An improved branch‐and‐cut algorithm for mixed‐integer nonlinear systems optimization problem

Abstract

in Wiley InterScience (www.interscience.wiley.com).Recent advances for global optimization and dynamic optimization of the mixed-integer systems have created an increasing demand for efficient and robust numerical algorithms for mixed-integer nonlinear programming (MINLP) problem. In this article, an improved branch-and-cut algorithm for 0-1 MINLP problems has been proposed by using our former critical finding (Zhu and Kuno, Ind Eng Chem Res. 2006;45:187-196) of the disjunctive cutting plane for 0-1 MINLP. By virtue of the polyhedral outer approximation of the mixed-integer nonlinear set of the original MINLP problem at each enumeration node, a lift-and-project cutting planes that is valid for the original MINLP problem and cuts the fractional solution away can be generated by using the cut generating, lifting, and strengthening approach for MILP problem of Balas et al. (Math Program. 1993;58:295-324). The specific implementation issues of the cutting planes are discussed and incorporated into the algorithmic development of a branchand-cut algorithm. The efficiency of the improved disjunctive cutting plane is demonstrated by the computational results for 11 systems optimization problems, and it is implied that the proposed branch-and-cut algorithm is very promising for practical MINLP problems as the interior-point solvers for nonlinear programming problems with many constraints are becoming mature gradually.

Disjunctive Inequalities: Applications And Extensions

“…Zhu et al [55] proposed a simple extension to MINLP where the CGLP is solved for disjunctions on binary variables.…”

Branch-and-Cut Algorithmic Framework for 0−1 Mixed-Integer Convex Nonlinear Programs

“…The cut can be produced in a reduced space because some binary variables are fixed at either their lower or upper bounds and can be eliminated before the CGLP is constructed; the detailed procedure is presented in our previous work. 5 Figure 2 describes the procedure of the branch-and-cut algorithm in pseudocode.…”

“…Mixed-integer optimization provides a powerful framework for mathematically modeling many process optimization problems involving both discrete and continuous variables. It has achieved wide and successful applications in process systems engineering, such as in process or product design problems and in batch-process scheduling problems. – Whereas mixed-integer linear programming (MILP) methods and codes have been available for more than 30 years, the number of methods and computer codes for solving mixed-integer nonlinear programming (MINLP) problems is still rather limited. SBB solves MINLP problems by implementing a branch-and-bound framework in which a sequential quadratic programming (SQP) algorithm is used to obtain the solution to the continuous relaxation NLP subproblem.…”

“…After observing that a valid cutting plane can improve the bounding property of a branch-and-bound algorithm, Zhu and Kuno proposed a branch-and-cut algorithm for 0−1 mixed-integer nonlinear programs, where the disjunctive cut is produced by solving a cut generation linear program (CGLP) based on the polyhedral approximation of the mixed-integer nonlinear set of the MINLP problem. Recently, Zhu et al improved this algorithm further by replacing the cut generation approach with the CGLP proposed by Balas et al for MILP problems. – However, most of the test problems used have been very small, and the optimal settings of the algorithmic parameters have not yet been exploited. In this article, the effectiveness of this improved branch-and-cut algorithm is evaluated systematically by larger operations research and chemical engineering problems.…”