Global Solution Approach for a Nonconvex MINLP Problem in Product Portfolio Optimization XIAOXIA LIN 1 , CHRISTODOULOS A. FLOUDAS 1 and JOSEF KALLRATH 2;3 1 Department of Chemical Engineering, Princeton University, Princeton, NJ 08544-5263, USA (e-mail: floudas@titan.princeton.edu) 2 BASF-AG, GVC/S(Scientific Computing)-B009, D-67056 Ludwigshafen, Germany 3 Department of Astronomy, University of Florida, Gainesville, FL 32661, USA (Received 1 April 2003; accepted in revised form 17 May 2004) Abstract. The rigorous and efficient determination of the global solution of a nonconvex MINLP problem arising from product portfolio optimization introduced by Kallrath (2003) is addressed. The objective of the optimization problem is to determine the optimal number and capacity of reactors satisfying the demand and leading to a minimal total cost. Based on the model developed by Kallrath (2003), an improved formulation is proposed, which consists of a concave objective function and linear constraints with binary and continuous variables. A variety of techniques are developed to tighten the model and accelerate the convergence to the optimal solution. A customized branch and bound approach that exploits the special math- ematical structure is proposed to solve the model to global optimality. Computational results for two case studies are presented. In both case studies, the global solutions are obtained and proved optimal very efficiently in contrast to available commercial MINLP solvers. Key words: Branch and bound, Concave objective function, Global optimization, Mixed- integer nonlinear programming (MINLP), Piece-wise linear underestimator, Portfolio opti- mization 1. Introduction The modeling of decision making in many processes, such as the design of chemical plants, often leads to nonconvex mixed-integer nonlinear pro- gramming (MINLP) problems. The solution of this class of problems is very challenging due to the presence of both the integer variables and the nonconvexities. A number of approaches have been proposed for the solu- tion of such problems within the branch and bound framework. For exam- ple, Adjiman et al. (2000) introduced a powerful theoretical and algorithmic framework based on the aBB global optimization approach for twice-differentiable nonlinear programming (NLP) problems (Adjiman et al., 1998). Adjiman et al. (2000) developed two broadly applicable algorithms for the solution of nonconvex MINLPs: a special structure mixed-integer aBB algorithm (SMIM-aBB) for problems with general non- convexities in the continuous variables and restricted participation of the Journal of Global Optimization (2005) 32: 417–431 Ó Springer 2005 DOI 10.1007/s10898-004-5903-5