Results 1  10
of
12
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 ..."
Abstract

Cited by 18 (14 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 17 (13 self)
 Add to MetaCart
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
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. ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
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. In: ”Integer programming and combinatorial optimization
 Eds., Lecture Notes in Computer Science
, 2008
"... Abstract. 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 p ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. 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. 1
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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
On Intervalsubgradient and Nogood Cuts
 OPERATIONS RESEARCH LETTERS
, 2010
"... Intervalgradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that nogood cuts ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Intervalgradient cuts are (nonlinear) valid inequalities for nonconvex NLPs defined for constraints g(x) ≤ 0 with g being continuously differentiable in a box [x, ¯x]. In this paper we define intervalsubgradient cuts, a generalization to the case of nondifferentiable g, and show that nogood cuts (which have the form ‖x−ˆx ‖ ≥ ε for some norm and positive constant ε) are a special case of intervalsubgradient cuts whenever the 1norm is used. We then briefly discuss what happens if other norms are used.
Nonlinearly Constrained MRFs: Exploring the Intrinsic Dimensions of HigherOrder Cliques
"... This paper introduces an efficient approach to integrating nonlocal statistics into the higherorder Markov Random Fields (MRFs) framework. Motivated by the observation that many nonlocal statistics (e.g., shape priors, color distributions) can usually be represented by a small number of parameter ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
This paper introduces an efficient approach to integrating nonlocal statistics into the higherorder Markov Random Fields (MRFs) framework. Motivated by the observation that many nonlocal statistics (e.g., shape priors, color distributions) can usually be represented by a small number of parameters, we reformulate the higherorder MRF model by introducing additional latent variables to represent the intrinsic dimensions of the higherorder cliques. The resulting new model, called NCMRF, not only provides the flexibility in representing the configurations of higherorder cliques, but also automatically decomposes the energy function into less coupled terms, allowing us to design an efficient algorithmic framework for maximum a posteriori (MAP) inference. Based on this novel modeling/inference framework, we achieve stateoftheart solutions to the challenging problems of classspecific image segmentation and templatebased 3D facial expression tracking, which demonstrate the potential of our approach. 1.
Accepted for publication in Computational Optimization and Applications
"... parallel interior point decomposition algorithm for block ..."