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... procedure or as a local search procedure. Some of the algorithms in this group include outer approximation (Duran and Grossmann, 1986; Fletcher and Leyffer, 1994), generalized Benders decomposition (=-=Geoffrion, 1972-=-), LP/NLP-based branch-and-bound (Quesada and Grossmann, 1992) and the extended cutting plane method (Westerlund and Pettersson, 1995; Westerlund et al., 1994). This approach is the basis for the I-OA...

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...r programming (MINLP) problems known as mixed integer conic quadratic programming problems. This class of problems arises from adding integrality requirements to conic quadratic programming problems (=-=Lobo et al., 1998-=-), and is used to model several applications from engineering and finance. Conic quadratic programming problems are also known as second order cone programming problems, and together with semidefinite...

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...ble solutions, to improve the polyhedral relaxations, to fathom nodes in a branch-and-bound procedure or as a local search procedure. Some of the algorithms in this group include outer approximation (=-=Duran and Grossmann, 1986-=-; Fletcher and Leyffer, 1994), generalized Benders decomposition (Geoffrion, 1972), LP/NLP-based branch-and-bound (Quesada and Grossmann, 1992) and the extended cutting plane method (Westerlund and Pe...

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...e nonlinear constraints. In particular, it is known that obtaining a tight polyhedral approximation of the Euclidean ball without using extra variables requires an exponential number of inequalities (=-=Ball, 1997-=-). To try to resolve this issue, current polyhedral based algorithms generate the relaxations dynamically. In the context of CP problems, an alternative polyhedral relaxation that is not based on grad...

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... of the LP based branch-and-bound procedure for mixed integer linear programming (MILP) problems and is the basis for the MICP solver in CPLEX 9.0 and 10.0 (ILOG, 2005) and the I-BB solver in Bonmin (=-=Bonami et al., 2005-=-). We refer to these algorithms as NLP based branch-and-bound algorithms. The second group uses polyhedral relaxations of the nonlinear constraints of MICPP, possibly together with the nonlinear relax...

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...he polyhedral relaxations, to fathom nodes in a branch-and-bound procedure or as a local search procedure. Some of the algorithms in this group include outer approximation (Duran and Grossmann, 1986; =-=Fletcher and Leyffer, 1994-=-), generalized Benders decomposition (Geoffrion, 1972), LP/NLP-based branch-and-bound (Quesada and Grossmann, 1992) and the extended cutting plane method (Westerlund and Pettersson, 1995; Westerlund e...

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...d Grossmann, 1986; Fletcher and Leyffer, 1994), generalized Benders decomposition (Geoffrion, 1972), LP/NLP-based branch-and-bound (Quesada and Grossmann, 1992) and the extended cutting plane method (=-=Westerlund and Pettersson, 1995-=-; Westerlund et al., 1994). This approach is the basis for the I-OA, I-QG and I-Hyb solvers in Bonmin (Bonami et al., 2005) and the MINLP solver FilMINT (Abhishek et al., 2006). We refer to these algo...

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...f the algorithms in this group include outer approximation (Duran and Grossmann, 1986; Fletcher and Leyffer, 1994), generalized Benders decomposition (Geoffrion, 1972), LP/NLP-based branch-and-bound (=-=Quesada and Grossmann, 1992-=-) and the extended cutting plane method (Westerlund and Pettersson, 1995; Westerlund et al., 1994). This approach is the basis for the I-OA, I-QG and I-Hyb solvers in Bonmin (Bonami et al., 2005) and ...

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...ami et al. (2005) and Grossmann (2002)). The first group only uses the nonlinear relaxation CPP in a branch-and-bound procedure (Borchers and Mitchell, 1994; Gupta and Ravindran, 1985; Leyffer, 2001; =-=Stubbs and Mehrotra, 1999-=-). This procedure is the direct analog of the LP based branch-and-bound procedure for mixed integer linear programming (MILP) problems and is the basis for the MICP solver in CPLEX 9.0 and 10.0 (ILOG,...

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... plane method (Westerlund and Pettersson, 1995; Westerlund et al., 1994). This approach is the basis for the I-OA, I-QG and I-Hyb solvers in Bonmin (Bonami et al., 2005) and the MINLP solver FilMINT (=-=Abhishek et al., 2006-=-). We refer to these algorithms as polyhedral relaxation based algorithms. For algorithms in the second group to perform efficiently, it is essential to have polyhedral relaxations of the nonlinear co...

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... relaxations they use (see for example Bonami et al. (2005) and Grossmann (2002)). The first group only uses the nonlinear relaxation CPP in a branch-and-bound procedure (Borchers and Mitchell, 1994; =-=Gupta and Ravindran, 1985-=-; Leyffer, 2001; Stubbs and Mehrotra, 1999). This procedure is the direct analog of the LP based branch-and-bound procedure for mixed integer linear programming (MILP) problems and is the basis for th...

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...ng on what type of continuous relaxations they use (see for example Bonami et al. (2005) and Grossmann (2002)). The first group only uses the nonlinear relaxation CPP in a branch-and-bound procedure (=-=Borchers and Mitchell, 1994-=-; Gupta and Ravindran, 1985; Leyffer, 2001; Stubbs and Mehrotra, 1999). This procedure is the direct analog of the LP based branch-and-bound procedure for mixed integer linear programming (MILP) probl...

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...raints of LPP and the integrality of the x variables. 12s4.2. Test Instances Our test set consists of three different portfolio optimization problems with cardinality constraints from the literature (=-=Ceria and Stubbs, 2006-=-; Lobo et al., 1998, 2007). For most portfolio optimization problems only the continuous variables are present in the nonlinear constraints and hence the convex hull of integer feasible solutions to t...