Abstract
An iteration of the stabilized sequential quadratic programming method consists in solving a certain quadratic program in the primal-dual space, regularized in the dual variables. The advantage with respect to the classical sequential quadratic programming is that no constraint qualifications are required for fast local convergence (i.e., the problem can be degenerate). In particular, for equality-constrained problems, the superlinear rate of convergence is guaranteed under the only assumption that the primal-dual starting point is close enough to a stationary point and a noncritical Lagrange multiplier (the latter being weaker than the second-order sufficient optimality condition). However, unlike for the usual sequential quadratic programming method, designing natural globally convergent algorithms based on the stabilized version proved quite a challenge and, currently, there are very few proposals in this direction. For equality-constrained problems, we suggest to use for the task linesearch for the smooth two-parameter exact penalty function, which is the sum of the Lagrangian with squared penalizations of the violation of the constraints and of the violation of the Lagrangian stationarity with respect to primal variables. Reasonable global convergence properties are established. Moreover, we show that the globalized algorithm preserves the superlinear rate of the stabilized sequential quadratic programming method under the weak conditions mentioned above. We also present some numerical experiments on a set of degenerate test problems.
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References
Wright, S.J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11, 253–275 (1998)
Hager, W.W.: Stabilized sequential quadratic programming. Comput. Optim. Appl. 12, 253–273 (1999)
Fischer, A.: Local behavior of an iterative framework for generalized equations with nonisolated solutions. Math. Program. 94, 91–124 (2002)
Wright, S.J.: Modifying SQP for degenerate problems. SIAM J. Optim. 13, 470–497 (2002)
Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering. Springer International Publishing, Switzerland (2014)
Fernández, D., Solodov, M.: Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems. Math. Program. 125, 47–73 (2010)
Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program. 122, 93–120 (2012)
Di Pillo, G., Grippo, L.: A new class of augmented Lagrangians in nonlinear programming. SIAM J. Control Optim. 17, 618–628 (1979)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
Bertsekas, D.P.: Enlarging the region of convergence of Newton’s method for constrained optimization. J. Optim. Theory Appl. 36, 221–252 (1982)
Andreani, R., Haeser, G., Schuverdt, M.L., Silva, P.J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. 135, 255–273 (2012)
Izmailov, A.F., Solodov, M.V., Uskov, E.I.: Global convergence of augmented Lagrangian methods applied to optimization problems with degenerate constraints, including problems with complementarity constraints. SIAM J. Optim. 22, 1579–1606 (2012)
Glad, S.T.: Properties of updating methods for the multipliers in augmented Lagrangian. J. Optim. Theory Appl. 28, 135–156 (1979)
Gill, P.E., Robinson, D.P.: A primal-dual augmented Lagrangian. Comput. Optim. Appl. 51, 1–25 (2012)
Gill, P.E., Robinson, D.P.: A globally convergent stabilized SQP method. SIAM J. Optim. 23, 1983–2010 (2013)
Gill, P.E., Kungurtsev, V., Robinson, D.P.: A regularized SQP method with convergence to second-order optimal points. UCSD Center for computational mathematics technical report CCoM-13-4 (2013)
Gill, P.E., Kungurtsev, V., Robinson, D.P.: A globally convergent stabilized SQP method: superlinear convergence. UCSD Center for Computational mathematics technical report CCoM-13-4 (2014)
Izmailov, A.F., Solodov, M.V., Uskov, E.I.: Combining stabilized SQP with the augmented Lagrangian algorithm. Comput. Optim. Appl. 62, 405–429 (2015)
Izmailov, A.F., Krylova, A.M., Uskov, E.I.: Hybrid globalization of stabilized sequential quadratic programming method. In: Bereznyov, V.A. (ed.) Theoretical and Applied Problems of Nonlinear Analysis, pp. 47–66. Computing Center RAS, Moscow (2011). (in Russian)
Fernández, D., Pilotta, E.A., Torres, G.A.: An inexact restoration strategy for the globalization of the sSQP method. Comput. Optim. Appl. 54, 595–617 (2013)
Martínez, J.M., Pilotta, E.A.: Inexact restoration methods for nonlinear programming: advances and perspectives. In: Qi, L., Teo, K.L., Yang, X.Q. (eds.) Optimization and Control with Applications,pp. 271–292. Springer, New York (2005)
Izmailov, A.F., Solodov, M.V.: On attraction of Newton-type iterates to multipliers violating second-order sufficiency conditions. Math. Program. 117, 271–304 (2009)
Izmailov, A.F., Solodov, M.V.: Critical Lagrange multipliers: what we currently know about them, how they spoil our lives, and what we can do about it. TOP 23, 1–26 (2015)
Izmailov, A.F., Kurennoy, A.S.: Abstract Newtonian frameworks and their applications. SIAM J. Optim. 23, 2369–2396 (2013)
Izmailov, A.F., Solodov, M.V.: On attraction of linearly constrained Lagrangian methods and of stabilized and quasi-Newton SQP methods to critical multipliers. Math. Program. 126, 231–257 (2011)
Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Optim. 12, 979–1006 (2002)
ALGENCAN. http://www.ime.usp.br/egbirgin/tango/
Bonnans, J.F., Gilbert, JCh., Lemaréchal, C., Sagastizábal, C.: Numerical Optimization: Theoretical and Practical Aspects, 2nd edn. Springer, Berlin (2006)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)
Mostafa, E.M.E., Vicente, L.N., Wright, S.J.: Numerical behavior of a stabilized SQP method for degenerate NLP problems. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction. Lecture Notes in Computer Science 2861, pp. 123–141. Springer, Berlin (2003)
Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)
Acknowledgments
This research is supported by the Russian Foundation for Basic Research Grant 14-01-00113, by the Russian Science Foundation Grant 15-11-10021, by CNPq Grants PVE 401119/2014-9 and 302637/2011-7 (Brazil), and by FAPERJ. The authors also thank the three anonymous referees for their evaluation and helpful comments.
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Izmailov, A.F., Solodov, M.V. & Uskov, E.I. Globalizing Stabilized Sequential Quadratic Programming Method by Smooth Primal-Dual Exact Penalty Function. J Optim Theory Appl 169, 148–178 (2016). https://doi.org/10.1007/s10957-016-0889-y
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DOI: https://doi.org/10.1007/s10957-016-0889-y
Keywords
- Stabilized sequential quadratic programming
- Superlinear convergence
- Global convergence
- Exact penalty function
- Second-order sufficiency
- Noncritical Lagrange multiplier