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A primal–dual modified log-barrier method for inequality constrained nonlinear optimization

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Abstract

We present a primal–dual modified log-barrier algorithm to solve inequality constrained nonlinear optimization problems. Basically, the algorithm is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing a modified log-barrier function to handle inequality constraints. The algorithm uses an outer/inner iteration scheme and the globalization is performed in the primal–dual space by means of a new primal–dual merit function. The robustness and efficiency of the algorithm is improved using quadratic extrapolation. The numerical performance of the new method is illustrated by comparing it with a primal–dual classical log-barrier method and two well-established interior-point solvers on two sets of problems from COPS and Hock–Schittkowski collections, including a set of problems that exhibits degeneracy.

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References

  1. Armand, P., Benoist, J., Omheni, R., Pateloup, V.: Study of a primal–dual algorithm for equality constrained minimization. Comput. Optim. Appl. 59(3), 405–433 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armand, P., Omheni, R.: A globally and quadratically convergent primal–dual augmented lagrangian algorithm for equality constrained optimization. Optim. Methods Softw. 32(1), 1–21 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Armand, P., Omheni, R.: A mixed logarithmic barrier-augmented lagrangian method for nonlinear optimization. J. Optim. Theory Appl. 173(2), 523–547 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Armand, P., Orban, D.: The squared slacks transformation in nonlinear programming. Sultan Qaboos University Journal for Science [SQUJS] 17(1), (2012)

  5. Ben-Tal, A., Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7(2), 347–366 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Breitfeld, M.G., Shanno, D.F.: Preliminary computational experience with modified log-barrier functions for large-scale nonlinear programming. In: Hager, W.W., Hearn, D.W., Pardalos, P.M. (eds.) Large Scale Optim. State Art, pp. 45–67. Springer, US (1994)

    MATH  Google Scholar 

  7. Breitfeld, M.G., Shanno, D.F.: Computational experience with penalty-barrier methods for nonlinear programming. Ann. Oper. Res. 62, 439–463 (1996). Interior point methods in mathematical programming

    Article  MathSciNet  MATH  Google Scholar 

  8. Byrd, R.H., Gilbert, J.C., Nocedal, J.: A trust region method based on interior point techniques for nonlinear programming. Math. Program. 89(1, Ser. A), 149–185 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conn, A.R., Gould, N., Toint, P.L.: A globally convergent lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Math. Comput. 66, 261–288 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dolan, E.D., Moré, J.J., Munson, T.S.: Benchmarking optimization software with COPS 3.0. Tech. rep., Argonne National Laboratory (2004)

  12. Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)

    MATH  Google Scholar 

  13. Forsgren, A., Gill, P.E.: Primal–dual interior methods for nonconvex nonlinear programming. SIAM J. Optim. 8(4), 1132–1152 (1998). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frisch, K.R.: The logarithmic potential method of convex programming. Memorandum, University Institute of Economics, Oslo (1955)

  15. Gay, D.M., Overton, M.L., Wright, M.A.H.: A primal-dual interior method for nonconvex nonlinear programming. In: Advances in nonlinear programming (Beijing, 1996), Appl. Optim., vol. 14, pp. 31–56. Kluwer Acad. Publ., Dordrecht (1998)

  16. Gill, P.E., Murray, W., Wright, M.H.: Practical optimization. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1981)

  17. Goldfarb, D., Polyak, R., Scheinberg, K., Yuzefovich, I.: A modified barrier-augmented Lagrangian method for constrained minimization. Comput. Optim. Appl. 14(1), 55–74 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hock, W., Schittkowski, K.: Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems, vol. 187. Springer, Berlin (1981)

  19. Murray, W.: Analytical expressions for the eigenvalues and eigenvectors of the Hessian matrices of barrier and penalty functions. J. Optim. Theory Appl. 7, 189–196 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  20. Murray, W., Wright, M.H.: Line search procedures for the logarithmic barrier function. SIAM J. Optim. 4(2), 229–246 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nash, S.G., Polyak, R., Sofer, A.: A numerical comparison of barrier and modified barrier methods for large-scale bound-constrained optimization. In: Large scale optimization (Gainesville, FL, 1993), pp. 319–338. Kluwer Acad. Publ., Dordrecht (1994)

  22. Nocedal, J., Wright, S.J.: Numerical optimization, Springer Series in Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)

    Google Scholar 

  23. Polyak, R.: Modified barrier functions (theory and methods). Math. Program. 54(2, Ser. A), 177–222 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. SAS Institute: The optmodel procedure. In: SAS/OR 14.3 Users Guide: Mathematical Programming, pp. 23–189 (2017)

  25. Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods. Math. Program. 87(2, Ser. B), 303–316 (2000). Studies in algorithmic optimization

    Article  MathSciNet  MATH  Google Scholar 

  26. Silva, R., Soares, J., Vicente, L.N.: Local analysis of the feasible primal-dual interior-point method. Comput. Optim. Appl. 40(1), 41–57 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1, Ser. A), 25–57 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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  1. SAS Institute, Cary, NC, USA

    Joshua Griffin & Riadh Omheni

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  1. Joshua Griffin
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  2. Riadh Omheni
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Correspondence to Riadh Omheni.

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Griffin, J., Omheni, R. A primal–dual modified log-barrier method for inequality constrained nonlinear optimization. Optim Lett 14, 2461–2477 (2020). https://doi.org/10.1007/s11590-020-01567-2

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