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 & 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, (2001). Google Scholar

[2]

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

[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. doi: 10.1007/s10107-006-0082-4. Google Scholar

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J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems and Control: Foundations and Applications, 2 (1990). Google Scholar

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

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

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

[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. doi: 10.1137/S1052623496306760. Google Scholar

[9]

O. A. Brezhneva and A. A. Tret'yakov, P-factor-approach to degenerate optimization problems,, IFIP Int. Fed. Inf. Process., 199 (2006), 83. Google Scholar

[10]

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

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A. Eberhard, Prox-regularity and subjets,, in, 47 (2001), 237. Google Scholar

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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. doi: 10.1080/02331934.2010.522713. Google Scholar

[13]

A. Eberhard, M. Nyblom and D. Ralph, Applying generalised convexity notions to jets,, in, 289 (1998), 111. Google Scholar

[14]

A. Eberhard and C. E. M. Pearce, A comparison of two approaches to second-order subdifferentiability concepts with applications to optimality conditions,, in, (2005), 35. doi: 10.1007/0-387-24255-4_2. Google Scholar

[15]

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

[16]

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

[17]

A. F. Izmailov, On optimality conditions in extremal problems with nonregular inequality constraints,, Mat. Zametki, 66 (1999), 89. doi: 10.1007/BF02674072. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory,", Grundlehren der Mathematischen Wissenschaften, 330 (2006). Google Scholar

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications,", Grundlehren der Mathematischen Wissenschaften, 331 (2006). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

R. T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization,, in, (1982), 125. Google Scholar

[28]

R. T. Rockafellar and J-B.Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften, 317 (1998). Google Scholar

[29]

A. Rubinov, "Abstract Convexity and Global Optimization,", Nonconvex Optimization and its Applications, 44 (2000). Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

show all references

References:
[1]

M. Andramonov, "Global Minimization of Some Classes of Generalized Convex Functions,", PhD Thesis, (2001). Google Scholar

[2]

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

[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. doi: 10.1007/s10107-006-0082-4. Google Scholar

[4]

J. P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems and Control: Foundations and Applications, 2 (1990). Google Scholar

[5]

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

[6]

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

[7]

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

[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. doi: 10.1137/S1052623496306760. Google Scholar

[9]

O. A. Brezhneva and A. A. Tret'yakov, P-factor-approach to degenerate optimization problems,, IFIP Int. Fed. Inf. Process., 199 (2006), 83. Google Scholar

[10]

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

[11]

A. Eberhard, Prox-regularity and subjets,, in, 47 (2001), 237. Google Scholar

[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. doi: 10.1080/02331934.2010.522713. Google Scholar

[13]

A. Eberhard, M. Nyblom and D. Ralph, Applying generalised convexity notions to jets,, in, 289 (1998), 111. Google Scholar

[14]

A. Eberhard and C. E. M. Pearce, A comparison of two approaches to second-order subdifferentiability concepts with applications to optimality conditions,, in, (2005), 35. doi: 10.1007/0-387-24255-4_2. Google Scholar

[15]

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

[16]

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

[17]

A. F. Izmailov, On optimality conditions in extremal problems with nonregular inequality constraints,, Mat. Zametki, 66 (1999), 89. doi: 10.1007/BF02674072. Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, I: Basic Theory,", Grundlehren der Mathematischen Wissenschaften, 330 (2006). Google Scholar

[23]

B. S. Mordukhovich, "Variational Analysis and Generalized Differentiation, II: Applications,", Grundlehren der Mathematischen Wissenschaften, 331 (2006). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

R. T. Rockafellar, Favorable classes of Lipschitz continuous functions in subgradient optimization,, in, (1982), 125. Google Scholar

[28]

R. T. Rockafellar and J-B.Wets, "Variational Analysis,", Grundlehren der Mathematischen Wissenschaften, 317 (1998). Google Scholar

[29]

A. Rubinov, "Abstract Convexity and Global Optimization,", Nonconvex Optimization and its Applications, 44 (2000). Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

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