# American Institute of Mathematical Sciences

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.

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##### 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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