Abstract
A nonsmooth and nonconvex general optimization problem is considered. Using the idea of pseudo-Jacobians, nonsmooth versions of the Robinson and Mangasarian–Fromovitz constraint qualifications are defined and relationships between them and the local error bound property are investigated. A new necessary optimality condition, based on the pseudo-Jacobians, is derived under the local error bound constraint qualification. These results are applied for computing and estimating the Fréchet and limiting subdifferentials of value functions. Moreover, several examples are provided to clarify the obtained results.
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References
Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory. Applications and Numerical Results. Kluwer Academic Publishers, Boston (1998)
Robinson, S.M.: Generalized equations and their solution, part \(\rm {II}\): Applications to nonlinear programming. Math. Program. Stud. 19, 200–221 (1982)
Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, New York (1985)
Bagirov, A.M., Jin, L., Karmitsa, N., Al Nuaimat, A., Sultanova, N.: Subgradient method for nonconvex nonsmooth optimization. J. Optim. Theory and Appl. 157, 416–435 (2013)
Burke, J.V., Lewis, A.C., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 571–779 (2005)
Jeyakumar, V., Luc, D.T.: Approximate Jacobian matrices for nonsmooth continuous maps and \(c^1\)-optimization. SIAM J. Control Optim. 36, 1815–1832 (1998)
Jeyakumar, V., Luc, D.T.: Nonsmooth calculus, minimality, and monotonicity of convexificators. J. Optim. Theory Appl. 101, 599–621 (1999)
Jeyakumar, V., Luc, D.T.: Nonsmooth Vector Functions and Continuous Optimization. Springer, New York (2007)
Jeyakumar, V., Yen, N.D.: Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization. SIAM J. Optim. 14, 1106–1127 (2004)
Zhang, J., Zhang, L., Wang, W.: On constraint qualifications in terms of approximate \(\rm {J}\)acobians for nonsmooth continuous optimization problems. Nonlinear Anal. 75, 2566–2680 (2012)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Clarke, F.H., Ledyaev, Y.S., Wolenski, R.S.P.: Nonsmooth Analysis and Control Theory. Springer, NewYork (1998)
Demp, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel progrmming. SIAM J. Optim. 22, 1309–1343 (2012)
Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York (1983)
Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Program. Stud. 19, 101–119 (1982)
Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)
Rockafellar, R.T.: Marginal values and second-order necessary conditions for optimality. Math. Program. 26, 245–286 (1983)
Dutta, J., Chandra, S.: Convexifcators, generalized convexity and vector optimzation. Optimization 53, 77–94 (2004)
Michel, P., Penot, J.P.: A generalized derivative for calm and stable functions. Differ. Integral Equ. 5, 433–454 (1992)
Mordukhovich, B.S., Shao, Y.H.: On nonconvex subdifferential calculus in Banach space. J. Convex Anal. 2, 211–227 (1995)
Treiman, J.S.: The linear nonconvex generalized gradient and Lagrange multipliers. SIAM J. Optim 5, 670–680 (1995)
Movahedian, N., Nobakhtian, S.: Constraint qualifications for nonsmooth mathematical programs with equilibrium constraints. Set-Valued Var. Anal. 17, 63–95 (2009)
Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. \(\text{ P }\)art 1: sufficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009)
Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. \(\text{ P }\)art 2: necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009)
Ioffe, A.D.: Variational Analysis of Regular Mappings. Theory and Applications. Springer, New York (2017)
Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Basic Theory. Springer, New York (2006)
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The third-named author was partially supported by a grant from IPM (No 96900422).
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Communicated by Dinh The Luc.
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Hejazi, M.A., Movahedian, N. & Nobakhtian, S. On Constraint Qualifications and Sensitivity Analysis for General Optimization Problems via Pseudo-Jacobians. J Optim Theory Appl 179, 778–799 (2018). https://doi.org/10.1007/s10957-018-1336-z
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DOI: https://doi.org/10.1007/s10957-018-1336-z
Keywords
- Nonsmooth analysis
- Pseudo-Jacobian
- Constraint qualification
- Necessary optimality condition
- Sensitivity analysis