A semismooth sequential quadratic programming method for lifted mathematical programs with vanishing constraints

A. F. Izmailov   ; A. L. Pogosyan

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 6, pp. 983-1006

Résumé

Mathematical programs with vanishing constraints are a difficult class of optimization problems with important applications to optimal topology design problems of mechanical structures. Recently, they have attracted increasingly more attention of experts. The basic difficulty in the analysis and numerical solution of such problems is that their constraints are usually nonregular at the solution. In this paper, a new approach to the numerical solution of these problems is proposed. It is based on their reduction to the socalled lifted mathematical programs with conventional equality and inequality constraints. Special versions of the sequential quadratic programming method are proposed for solving lifted problems. Preliminary numerical results indicate the competitiveness of this approach.

Bibliographie

[1] Achtziger W., Kanzow C., “Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications”, Math. Program., 114:1 (2007), 69–99 | DOI | MR

[2] Achtziger W., Hoheisel T., Kanzow C., A smoothing-regularization approach to mathematical programs with vanishing constraints, Preprint 284, Inst. Math. Univ. Würzburg, Würzburg, 2008

[3] Hoheisel T., Kanzow C., “First- and second-order optimality conditions for mathematical programs with vanishing constraints”, Appl. Math., 52 (2007), 495–514 | DOI | MR | Zbl

[4] Hoheisel T., Kanzow C., “Stationarity conditions for mathematical programs with vanishing constraints using weak constraint qualifications”, J. Math. Analys. Appl., 337 (2008), 292–310 | DOI | MR | Zbl

[5] Hoheisel T., Kanzow C., “On the Abadie and Guignard constraint qualifications for mathematical programs with vanishing constraints”, Optimizat., 58:4 (2009), 431–448 | DOI | MR | Zbl

[6] Izmailov A. F., Solodov M. V., “Mathematical programs with vanishing constraints: optimality conditions, sensitivity, and a relaxation method”, J. Optimizat. Theory Appl., 142:3 (2009), 501–532 | DOI | MR | Zbl

[7] Izmailov A. F., Pogosyan A. L., “Usloviya optimalnosti i nyutonovskie metody dlya zadach optimizatsii s ischezayuschimi ogranicheniyami”, Zh. vychisl. matem. i matem. fiz., 49:7 (2009), 1184–1196 | MR | Zbl

[8] Izmailov A. F., Pogosyan A. L., “O metodakh aktivnogo mnozhestva dlya zadach optimizatsii s ischezayuschimi ogranicheniyami”, Teor. i prikl. zadachi nelineinogo analiza, VTs RAN, M., 2009, 18–49

[9] Hoheisel T., Kanzow C., Schwartz A., “Convergence of a local regularization approach for mathematical programs with complementarity or vanishing constraints”, Optimizat. Meth. Software, 2011 | DOI | Zbl

[10] Luo Z.-Q., Pang J.-S., Ralph D., Mathematical programs with equilibrium constraints, Cambridge Univ. Press, Cambridge, 1996

[11] Outrata J. V., Kocvara M., Zowe J., “Nonsmooth approach to optimization problems with equilibrium constraints”, Theory, applications and numerical results, Kluwer Acad. Publs, Boston, 1998 | Zbl

[12] Izmailov A. F., “Zadachi optimizatsii s komplementarnymi ogranicheniyami: regulyarnost, usloviya optimalnosti i chuvstvitelnost”, Zh. vychisl. matem. i matem. fiz., 44:7 (2004), 1209–1228 | MR | Zbl

[13] Izmailov A. F., Chuvstvitelnost v optimizatsii, Fizmatlit, M., 2006

[14] Stein O., “Lifting mathematical programs with complementarity constraints”, Math. Program., 2010 | DOI

[15] Izmailov A. F., Pogosyan A. L., Solodov M. V., “Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints”, Comput. Optimizat. Appl., 2010 | DOI | MR

[16] Izmailov A. F., Solodov M. V., Chislennye metody optimizatsii, Izd. 2-e, pererab. i dop., Fizmatlit, M., 2008

[17] Qi L., “Superlinearly convergent approximate Newton methods for LC${}^1$ optimization problems”, Math. Program., 64 (1994), 277–294 | DOI | Zbl

[18] Han J., Sun D., “Superlinear convergence of approximate Newton methods for LC${}^1$ optimization problems without strict complementarity”, Recent Advances in Nonsmooth Optimization, 58, World Scient. Publs Co., Singapur, 1993, 353–367

[19] Izmailov A. F., Pogosyan A. L., Solodov M. V., Semismooth SQP method for equality-constrained optimization problems with an application to the lifted reformulation of mathematical programs with complementrarity constraints, Preprint A 675/2010/IMPA, Rio de Janeiro, 2010 Available at http://www.preprint.impa.br:80/Shadows/SERIE A/2010/675.html

[20] Bonnans J. F., “Local analysis of Newton-type methods for variational inequalities and nonlinear programming”, Appl. Math. Optimizat., 29 (1994), 161–186 | DOI | MR | Zbl

[21] Nocedal J., Wright S. J., Numerical optimization, Sec. ed., Springer, New York, 2006

[22] Fernández D., Izmailov A. F., Solodov M. V., “Sharp primal superlinear convergence results for some Newtonian methods for constrained optimization”, SIAM J. Optimizat., 20:6 (2010), 3312–3334 | DOI | Zbl

[23] Tseng P., “Growth behavior of a class of merit functions for the nonlinear complementarity problem”, J. Optimizat. Theory Appl., 89:1 (1996), 17–37 | DOI | MR | Zbl

[24] Bonnans J. F., Gilbert J. Ch., Lemaréchal C., Sagastizábal C., Numerical optimization: theoretical and practical aspects, Sec. ed., Springer, Berlin, 2006

[25] Dolan E., Moré J., “Benchmarking optimization software with performance profiles”, Math. Program., 91:2 (2002), 201–213 | DOI | MR | Zbl

[26] Darina A. N., Izmailov A. F., “Polugladkii metod Nyutona dlya zadachi kvadratichnogo programmirovaniya s prostymi ogranicheniyami”, Zh. vychisl. matem. i matem. fiz., 9:10 (2009), 1785–1795