2012, 2(2): 301-331. doi: 10.3934/naco.2012.2.301

A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian

1. 

School of Mathematical and Geospatial Sciences, Royal Melbourne Institute of Technology, G.P.O. Box 2476V, Melbourne, Australia 3001

2. 

School of Mathematical Sciences, The University of Adelaide, Australia SA 5005

Received  December 2011 Revised  May 2012 Published  May 2012

A reformulation of a standard smooth mathematical program in terms of a nonlinear Lagrangian is used in conjunction with the calculus of subhessians to derive a set of sufficient optimality conditions that are applicable to some nonregular problems. These conditions are cast solely in terms of the first-- and second--order derivatives of the constituent functions and generalize standard second--order sufficiency conditions to a wide class of potentially nonregular problems.
Citation: A. C. Eberhard, C.E.M. Pearce. A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 301-331. doi: 10.3934/naco.2012.2.301
References:
[1]

M. Andramonov, "Global Minimization of Some Classes of Generalized Convex Functions," PhD Thesis, University of Ballarat, Australia, 2001.

[2]

A. V. Arutyunov and A. F. Izmailov, Tangent vectors to a zero set at abnormal points, J. Math. Anal. Appl., 289 (2004), 66-76. doi: 10.1016/j.jmaa.2003.08.023.

[3]

A. V. Arutyunov, E. R. Avakov and A. F. Izmailov, Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications, Math. Prog. Ser. A, 114 (2008), 37-68. doi: 10.1007/s10107-006-0082-4.

[4]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems and Control: Foundations and Applications, 2, Birkhäuser, Boston-Basel-Berlin, (1990).

[5]

A. Auslender, Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim., 22 (1984), 239-254. doi: 10.1137/0322017.

[6]

A. Ben-Tal, Second-order and related extremality conditions in nonlinear programming, J. Optim. Theory Appl., 31 (1980), 143-165. doi: 10.1007/BF00934107.

[7]

A. Ben-Tal and J. Zowe, Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems, Math. Programming, 24 (1982), 70-91. doi: 10.1007/BF01585095.

[8]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second order optimality conditions based on parabolic second order tangent sets, SIAM J. Optim., 9 (1999), 466-492. doi: 10.1137/S1052623496306760.

[9]

O. A. Brezhneva and A. A. Tret'yakov, P-factor-approach to degenerate optimization problems, IFIP Int. Fed. Inf. Process., Springer, New York, System Modeling and Optimization, 199 (2006), 83-90.

[10]

O. A. Brezhneva and A. A. Tret'yakov, pth order optimality condition for nonregular optimization problems, Dokl. Math., 77 (2008), 163-165. doi: 10.1134/S1064562408020014.

[11]

A. Eberhard, Prox-regularity and subjets, in "Optimization and Related Topics" (ed. A. Rubinov), Appl. Optim., Kluwer Academic Pub., 47 (2001), 237-313.

[12]

A. C. Eberhard and B. S Mordukhovich, First-order and second-order optimality conditions for nonsmooth constrained problems via convolution smoothing, Optimization, 60 (2011), 253-257. doi: 10.1080/02331934.2010.522713.

[13]

A. Eberhard, M. Nyblom and D. Ralph, Applying generalised convexity notions to jets, in "Generalized Convexity, Generalized Monotonicity: Recent Results" ( eds J.P. Crouzeix et al.), Kluwer Academic Pub., 289 (1998), 111-157.

[14]

A. Eberhard and C. E. M. Pearce, A comparison of two approaches to second-order subdifferentiability concepts with applications to optimality conditions, in "Optimization and Control with Applications" (eds L. Qi, K. L. Teo and X. Yang), Kluwer Academic Pub., (2005), 35-100. doi: 10.1007/0-387-24255-4_2.

[15]

A. Eberhard and R. Wenczel, Some sufficient optimality conditions in nonsmooth analysis, SIAM J. Optim., 20 (2009), 251-296. doi: 10.1137/07068059X.

[16]

A. Eberhard and R. Wenczel, A study of tilt-stable optimality and sufficient conditions, Nonlin. Anal., 75 (2012), 1260-1281.

[17]

A. F. Izmailov, On optimality conditions in extremal problems with nonregular inequality constraints, Mat. Zametki, 66 (1999), 89-101, Transl. Math. Notes 66 (1999), 72-81. doi: 10.1007/BF02674072.

[18]

A. F. Izmailov and M. V. Solodov, Optimality conditions for irregular inequality-constrained problems, SIAM J. Control Optim., 40 (2001), 1280-1295. doi: 10.1137/S0363012999357549.

[19]

U. Ledzewicz and H. Schaettler, Second-order conditions for extremum problems with nonregular equality constraints, J. Optim. Theory Appl., 86 (1995), 113-144. doi: 10.1007/BF02193463.

[20]

Z-Q Lou, J-S Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, 1996.

[21]

R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optimization, 15 (1977), 957-972. doi: 10.1137/0315061.

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory," Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 330 2006.

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications," Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 331 2006.

[24]

J.-P. Penot, Optimality conditions in mathematical programming and composite optimization, Math. Programming, 67 (1994), 225-245. doi: 10.1007/BF01582222.

[25]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: 10.1090/S0002-9947-96-01544-9.

[26]

R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local minimum, SIAM J. Optim., 8 (1998), 287-299. doi: 10.1137/S1052623496309296.

[27]

R. T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, in "Progress in Nondifferentiable Optimization" (ed. E. Nurminsk), IIASA Collaborative Proc. Ser. CP-82,8, Internat. Inst. Appl. Systems Anal., Laxenberg, Austria, (1982), 125-143.

[28]

R. T. Rockafellar and J-B.Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, Springer, 317 1998.

[29]

A. Rubinov, "Abstract Convexity and Global Optimization," Nonconvex Optimization and its Applications, Kluwer Academic Publishers, 44 2000.

[30]

J. E. Spingarn, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc., 264 (1981), 77-89. doi: 10.1090/S0002-9947-1981-0597868-8.

[31]

M. Studniarski, Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim., 25 (1986), 1044-1049. doi: 10.1137/0324061.

[32]

D. E. Ward, Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl., 80 (1994), 551-571. doi: 10.1007/BF02207780.

[33]

D. E. Ward, A comparison of second-order epiderivatives: calculus and optimality conditions, J. Math. Anal. Appl., 193 (1995), 465-482. doi: 10.1006/jmaa.1995.1247.

show all references

References:
[1]

M. Andramonov, "Global Minimization of Some Classes of Generalized Convex Functions," PhD Thesis, University of Ballarat, Australia, 2001.

[2]

A. V. Arutyunov and A. F. Izmailov, Tangent vectors to a zero set at abnormal points, J. Math. Anal. Appl., 289 (2004), 66-76. doi: 10.1016/j.jmaa.2003.08.023.

[3]

A. V. Arutyunov, E. R. Avakov and A. F. Izmailov, Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications, Math. Prog. Ser. A, 114 (2008), 37-68. doi: 10.1007/s10107-006-0082-4.

[4]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems and Control: Foundations and Applications, 2, Birkhäuser, Boston-Basel-Berlin, (1990).

[5]

A. Auslender, Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim., 22 (1984), 239-254. doi: 10.1137/0322017.

[6]

A. Ben-Tal, Second-order and related extremality conditions in nonlinear programming, J. Optim. Theory Appl., 31 (1980), 143-165. doi: 10.1007/BF00934107.

[7]

A. Ben-Tal and J. Zowe, Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems, Math. Programming, 24 (1982), 70-91. doi: 10.1007/BF01585095.

[8]

J. F. Bonnans, R. Cominetti and A. Shapiro, Second order optimality conditions based on parabolic second order tangent sets, SIAM J. Optim., 9 (1999), 466-492. doi: 10.1137/S1052623496306760.

[9]

O. A. Brezhneva and A. A. Tret'yakov, P-factor-approach to degenerate optimization problems, IFIP Int. Fed. Inf. Process., Springer, New York, System Modeling and Optimization, 199 (2006), 83-90.

[10]

O. A. Brezhneva and A. A. Tret'yakov, pth order optimality condition for nonregular optimization problems, Dokl. Math., 77 (2008), 163-165. doi: 10.1134/S1064562408020014.

[11]

A. Eberhard, Prox-regularity and subjets, in "Optimization and Related Topics" (ed. A. Rubinov), Appl. Optim., Kluwer Academic Pub., 47 (2001), 237-313.

[12]

A. C. Eberhard and B. S Mordukhovich, First-order and second-order optimality conditions for nonsmooth constrained problems via convolution smoothing, Optimization, 60 (2011), 253-257. doi: 10.1080/02331934.2010.522713.

[13]

A. Eberhard, M. Nyblom and D. Ralph, Applying generalised convexity notions to jets, in "Generalized Convexity, Generalized Monotonicity: Recent Results" ( eds J.P. Crouzeix et al.), Kluwer Academic Pub., 289 (1998), 111-157.

[14]

A. Eberhard and C. E. M. Pearce, A comparison of two approaches to second-order subdifferentiability concepts with applications to optimality conditions, in "Optimization and Control with Applications" (eds L. Qi, K. L. Teo and X. Yang), Kluwer Academic Pub., (2005), 35-100. doi: 10.1007/0-387-24255-4_2.

[15]

A. Eberhard and R. Wenczel, Some sufficient optimality conditions in nonsmooth analysis, SIAM J. Optim., 20 (2009), 251-296. doi: 10.1137/07068059X.

[16]

A. Eberhard and R. Wenczel, A study of tilt-stable optimality and sufficient conditions, Nonlin. Anal., 75 (2012), 1260-1281.

[17]

A. F. Izmailov, On optimality conditions in extremal problems with nonregular inequality constraints, Mat. Zametki, 66 (1999), 89-101, Transl. Math. Notes 66 (1999), 72-81. doi: 10.1007/BF02674072.

[18]

A. F. Izmailov and M. V. Solodov, Optimality conditions for irregular inequality-constrained problems, SIAM J. Control Optim., 40 (2001), 1280-1295. doi: 10.1137/S0363012999357549.

[19]

U. Ledzewicz and H. Schaettler, Second-order conditions for extremum problems with nonregular equality constraints, J. Optim. Theory Appl., 86 (1995), 113-144. doi: 10.1007/BF02193463.

[20]

Z-Q Lou, J-S Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, 1996.

[21]

R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optimization, 15 (1977), 957-972. doi: 10.1137/0315061.

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory," Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 330 2006.

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications," Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, 331 2006.

[24]

J.-P. Penot, Optimality conditions in mathematical programming and composite optimization, Math. Programming, 67 (1994), 225-245. doi: 10.1007/BF01582222.

[25]

R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: 10.1090/S0002-9947-96-01544-9.

[26]

R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local minimum, SIAM J. Optim., 8 (1998), 287-299. doi: 10.1137/S1052623496309296.

[27]

R. T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization, in "Progress in Nondifferentiable Optimization" (ed. E. Nurminsk), IIASA Collaborative Proc. Ser. CP-82,8, Internat. Inst. Appl. Systems Anal., Laxenberg, Austria, (1982), 125-143.

[28]

R. T. Rockafellar and J-B.Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, Springer, 317 1998.

[29]

A. Rubinov, "Abstract Convexity and Global Optimization," Nonconvex Optimization and its Applications, Kluwer Academic Publishers, 44 2000.

[30]

J. E. Spingarn, Submonotone subdifferentials of Lipschitz functions, Trans. Amer. Math. Soc., 264 (1981), 77-89. doi: 10.1090/S0002-9947-1981-0597868-8.

[31]

M. Studniarski, Necessary and sufficient conditions for isolated local minima of nonsmooth functions, SIAM J. Control Optim., 25 (1986), 1044-1049. doi: 10.1137/0324061.

[32]

D. E. Ward, Characterizations of strict local minima and necessary conditions for weak sharp minima, J. Optim. Theory Appl., 80 (1994), 551-571. doi: 10.1007/BF02207780.

[33]

D. E. Ward, A comparison of second-order epiderivatives: calculus and optimality conditions, J. Math. Anal. Appl., 193 (1995), 465-482. doi: 10.1006/jmaa.1995.1247.

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