Branch and Bound Experiments in Convex Nonlinear Integer Programming

Gupta, Omprakash K.; Ravindran, A. Ravi

doi:10.1287/mnsc.31.12.1533

Cited by 409 publications

(212 citation statements)

References 10 publications

Supporting

Mentioning

198

Contrasting

Unclassified

Order By: Relevance

“…Finally, the third dice has also 58.3% probability of beating the first one. The linear GDP formulation of this problem is shown in (11), and it is formulated as minimization of the number of outcomes in which dice n + 1 beats dice n (instead of maximizing the number of outcomes in which dice n beats dice n + 1).…”

Section: Design Of Nontransitive Dice (Dice)mentioning

confidence: 99%

See 1 more Smart Citation

Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

Trespalacios

Grossmann

2015

INFORMS Journal on Computing

View full text Add to dashboard Cite

In this work, we propose an algorithmic approach to improve mixed-integer models that are originally formulated as convex Generalized Disjunctive Programs (GDP). The algorithm seeks to obtain an improved continuous relaxation of the MILP/MINLP reformulation of the GDP, while limiting the growth in the problem size. There are three main stages that form the basis of the algorithm. The first one is a pre-solve, consequence of the logic nature of GDP, which allows us to reduce the problem size, find good relaxation bounds and identify properties that help us determine where to apply a basic step. The second stage is the iterative application of basic steps, selecting where to apply them, and monitoring the improvement of the formulation. Finally, we use a hybrid reformulation of GDP that seeks to exploit both of the advantages attributed to the two common GDP-to-MILP/MINLP transformations, the Big-M and Hull reformulation. We illustrate the application of this algorithm with several examples. The results show the improvement in the problem formulations by generating models with improved relaxed solutions and relatively small growth in the number of continuous variables and constraints. The algorithm generally leads to reduction in the solution times.

show abstract

Section: Design Of Nontransitive Dice (Dice)mentioning

confidence: 99%

“…Problem (12) has two main difference with (11). The first one is that the second disjunction of (11) is formulated as an MILP constraint, the third global constraint in (12).…”

Section: Design Of Nontransitive Dice Hybrid Gdp-minlp (Diceh)mentioning

confidence: 99%

Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

Trespalacios

Grossmann

2015

INFORMS Journal on Computing

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

“…Indeed, this approach was experimented with in [63]. The Branch-andBound Algorithm for mixed-integer nonlinear programming has been implemented as MINLP-BB [88], with continuous nonlinear-programming subproblem relax-ations solved with the active-set solver filterSQP and also as SBB, with associated subproblems solved with any of CONOPT, SNOPT and MINOS.…”

Section: General Algorithmsmentioning

confidence: 99%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

View full text Add to dashboard Cite

Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems.It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

show abstract

Branch and Bound Experiments in Convex Nonlinear Integer Programming

Cited by 409 publications

References 10 publications

Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs

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

Nonlinear Integer Programming

Contact Info

Product

Resources

About