An improved branch and bound algorithm for mixed integer nonlinear programs

Borchers, Brian; Mitchell, John E.

doi:10.1016/0305-0548(94)90024-8

Cited by 190 publications

(96 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Gupta and Ravindran [63] suggested an implementation of the branch-and-bound method for convex MINLPs and investigated different search strategies. Other early works related to NLP-Based branch-and-bound (NLP-BB for short) for convex MINLP include [92], [28], and [79].…”

Section: Nlp-based Branch and Boundmentioning

confidence: 99%

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Bonami

Kılınç

Linderoth

2011

The IMA Volumes in Mathematics and Its Applications

195

181

View full text Add to dashboard Cite

Abstract. This paper provides a survey of recent progress and software for solving mixed integer nonlinear programs (MINLP) wherein the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in very years. By exploiting analogies to the case of well-known techniques for solving mixed integer linear programs and incorporating these techniques into the software, significant improvements have been made in our ability to solve the problems.Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.1. Introduction. Mixed-Integer Nonlinear Programs (MINLP) are optimization problems where some of the variables are constrained to take integer values and the objective function and feasible region of the problem are described by nonlinear functions. Such optimization problems arise in many real world applications. Integer variables are often required to model logical relationships, fixed charges, piecewise linear functions, disjunctive constraints and the non-divisibility of resources. Nonlinear functions are required to accurately reflect physical properties, covariance, and economies of scale.In all its generality, MINLP forms a particularly broad class of challenging optimization problems as it combines the difficulty of optimizing over integer variables with handling of nonlinear functions. Even if we restrict our model to contain only linear functions, MINLP reduces to a Mixed-Integer Linear Program (MILP), which is an NP-Hard problem [56]. On the other hand, if we restrict our model to have no integer variable but allow for general nonlinear functions in the objective or the constraints, then MINLP reduces to a Nonlinear Program (NLP) which is also known to be NP-Hard [91]. Combining both integrality and nonlinearity can lead to examples of MINLP that are undecidable [68].

show abstract

Section: Nlp-based Branch and Boundmentioning

confidence: 99%

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Bonami

Kılınç

Linderoth

2011

The IMA Volumes in Mathematics and Its Applications

195

181

View full text Add to dashboard Cite

show abstract

“…Solution strategies explicitly addressing mixed-integer nonlinear programs (MINLPs) have appeared rather sporadically in the literature and initially dealt mostly with problems that are convex when integrality restrictions are dropped and/or contain nonlinearities that are quadratic [24,22,16,23,10,6,7,5]. A few recent works have indicated that the application of deterministic branchand-bound algorithms to the global optimization of general classes of MINLPs is promising [30,37,12,36].…”

Section: Introductionmentioning

confidence: 99%

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Tawarmalani

Sahinidis²

2004

Mathematical Programming

473

285

View full text Add to dashboard Cite

This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm.In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems.In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0 − 1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

show abstract

“…The nonlinear branch and bound [29][30][31][32][33] is an extension to the well known linear branch and bound [5]. To find optimality, the method performs an extensive tree search on the integer variables.…”

Section: Nonlinear Branch and Boundmentioning

confidence: 99%

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

Trespalacios

Grossmann

2014

Chemie Ingenieur Technik

152

View full text Add to dashboard Cite

This work presents a review of the applications of mixed-integer nonlinear programming (MINLP) in process systems engineering (PSE). A review on the main deterministic MINLP solution methods is presented, including an overview of the main MINLP solvers. Generalized disjunctive programming (GDP) is an alternative higher-level representation of MINLP problems. This work reviews some methods for solving GDP models, and techniques for improving MINLP methods through GDP. The paper also provides a high-level review of the applications of MINLP in PSE, particularly in process synthesis, planning and scheduling, process control and molecular computing.

show abstract

An improved branch and bound algorithm for mixed integer nonlinear programs

Cited by 190 publications

References 15 publications

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Algorithms and Software for Convex Mixed Integer Nonlinear Programs

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Review of Mixed‐Integer Nonlinear and Generalized Disjunctive Programming Methods

Contact Info

Product

Resources

About