A Disjunctive Cutting-Plane-Based Branch-and-Cut Algorithm for 0−1 Mixed-Integer Convex Nonlinear Programs

Abstract

In this paper, a disjunctive cutting-plane-based branch-and-cut algorithm is developed to solve the 0-1 mixedinteger convex nonlinear programming (MINLP) problems. In a branch-and-bound framework, the 0-1 MINLP problem is approximated with a 0-1 mixed-integer linear program at each node, and then the lift-and-project technology is used to generate valid cuts to accelerate the branching process. The cut is produced by solving a linear program that is transformed from a projection problem, in terms of the disjunction on a free binary variable, and its dual solutions are applied to lift the cut to become valid throughout the enumeration tree. A strengthening process is derived to improve the coefficients of the cut by imposing integrality on the left free binary variables. Finally, the computational results on four test problems indicate that the added cutting planes can reduce the branching process greatly and show that the proposed algorithm is very promising for largescale 0-1 MINLP problems, because a linear program is always computationally less expensive than a nonlinear program.

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

“…No such extension is known in the case of MINLP. Zhu and Kuno [114] have suggested to replace the true nonlinear convex hull by a linear approximation taken about the solution to a linearized master problem like MP(K).…”

A cutting-plane method for Mixed-Logical Semidefinite Programs with an application to multi-vehicle robust path planning

“…There are specialized cut generation schemes for Mixed 0-1 SDPs [15], [16], but these methods are also based on the convex hull approach mentioned above and therefore inherit its limitations. The cut-generation itself is even more computationally intensive than in the MILP case.…”

“…We require the control inputs x to be chosen in such a way that the state constraints (10), (13a) and (15a) hold robustly for all ζ ∈ Z and ξ ∈ Ξ when the functions in (16) are substituted for the vehicle positions and velocities. Similarly, we aim at minimizing the objective criterion (9) in view of the worstcase realization of ζ ∈ Z and ξ ∈ Ξ when the functions in (16) are substituted for the vehicle positions and velocities. By using now standard techniques due to Löfberg [22] and El Ghaoui et al [23], the resulting robust control problem over x and y can be reformulated as an MISDP of the type (2).…”

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

“…But, the disjunctive cutting plane developed by Stubbs and Mehrotra19 is obtained by solving a convex projection problem, which is computationally expensive and always encounters nondifferential problems. Recently, Zhu and Kuno23 developed a new disjunctive cut generation approach where the disjunctive cut is produced by solving a linear programming problem rather than a nonlinear programming problem. The critical finding of Zhu and Kuno23 approach is that a lift‐and‐project cutting plane that is valid for the original MINLP problem and cuts the fractional solution away can be generated based on the polyhedral outer approximation of the mixed‐integer nonlinear set of the original MINLP problem at the current node.…”

“…Recently, Zhu and Kuno23 developed a new disjunctive cut generation approach where the disjunctive cut is produced by solving a linear programming problem rather than a nonlinear programming problem. The critical finding of Zhu and Kuno23 approach is that a lift‐and‐project cutting plane that is valid for the original MINLP problem and cuts the fractional solution away can be generated based on the polyhedral outer approximation of the mixed‐integer nonlinear set of the original MINLP problem at the current node. In this article, the cut generating, lifting, and strengthening approach proposed by Balas et al20 for MILP problem is used to replace the cut generating linear programming in our former work, and the specific implementation issues are discussed and incorporated into the algorithmic development to handle large‐scale MINLP problems.…”