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41
Cones Of Matrices And Successive Convex Relaxations Of Nonconvex Sets
, 2000
"... . Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each ..."
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Cited by 49 (20 self)
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. Let F be a compact subset of the ndimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2, . . . ) of R n such that (a) the convex hull of F # C k+1 # C k (monotonicity), (b) # # k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding LovaszSchrijver liftandproject procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using th...
An exact reformulation algorithm for large nonconvex NLPs involving bilinear terms
 Journal of Global Optimization
, 2005
"... Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subse ..."
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Cited by 21 (9 self)
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Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of ReformulationLinearization Technique constraints) which do not a#ect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an e#cient graphtheoretical algorithm for e#ecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branchand Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.
Reformulations in Mathematical Programming: A Computational Approach
"... Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathema ..."
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Cited by 17 (13 self)
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Summary. Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of blackbox functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization. 1
REFORMULATIONS IN MATHEMATICAL PROGRAMMING: DEFINITIONS AND SYSTEMATICS
, 2008
"... A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations c ..."
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Cited by 17 (13 self)
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A reformulation of a mathematical program is a formulation which shares some properties with, but is in some sense better than, the original program. Reformulations are important with respect to the choice and efficiency of the solution algorithms; furthermore, it is desirable that reformulations can be carried out automatically. Reformulation techniques are very common in mathematical programming but interestingly they have never been studied under a common framework. This paper attempts to move some steps in this direction. We define a framework for storing and manipulating mathematical programming formulations, give several fundamental definitions categorizing reformulations in essentially four types (optreformulations, narrowings, relaxations and approximations). We establish some theoretical results and give reformulation examples for each type.
Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta mathematica vietnamica
"... Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. In this paper, we present some general as well as explicit characterizations of the convex envelope of multilinear functions defined over a unit hypercube. A new approach is used to derive this characterization via a related ..."
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Cited by 15 (1 self)
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Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. In this paper, we present some general as well as explicit characterizations of the convex envelope of multilinear functions defined over a unit hypercube. A new approach is used to derive this characterization via a related convex hull representation obtained by applying the ReformulationLinearization Technique (RLT) of Sherali and Adams (1990, 1994). For the special cases of multilinear functions having coefficients that are either all +1 or all −1, we develop explicit formulae for the corresponding convex envelopes. Extensions of these results are given for the case when the multilinear function is defined over discrete sets, including explicit formulae for the foregoing special cases when this discrete set is represented by generalized upper bounding (GUB) constraints in binary variables. For more general cases of multilinear functions, we also discuss how this construct can be used to generate suitable relaxations for solving nonconvex optimization problems that include such structures. 1.
Towards Implementations of Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
, 1999
"... Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically ..."
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Cited by 12 (6 self)
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Recently Kojima and Tuncel proposed new successive convex relaxation methods and their localizeddiscretized variants for general nonconvex quadratic optimization problems. Although an upper bound of the optimal objective function value within a previously given precision can be found theoretically by solving a finite number of linear programs, several important implementation issues remain unsolved. In this paper, we discuss those issues, present practically implementable algorithms and report numerical results.
Deterministic Global Optimization In Design, Control, And Computational Chemistry
 IMA Volumes in Mathematics and its Applications : Large Scale Optimization with Applications, Part II
, 1997
"... . This paper presents an overview of the deterministic global optimization approaches and their applications in the areas of Process Design, Control, and Computational Chemistry. The focus is on (i) decompositionbased primal dual methods, (ii) methods for generalized geometric programming problems, ..."
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Cited by 10 (7 self)
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. This paper presents an overview of the deterministic global optimization approaches and their applications in the areas of Process Design, Control, and Computational Chemistry. The focus is on (i) decompositionbased primal dual methods, (ii) methods for generalized geometric programming problems, and (iii) global optimization methods for general nonlinear programming problems. The classes of mathematical problems that are addressed range from indefinite quadratic programming to concave programs, to quadratically constrained problems, to polynomials, to general twice continuously differentiable nonlinear optimization problems. For the majority of the presented methods nondistributed global optimization approaches are discussed with the exception of decompositionbased methods where a distributed global optimization approach is presented. 1. Background. A significant effort has been expended in the last five decades toward theoretical and algorithmic studies of applications that arise...
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Cited by 9 (7 self)
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
Global Optimization of Nonconvex Nonlinear Programs Using Parallel Branch and Bound
, 1995
"... A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented. The algorithm is directly suitable for a wide class of problems arising in chemical engineering design. It can solve problems defined using algebraic functions ..."
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Cited by 8 (0 self)
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A branch and bound algorithm for computing globally optimal solutions to nonconvex nonlinear programs in continuous variables is presented. The algorithm is directly suitable for a wide class of problems arising in chemical engineering design. It can solve problems defined using algebraic functions and twice differentiable transcendental functions, in which finite upper and lower bounds can be placed on each variable. The algorithm uses rectangular partitions of the variable domain and a new bounding program based on convex/concave envelopes and positive definite combinations of quadratic terms. The algorithm is deterministic and obtains convergence with final regions of finite size. The partitioning strategy uses a sensitivity analysis of the bounding program to predict the best variable to split and the split location. Two versions of the algorithm are considered, the first using a local NLP algorithm (MINOS) and the second using a sequence of lower bounding programs in the search fo...
Optimal running and planning of a biomassbased energy production process
, 2008
"... We propose ..."