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The Computational Infrastructure for Operations Research (COIN-OR) branchand-cut solver (CBC) is an open-source mixed-integer program (MIP) solver. The performance of branch-and-cut algorithms can vary greatly with problem-specific customization, such as dictating that the order nodes in the search tree are traversed. CBC provides operations research professionals with a well-tested, robust, reusable code base for experimenting with advanced customizations of branch-and-cut algorithms. The CBC design makes the most commonly desired customizations readily possible: (a) dynamically selecting the next node in the search tree for processing, (b) using specialized criteria for determining which variable(s) to branch on, (c) calling tailormade heuristics to generate MIP-feasible solutions quickly, (d) including standard (or user-provided) cut generation in solving the linear program (LP) relaxations of the MIP, and (e) invoking customized subproblem solvers. CBC is written in C++ and is intended to be used primarily as a callable library. CBC requires a linear program (LP) solver. CBC uses the COIN-OR open solver interface (OSI) to communicate with the user's choice of LP solver. CBC can use any LP solver with an OSI. The LP solver expected to be used most commonly is the freely available COIN-OR LP solver (CLP). CBC can be used as a branch-and-bound solver or as a branch-and-cut solver. For cut generators, CBC relies on the COIN-OR Cut Generation Library (CGL). CBC can use any cut generator written to CGL standards. CBC is an active open-source project led by John Forrest. The full CBC source code is available under the Common Public License for industrial and academic use at www.coin-or.org. This chapter introduces CBC and illustrates how to implement a variety of common branch-and-cut customizations in CBC. The chapter assumes familiarity with C++, fundamentals of mixed-integer programming, and basic knowledge of OSI. Keywords software; branch and bound; cutting planes; mixed-integer programming; open-source software * C International Business Machines Corporation 2005. Reproduced by permission of International Business Machines, Armonk, NY. 1 The complete acronym is "COIN-OR" which stands for the Compuational Infrastructure for Operations Research. For simplicity (and in keeping with the directory and function names) we will simply use "COIN.

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We study a conditional logic approach for tightening the continuous relaxation of a mixed 0-1 linear program. The procedure first constructs quadratic inequalities by computing pairwise products of constraints, and then surrogates modified such inequalities to produce valid linear restrictions. Strength is achieved by adjusting the coefficients on the quadratic restrictions. The approach is a unifying framework for published coefficient adjustment methods, and generalizes the process of sequential lifting. We give illustrative examples and discuss various extensions, including the use of more complex conditional logic constructs that compute and surrogate polynomial expressions, and the application to general integer programs.

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