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Numerical experience with lower bounds for MIQP branchandbound
, 1995
"... The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduc ..."
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Cited by 47 (0 self)
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The solution of convex Mixed Integer Quadratic Programming (MIQP) problems with a general branchandbound framework is considered. It is shown how lower bounds can be computed efficiently during the branchandbound process. Improved lower bounds such as the ones derived in this paper can reduce the number of QP problems that have to be solved. The branchandbound approach is also shown to be superior to other approaches to solving MIQP problems. Numerical experience is presented which supports these conclusions. Key words : Integer Programming, Mixed Integer Quadratic Programming, BranchandBound AMS subject classification: 90C10, 90C11, 90C20 1 Introduction One of the most successful methods for solving mixedinteger nonlinear problems is branchandbound. Land and Doig [16] first introduced a branchandbound algorithm for the travelling salesman problem. Dakin [3] introduced the now common branching dichotomy and was the first to realize that it is possible to so...
Filter Pattern Search Algorithms for Mixed Variable Constrained Optimization Problems
 SIAM Journal on Optimization
, 2004
"... A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for gene ..."
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Cited by 36 (8 self)
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A new class of algorithms for solving nonlinearly constrained mixed variable optimization problems is presented. This class combines and extends the AudetDennis Generalized Pattern Search (GPS) algorithms for bound constrained mixed variable optimization, and their GPSfilter algorithms for general nonlinear constraints. In generalizing existing algorithms, new theoretical convergence results are presented that reduce seamlessly to existing results for more specific classes of problems. While no local continuity or smoothness assumptions are required to apply the algorithm, a hierarchy of theoretical convergence results based on the Clarke calculus is given, in which local smoothness dictate what can be proved about certain limit points generated by the algorithm. To demonstrate the usefulness of the algorithm, the algorithm is applied to the design of a loadbearing thermal insulation system. We believe this is the first algorithm with provable convergence results to directly target this class of problems.
Integrating SQP and branchandbound for Mixed Integer Nonlinear Programming
 Computational Optimization and Applications
, 1998
"... This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving ..."
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Cited by 24 (0 self)
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This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems. In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branchandbound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver. A numerical comparison of the new method with nonlinear branchandbound is presented and a factor of about 3 improvement over branchandbound is observed...
New Algorithms for Nonlinear Generalized Disjunctive Programming
 Computers and Chemical Engineering Journal
, 2000
"... Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the disc ..."
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Cited by 22 (17 self)
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Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative model to MINLP for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing discrete decisions in the continuous space with disjunctions, and constraints in the discrete space with logic propositions. In this paper, we describe a new convex nonlinear relaxation of the nonlinear GDP problem that relies on the use of the convex hull of each of the disjunctions involving nonlinear inequalities. The proposed nonlinear relaxation is used to reformulate the GDP problem as a tight MINLP problem, and for deriving a branch and bound method. Properties of these methods are given, and the relation of this method with the Logic Based OuterApproximation method is established. Numerical results are presented for problems in jobshop scheduling, synthesis of process networks, optimal positioning of new products and batch process design.
A lifted linear programming branchandbound algorithm for mixed integer conic quadratic programs
, 2007
"... This paper develops a linear programming based branchandbound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by BenTal and Nemirovski. The algorithm is different from o ..."
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Cited by 16 (0 self)
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This paper develops a linear programming based branchandbound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by BenTal and Nemirovski. The algorithm is different from other linear programming based branchandbound algorithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations. Key words: nonlinear integer programming; branch and bound; portfolio optimization History: February 2007. 1.
Global Optimization of MINLP Problems in Process Synthesis and Design
 Computers & Chemical Engineering
, 1997
"... : Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of ..."
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Cited by 15 (6 self)
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: Two new methodologies for the global optimization of MINLP models, the Special structure Mixed Integer Nonlinear ffBB, SMINffBB, and the General structure Mixed Integer Nonlinear ffBB, GMINffBB, are presented. Their theoretical foundations provide guarantees that the global optimum solution of MINLPs involving twicedifferentiable nonconvex functions in the continuous variables can be identified. The conditions imposed on the functionality of the binary variables differ for each method : linear and mixed bilinear terms can be treated with the SMINffBB; mixed nonlinear terms whose continuous relaxation is twicedifferentiable are handled by the GMINffBB. While both algorithms use the concept of a branch & bound tree, they rely on fundamentally different bounding and branching strategies. In the GMINffBB algorithm, lower (upper) bounds at each node result from the solution of convex (nonconvex) MINLPs derived from the original problem. The construction of convex lower bound...
Global Optimization of MixedInteger Nonlinear Problems
 AIChE J
"... Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities ..."
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Cited by 14 (2 self)
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Two novel deterministic global optimization algorithms for nonconvex mixedinteger problems (MINLPs) are proposed, using the advances of the ffBB algorithm for nonconvex NLPs Adjiman et al. (1998a). The Special Structure MixedInteger ffBB algorithm (SMINffBB addresses problems with nonconvexities in the continuous variables and linear and mixedbilinear participation of the binary variables. The General Structure MixedInteger ffBB algorithm (GMINffBB), is applicable to a very general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are developed using the concepts of branchandbound, but they differ in their approach to each of the required steps. The SMINffBB algorithm is based on the convex underestimation of the continuous functions while the GMINffBB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval based variable bound updates to enhance effici...
Nonlinear integer programming
 DISC. OPTIM
, 2009
"... 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 chapt ..."
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Cited by 13 (5 self)
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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
Algorithms and software for convex mixed integer nonlinear programs, IMA Volumes
"... Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have ..."
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Cited by 11 (2 self)
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Abstract. This paper provides a survey of recent progress and software for solving convex mixed integer nonlinear programs (MINLP)s, where 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 recent years. By exploiting analogies to wellknown techniques for solving mixed integer linear programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems. Key words. Mixed Integer Nonlinear Programming; Branch and Bound; AMS(MOS) subject classifications.
A Computational Comparison of Branch and Bound and Outer Approximation Algorithms for 01 Mixed Integer Nonlinear Programs
 Computers and Operations Research
, 1996
"... this paper we compare the performance of two codes for convex 01 mixed integer nonlinear programs on a number of test problems. The first code uses a branch and bound algorithm. The second code is a commercially available implementation of an outer approximation algorithm. The comparison demons ..."
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Cited by 10 (2 self)
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this paper we compare the performance of two codes for convex 01 mixed integer nonlinear programs on a number of test problems. The first code uses a branch and bound algorithm. The second code is a commercially available implementation of an outer approximation algorithm. The comparison demonstrates that both approaches are generally capable of solving the test problems. However, there are significant differences in the robustness of the two codes and their performance on different classes of problems