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A Penalty-Barrier Algorithm: SPENBAR

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Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 121))

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Abstract

Let us consider the following general nonlinear optimization problem:subject to:where x ∈ ℝ n,E ≜ {1,  … , m e } and I c  ≜ {1,  … , m}. The functions c i  : ℝ n → ℝ, i ∈ I c  ∪ E, are assumed to be twice continuously differentiable on ℝ n. I l  , I u  ⊆ {1,  … , n}. To simplify the presentation of the algorithm, the simple bounds on the variables are also denoted c i (x). Define I sb as the set of indices such that for all j ∈ I l  ∪ I u there is an i ∈ I sb with the property:

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References

  • Andrei, N. (1996a). Computational experience with a modified penalty-barrier method for large-scale nonlinear constrained optimization. (Working Paper No. AMOL-96-1, Research Institute for Informatics-ICI, Bucharest, February 6, 1996).

    Google Scholar 

  • Andrei, N. (1996b). Computational experience with a modified penalty-barrier method for large-scale nonlinear, equality and inequality constrained optimization. (Technical Paper No. AMOL-96-2, Research Institute for Informatics-ICI, Bucharest, February 12, 1996).

    Google Scholar 

  • Andrei, N. (1996c). Computational experience with “SPENBAR” a sparse variant of a modified penalty-barrier method for large-scale nonlinear, equality and inequality, constrained optimization. (Working Paper No. AMOL-96-3, Research Institute for Informatics-ICI, Bucharest, March 10, 1996).

    Google Scholar 

  • Andrei, N. (1996d). Computational experience with “SPENBAR” a sparse variant of a modified penalty-barrier method for large-scale nonlinear, equality and inequality, constrained optimization. (Technical Paper No. AMOL-96-4, Research Institute for Informatics-ICI, Bucharest, March 11, 1996).

    Google Scholar 

  • Andrei, N. (1996e). Numerical examples with “SPENBAR”for large-scale nonlinear, equality and inequality, constrained optimization with zero columns in Jacobian matrices. (Technical Paper No. AMOL-96-5, Research Institute for Informatics-ICI, Bucharest, March 29, 1996).

    Google Scholar 

  • Andrei, N. (1998a). Penalty-barrier algorithms for nonlinear optimization. Preliminary computational results. Studies in Informatics and Control, 7(1), 15–36.

    Google Scholar 

  • Andrei, N. (2003). Modele, Probleme de Test şi Aplicaţii de Programare Matematică. [Models, Test Problems and Applications for Mathematical Programming] Editura Tehnică, Bucureşti.

    Google Scholar 

  • Andrei, N. (2015). Critica Raţiunii Algoritmilor de Optimizare cu Restricţii. [Criticism of the Constrained Optimization Algorithms Reasoning]. Bucureşti, Balkans: Editura Academiei Române

    Google Scholar 

  • Ben-Tal, A., & Zibulevsky, M. (1993). Penalty-barrier multiplier methods for large-scale convex programming problems. (Research Report 6/93. Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel).

    Google Scholar 

  • Ben-Tal, A., Yuzefovich, I., & Zibulevsky, M. (1992). Penalty/barrier multiplier methods for minimax and constrained smooth convex problems. (Research Report 9/92, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel).

    Google Scholar 

  • Bertsekas, D. P. (1982b). Constrained optimization and lagrange multiplier methods. New York, NY, USA: Academic.

    MATH  Google Scholar 

  • Birgin, E. G., Martínez, J. M., & Raydan, M. (2000). Nonmonotone spectral projected gradient methods on convex sets. SIAM Journal on Optimization, 10, 1196–1211.

    Article  MathSciNet  MATH  Google Scholar 

  • Birgin, E. G., Martínez, J. M., & Raydan, M. (2001). Algorithm 813: SPG - software for convex-constrained optimization. ACM Transactions on Mathematical Software, 27, 340–349.

    Article  MATH  Google Scholar 

  • Breitfeld, M. G., & Shanno, D. F. (1994a). Preliminary computational experience with modified log-barrier functions for large-scale nonlinear programming. In W. W. Hager, D. W. Hearn, & P. M. Pardalos (Eds.), Large scale optimization, state of the art (pp. 45–67). Dordrecht/Boston/London: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Breitfeld, M.G., & Shanno, D.F. (1994b). Computational experience with penalty-barrier methods for nonlinear programming. (RUTCOR Research Report, RRR 17–93, August 1993, Revised March 1994. Rutgers Center for Operations Research, Rutgers University, New Brunswick, New Jersey 08903, March 1994).

    Google Scholar 

  • Breitfeld, M. G., & Shanno, D. F. (1994c). A globally convergent penalty-barrier algorithm for nonlinear programming and its computational performance. (RUTCOR Research Report, RRR 12–94, April 1994, Rutgers Center for Operations Research, Rutgers University, New Brunswick, New Jersey 08903, March 1994).

    Google Scholar 

  • Brown, A. A., & Bartholomew-Biggs, M. C. (1987). ODE vs SQP methods for constrained optimisation. (Technical Report No. 179, Numerical Optimisation Centre, The Hatfield Polytechnic, Hatfield, June 1987).

    Google Scholar 

  • Byrd, R. H., Lu, P., & Nocedal, J. (1995a). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific and Statistical Computing, 16(5), 1190–1208.

    Article  MathSciNet  MATH  Google Scholar 

  • Conn, A. R., Gould, N. I. M., & Toint, P. L. (1991). A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM Journal on Numerical Analysis, 28, 545–572.

    Article  MathSciNet  MATH  Google Scholar 

  • Conn, A. R., Gould, N. I. M., & Toint, Ph. L. (1992a). A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. (Technical Report 92/07, Department of Mathematics, Faculté Universitaires de Namur, Namur, Belgium).

    Google Scholar 

  • Dembo, R. S., & Steihaug, T. (1983). Truncated newton algorithms for large-scale unconstrained optimization. Mathematical Programming, 26, 190–212.

    Article  MathSciNet  MATH  Google Scholar 

  • Dolan, E. D., Moré, J. J., & Munson, T. S. (2004). Benchmarking optimization software with COPS 3.0. (Preprint ANL/MCS-TM-273, Argonne National Laboratory, Argonne, Illinois, February 2004).

    Google Scholar 

  • Fiacco, A. V., & McCormick, G. P. (1968). Nonlinear programming. Sequential unconstrained minimization techniques. John Wiley & Sons, Inc., New York, 1968. [Reprinted as: Volume 4 of SIAM Classics in Applied Mathematics. SIAM Publications, Philadelphia, PA 19104–2688, 1990].

    Google Scholar 

  • Frisch, K. R. (1955). The logarithmic potential method for convex programming. (Manuscript. Institute of Economics, University of Oslo, Oslo, May 1955.)

    Google Scholar 

  • Gill, P. E., & Murray, W. (1974b). Newton-type methods for unconstrained and linearly constrained optimization. Mathematical Programming, 7, 311–350.

    Article  MathSciNet  MATH  Google Scholar 

  • Hestens, M. R. (1969). Multiplier and gradient methods. Journal of Optimization Theory and Applications, 4, 303–320.

    Article  MathSciNet  Google Scholar 

  • Jensen, D. L., & Polyak, R. (1992). The convergence of a modified barrier method for convex programming. (Research Report RC 18570, IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York, 1992).

    Google Scholar 

  • Jittorntrum, K., & Osborne, M. (1980). A modified barrier function method with improved rate of convergence for degenerate problems. Journal of the Australian Mathematical Society (Series B), 21, 305–329.

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, D. C., & Nocedal, J. (1989). On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45, 503–528.

    Article  MathSciNet  MATH  Google Scholar 

  • Murtagh, B. A., & Saunders, M. A. (1978). Large-scale linearly constrained optimization. Mathematical Programming, 14, 41–72.

    Article  MathSciNet  MATH  Google Scholar 

  • Murtagh, B. A., & Saunders, M. A. (1980). MINOS/AUGMENTED user’s manual. (Technical Report SOL 80–14, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California, CA 94305).

    Google Scholar 

  • Murtagh, B. A., & Saunders, M. A. (1982). A projected lagrangian algorithm and its implementation for sparse nonlinear constraints. Mathematical Programming Study, 16, 84–117.

    Article  MathSciNet  MATH  Google Scholar 

  • Murtagh, B. A., & Saunders, M. A. (1995). MINOS 5.4 user’s guide. (Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California, CA 94305, February 1995).

    Google Scholar 

  • Nash, S. G. (1984a). Newton type minimization via the Lanczos method. SIAM Journal on Numerical Analysis, 21, 770–788.

    Article  MathSciNet  MATH  Google Scholar 

  • Nash, S. G. (1984b). User’s guide for TN/TNBC: Fortran routines for nonlinear optimization. (Report 397, Baltimore, MD: Mathematical Sciences Department, The John Hopkins University).

    Google Scholar 

  • Nash, S. G. (1985). Preconditioning of truncated-Newton methods. SIAM Journal on Scientific and Statistical Computing, 6, 599–616.

    Article  MathSciNet  MATH  Google Scholar 

  • Nash, S. G., & Nocedal, J. (1991). A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization. SIAM Journal of Optimization, 1, 358–372.

    Article  MathSciNet  MATH  Google Scholar 

  • Nash, S. G., Polyak, R., & Sofer, A. (1994). A numerical comparison of barrier and modified-barrier methods for large-scale bound-constrained optimization. In W. W. Hager, D. W. Hearn, & P. M. Pardlos (Eds.), Large scale optimization: State of the art (pp. 319–338). Dordrecht/Boston/London: Kluwer Academic Publishers.

    Chapter  Google Scholar 

  • Polyak, R. (1992). Modified barrier functions (theory and methods). Mathematical Programming, 54, 177–222.

    Article  MathSciNet  MATH  Google Scholar 

  • Powell, M. J. D. (1969). A method for nonlinear constraints in optimization problems. In R. Fletcher (Ed.), Optimization (pp. 283–297). New York, NY, USA: Academic.

    Google Scholar 

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Authors and Affiliations

  1. Research Institute for Informatics, Center for Advanced Modeling & Optimization, Bucharest, Romania

    Neculai Andrei

  2. Academy of Romanian Scientists, Bucharest, Romania

    Neculai Andrei

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  1. Neculai Andrei
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Andrei, N. (2017). A Penalty-Barrier Algorithm: SPENBAR. In: Continuous Nonlinear Optimization for Engineering Applications in GAMS Technology. Springer Optimization and Its Applications, vol 121. Springer, Cham. https://doi.org/10.1007/978-3-319-58356-3_8

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