantage for cases where DEE/A* performed well. In the second part of the thesis, we explore several ways of improving the energy lower bounds by using Lagrangian relaxation. Through computational experiments, solving the dual problem derived from cyclic subgraphs, such as triplets, is shown to produce stronger lower bounds than using the tree-reweighted max-product algorithm.(cont.) In the second approach, the Lagrangian relaxation is tightened through addition of violated valid inequalities. Finally, we suggest a way of computing individual lower bounds using the dual method. The preliminary results from evaluating BroMAP employing the dual bounds suggest that the use of the strengthened bounds does not in general improve the running time of BroMAP due to the longer running time of the dual method.by Eun-Jong Hong.Ph.D

A New Lagrangian Based Branch And Bound Algorithm For The 0-1 Knapsack Problem

This paper describes a new Branch and Bound algorithm for the 0-1 Knapsack Problem (KP). The algorithm is based on the use of a Lagrangean Relax-and-Cut procedure that allows exponentially many Fractional Gomory Cuts and Extended Cover Inequalities to be candidates to Lagrangean dualization. In doing so, the upper bounds thus obtained are stronger than the standard Linear Programming relaxation bound for KP. The algorithm is aimed at solving instances with coefficients as large as 1015, a class of KP instance for which existing solution algorithms might not be directly applicable. © 2010 Elsevier B.V.36C623630Guignard, M., Efficient cuts in Lagrangean Relax and Cut Schemes (1998) European Journal on Operational Research, 105 (1), pp. 216-223Lucena, A., Lagrangian Relax-and-cut algorithms (2005) Handbook on Telecommunications, pp. 129-146. , Kluwer Academic, Boston, MA, P. Pardalos, M.G.C. Resende (Eds.)Lucena, A., Non Delayed Relax-and-Cut Algorithms (2005) Annals of Operations Research, 140, pp. 375-410Martello, S., Pisinger, D., Toth, P., Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem (1997) Management Science, 45, pp. 414-424Martello, S., Toth, P., (1990) Knapsack Problems: Algorithms and Computer Implementations, , Wiley IntersciencePisinger, D., A hybrid method for the 0 -1 Knapsack Problem (1985) Methods of Operations Research, 49, pp. 277-329Pisinger, D., Where are the hard knapsack problems? (2005) Computers and Operations Research, 32, pp. 2271-228