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23
Reformulations in Mathematical Programming: A Computational Approach
"... 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 ex ..."
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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.
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 21 (17 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.
Automated hierarchy discovery for planning in partially observable domains
 Advances in Neural Information Processing Systems 19
, 2006
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ..."
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Cited by 12 (2 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.
Disjunctive Cuts for NonConvex Mixed Integer Quadratically Constrained Problems
 EDS., LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of nonconvexities: integer variables and nonconvex quadratic constraints. To produce s ..."
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Cited by 9 (1 self)
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This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of nonconvexities: integer variables and nonconvex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the liftandproject methodology. In particular, we propose new methods for generating valid inequalities by using the equation Y = xx T. We use the concave constraint 0 � Y − xx T to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y −xx T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint Y − xx T � 0 to derive convex quadratic cuts and combine both approaches in a cutting plane algorithm. We present preliminary computational results to illustrate our findings.
Practical MixedInteger Optimization for Geometry Processing
"... Abstract. Solving mixedinteger problems, i.e., optimization problems where some of the unknowns are continuous while others are discrete, is NPhard. Unfortunately, realworld problems like e.g., quadrangular remeshing usually have a large number of unknowns such that exact methods become unfeasibl ..."
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Abstract. Solving mixedinteger problems, i.e., optimization problems where some of the unknowns are continuous while others are discrete, is NPhard. Unfortunately, realworld problems like e.g., quadrangular remeshing usually have a large number of unknowns such that exact methods become unfeasible. In this article we present a greedy strategy to rapidly approximate the solution of large quadratic mixedinteger problems within a practically sufficient accuracy. The algorithm, which is freely available as an open source library implemented in C++, determines the values of the discrete variables by successively solving relaxed problems. Additionally the specification of arbitrary linear equality constraints which typically arise as side conditions of the optimization problem is possible. The performance of the base algorithm is strongly improved by two novel extensions which are (1) simultaneously estimating sets of discrete variables which do not interfere and (2) a fillin reducing reordering of the constraints. Exemplarily the solver is applied to the problem of quadrilateral surface remeshing, enabling a great flexibility by supporting different types of user guidance within a realtime modeling framework for input surfaces of moderate complexity. Keywords: MixedInteger Optimization, Constrained Optimization 1
LaGO  a (heuristic) Branch and Cut algorithm for nonconvex MINLPs
 Preprint, Institut für Mathematik, HumboldtUniversität zu Berlin (ISSN 08630976
"... We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixedinteger nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation tech ..."
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We present a Branch and Cut algorithm of the software package LaGO to solve nonconvex mixedinteger nonlinear programs (MINLPs). A linear outer approximation is constructed from a convex relaxation of the problem. Since we do not require an algebraic representation of the problem, reformulation techniques for the construction of the convex relaxation cannot be applied, and we are restricted to sampling techniques in case of nonquadratic nonconvex functions. The linear relaxation is further improved by mixedintegerrounding cuts. Also box reduction techniques are applied to improve efficiency. Numerical results on medium size test problems are presented to show the efficiency of the method.
Solving Planning and Design Problems in the Process Industry Using Mixed Integer and Global Optimization
, 2004
"... This contribution gives an overview on the stateoftheart and recent advances in mixed integer optimization to solve planning and design problems in the process industry. In some case studies specific aspects are stressed and the typical difficulties of real world problems are addressed. Mixed in ..."
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Cited by 6 (0 self)
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This contribution gives an overview on the stateoftheart and recent advances in mixed integer optimization to solve planning and design problems in the process industry. In some case studies specific aspects are stressed and the typical difficulties of real world problems are addressed. Mixed integer linear optimization is widely used to solve supply chain planning problems. Some of the complicating features such as origin tracing and shelf life constraints are discussed in more detail. If properly done the planning models can also be used to do product and customer portfolio analysis. We also stress the importance of multicriteria optimization and correct modeling for optimization under uncertainty. Stochastic programming for continuous LP problems is now part of most optimization packages, and there is encouraging progress in the field of stochastic MILP and robust MILP. Process and network design problems often lead to nonconvex mixed integer nonlinear programming models. If the time to compute the solution is not bounded, there are already a commercial solvers available which can compute the global optima of such problems within hours. If time is more restricted, then tailored solution techniques are required.
On the Impact of Power Allocation on Coalition Formation in Cooperative Wireless Networks
 IEEE 8th International Conference on Wireless and Mobile Computing, Networking and Communications
, 2012
"... Abstract—In this paper, the impact of cooperative power allocation on distributed altruistic coalition formation in cooperative relay networks is studied. Particularly, equal power allocation (EPA), maxmin rate (MMR) and sumofrates maximizing (SRM) power allocation criteria are considered. A dist ..."
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Abstract—In this paper, the impact of cooperative power allocation on distributed altruistic coalition formation in cooperative relay networks is studied. Particularly, equal power allocation (EPA), maxmin rate (MMR) and sumofrates maximizing (SRM) power allocation criteria are considered. A distributed mergeandsplit algorithm is proposed to allow network nodes to form coalitions and improve their total achievable rate. The proposed algorithm is compared with that of centralized power control and coalition formation, and is shown to yield a good tradeoff between network sumrate and computational complexity. Finally, numerical results illustrate that the SRM power allocation criterion promotes altruistic coalition formation and results in the largest coalitions among the different power allocation criteria. Index Terms—Coalition formation, cooperation, decodeandforward (DF), network coding, power allocation I.
Optimizing the design of complex energy conversion systems by Branch and Cut
, 2010
"... The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combinedcyclebased cogeneration plant. The optimization problem is formulated as a nonconvex mixedinteger nonlinear problem (MINLP) and s ..."
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The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combinedcyclebased cogeneration plant. The optimization problem is formulated as a nonconvex mixedinteger nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The algorithm generates a convex relaxation of the MINLP and applies a Branch and Cut algorithm to the relaxation. Numerical results for different demands for electric power and process steam are discussed and a sensitivity analysis is performed. 1
Experiments with a feasibility pump approach for nonconvex MINLPs
 Symposium on Experimental Algorithms, volume 6049 of LNCS
, 2010
"... Abstract. We present a new Feasibility Pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programming problems. Differences with the previously proposed Feasibility Pump algorithms and difficulties arising from nonconvexities in the models are extensively discussed. The main methodologica ..."
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Abstract. We present a new Feasibility Pump algorithm tailored for nonconvex Mixed Integer Nonlinear Programming problems. Differences with the previously proposed Feasibility Pump algorithms and difficulties arising from nonconvexities in the models are extensively discussed. The main methodological innovations of this variant are: (a) the first subproblem is a nonconvex continuous Nonlinear Program, which is solved using global optimization techniques; (b) the solution method for the second subproblem is complemented by a tabu list. We exhibit computational results showing the good performance of the algorithm on instances taken from the MINLPLib. 1