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Interiorpoint methods for nonconvex nonlinear programming: Filter methods and merit functions
 Computational Optimization and Applications
, 2002
"... Abstract. In this paper, we present global and local convergence results for an interiorpoint method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound ..."
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Abstract. In this paper, we present global and local convergence results for an interiorpoint method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interiorpoint method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Qquadratically for a large class of problems. The proposed approach is the first to simultaneously have all of the following properties while solving a general nonconvex nonlinear programming problem: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the KarushKuhnTucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. Numerical testing on a set of general nonlinear programming problems, including degenerate problems and infeasible problems, confirm the theoretical results. We also provide comparisons to a highlyefficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1.
Mathematical programming techniques for designing minimum cost pipeline networks for CO2 sequestration
"... It has been proposed that the CO2 produced at sources such as fossil energy conversion facilities and industrial process plants could be captured, compressed to supercritical pressures, transported via pipelines and stored in underground geologic formations such as depleted oil and natural gas reser ..."
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It has been proposed that the CO2 produced at sources such as fossil energy conversion facilities and industrial process plants could be captured, compressed to supercritical pressures, transported via pipelines and stored in underground geologic formations such as depleted oil and natural gas reservoirs and deep saline aquifers. In this paper, we describe the initial phases of a project using mathematical programming techniques to find the minimum cost network for transporting the CO2 from the sources to the sinks. A nonlinear model of the CO2 pipeline system is described, with appropriate constraints, and the lowest cost system is found using a stateoftheart nonlinear optimization software called LOQO. We also present ways to model the behavior of the system over time, and to model the impact of uncertainties. In future work, we plan to extend these methods to look at the larger system of fossil energy complexes with CO2 capture and sequestration and distribution of hydrogen.
Global convergence of a primaldual interiorpoint method for nonlinear programming ∗
"... Many recent convergence results obtained for primaldual interiorpoint methods for nonlinear programming, use assumptions of the boundedness of generated iterates. In this paper we replace such assumptions by new assumptions on the NLP problem, develop a modification of a primaldual interiorpoint ..."
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Many recent convergence results obtained for primaldual interiorpoint methods for nonlinear programming, use assumptions of the boundedness of generated iterates. In this paper we replace such assumptions by new assumptions on the NLP problem, develop a modification of a primaldual interiorpoint method implemented in software package loqo and analyze convergence of the new method from any initial guess. Keywords. Interiorpoint method, primaldual, convergence analysis. 1
Appendix: Details on Experiments (Counting . . .
, 2009
"... Here we present details for experiments with the Gaussian estimation of Barvinok and Hartigan. As described in the article, we needed the maximum objective values and the unique optimal solutions of the strictly concave optimization problems from Theorems 1 and 2 in order to calculate the Gaussian e ..."
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Here we present details for experiments with the Gaussian estimation of Barvinok and Hartigan. As described in the article, we needed the maximum objective values and the unique optimal solutions of the strictly concave optimization problems from Theorems 1 and 2 in order to calculate the Gaussian estimations. These values were obtained using LOQO and the rest of the calculation was implemented inside MAPLE version 12. There is no need for any sophisticated computation, one can easily calculate the covariance matrices and the lattice index using a Smith normal form calculation. Overall the evaluation step takes a negligible amount of time in all instances, so we do not record any time of computation.
programming
, 2005
"... Convergence analysis of a primaldual interiorpoint method for nonlinear ..."
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Convergence analysis of a primaldual interiorpoint method for nonlinear