Abstract
In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming.
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References
Anitescu, M.: Nonlinear programs with unbounded Lagrange multiplier sets. Technical Report ANL/MCS-P793-0200, Argonne National Labs
Benson, H.Y.: Numerical testing results for the primal-dual penalty approach. http://www.pages.drexel.edu/~hvb22/penaltyweb
Benson, H.Y., Shanno, D.F.: An exact primal-dual penalty method approach to warmstarting interior-point methods for linear programming. Comput. Optim. Appl. (2006, to appear)
Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: A comparative study of large scale nonlinear optimization algorithms. In: Proceedings of the Workshop on High Performance Algorithms and Software for Nonlinear Optimization, Erice, Italy (2001)
Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: filter methods and merit functions. Comput. Optim. Appl. 23(2), 257–272 (2002)
Benson, H.Y., Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: jamming and comparative numerical testing. Math. Program. A 99(1), 35–48 (2004)
Benson, H.Y., Sen, A., Shanno, D.F., Vanderbei, R.J.: Interior point algorithms, penalty methods and equilibrium problems. Comput. Optim. Appl. 34(2), 155–182 (2006)
Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib—a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15(1), 114–119 (2003)
Fiacco, A.V., McCormick, G.P.: Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Research Analysis Corporation, McLean (1968). Republished in 1990 by SIAM, Philadelphia
Fletcher, R.: Practical Methods of Optimization. Wiley, Chichester (1987)
Gill, P.E., Murray, W., Saunders, M.A.: User’s guide for SNOPT 5.3: a Fortran package for large-scale nonlinear programming. Technical Report, Systems Optimization Laboratory, Stanford University, Stanford, CA (1997)
Gould, N.I.M., Orban, D., Toint, P.L.: An interior-point l1-penalty method for nonlinear optimization. Technical Report RAL-TR-2003-022, Rutherford Appleton Laboratory Chilton, Oxfordshire, UK (November 2003)
Gould, N.I.M., Orban, D., Toint, P.L.: CUTEr and SifDec: a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)
Hock, W., Schittkowski, K.: Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems, vol. 187. Springer, Heidelberg (1981)
Kuriger, G., Ravindran, A.R.: Intelligent search methods for nonlinear goal programming problems. INFOR: Inf. Syst. Oper. Res. 43(2), 79–92 (2005)
Leyffer, S., Lopez-Calva, G., Nocedal, J.: Interior methods for mathematical programs with complementarity constraints. Technical Report OTC 2004-10, Northwestern University, Evanston, IL (December 2004)
Matsumoto, M., Nishimura, T.: Mersenne twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model. Comput. Simul. 8(1), 3–30 (1998)
Nocedal, J., Waltz, R.A.: Knitro 2.0 user’s manual. Technical Report OTC 02-2002, Optimization Technology Center, Northwestern University (January 2002)
Nocedal, J., Morales, J.L., Waltz, R., Liu, G., Goux, J.P.: Assessing the potential of interior-point methods for nonlinear optimization. Technical Report OTC-2001-6, Optimization Technology Center (2001)
Shanno, D.F., Vanderbei, R.J.: Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods. Math. Program. 87(2), 303–316 (2000)
Ulbrich, M., Ulbrich, S., Vicente, L.: A globally convergent primal-dual interior point filter method for nonconvex nonlinear programming. Math. Program. 100, 379–410 (2004)
Vanderbei, R.J.: AMPL models. http://orfe.princeton.edu/~rvdb/ampl/nlmodels
Vanderbei, R.J., Shanno, D.F.: An interior-point algorithm for nonconvex nonlinear programming. Comput. Optim. Appl. 13, 231–252 (1999)
Wächter, A., Biegler, L.: Failure of global convergence for a class of interior point methods for nonlinear programming. Math. Program. 88(3), 565–587 (2000)
Wächter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Technical Report RC 23149, IBM T.J. Watson Research Center, Yorktown, USA (March 2004)
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Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.
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Benson, H.Y., Shanno, D.F. Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts. Comput Optim Appl 40, 143–189 (2008). https://doi.org/10.1007/s10589-007-9089-x
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DOI: https://doi.org/10.1007/s10589-007-9089-x