×

Optimality conditions for a bilevel optimization problem in terms of KKT multipliers and convexificators. (English) Zbl 1469.90139

The authors use the optimistic approach for the following bilevel optimization problem (P) to prove necessary optimality conditions for local solutions. (P) has the form \[ \min_{x,y}F(x,y) \: \text{s.t.} \: G_j(x,y)\leq 0, j \in J, y \in \psi(x), \] where, for each \(x\in \mathbb {R}^{n_1}, \psi(x)\) is the set of optimal solutions of the parametric optimization problem \[ \min_y f(x,y) \; \text{s.t.} \; g_i(x,y)\leq 0, i\in I, \] where \(F, f, G_j, g_i: \mathbb {R}^{n_1} \times \mathbb {R}^{n_2} \rightarrow \mathbb {R}^{1}; I, J \) finite, \(F, G_j\) locally Lipschitz, \(f,g_i\) convex continuous.
In Section 2 the authors recall some basic constructions from nonsmooth analysis as f.i. inner semicompactness, inner semicontinuity, properties of convexificators, nonsmooth Abadie constraint qualification, nonsmooth generalized Guignard constraint qualification and especially on the Lipschitz continuity of the value function \(V\) of the second level \[V(x)=\min_y = \{f(x,y): g_i(x,y)\leq 0, \quad i\in I= \{1,\dots,q\}, \quad y \in \mathbb{R}^{n_2}\}.\] Section 3 starts with a remark on the existence of a solution of (P) (taken from the literature) followed by two theorems with optimality conditions for local optimal solutions and (considering the optimistic approach) for local weak efficient solutions and in both cases with Kuhn-Tucker-multipliers in \(\mathbb {R}^{q}_{+}\) and nonnegative coefficients with respect to the convexificators at the solution and validity of both a lower level regularity condition for a solution and of the generalized Guignard constraint qualification. The subdifferential of \(V\) at a solution is used in the proof. The found new optimality conditions extend the common KKT-conditions.

MSC:

90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bard, J. F. (1998). Practical Bilevel Optimization-Algorithms and Applications.Nonconvex Optimization and Its Applications, 30, 232-268. Springer: Boston.doi: 10.1007/978-1-4757-2836-1 7 · Zbl 0943.90078
[2] Clarke, F. H. (1990). Optimization and Nonsmooth Analysis.Classics in Applied Mathematics. Monograph published by SIAM.doi: 10.1137/1.9781611971309 · Zbl 0696.49002
[3] Dempe, S. (2002). Foundations of Bilevel Programming.Nonconvex Optimization and Its Applications, 61. Springer: Boston.doi: 10.1007/b101970 · Zbl 1038.90097
[4] Dempe, S. (2003). Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints.Optimization, 52(3), 333-359.doi: 10.1080/0233193031000149894 · Zbl 1140.90493
[5] Dempe, S., Dutta, J. and Mordukhovich, B. S. (2007). New necessary optimality conditions in optimistic bilevel programming.Optimization, 56(5-6), 577-604.doi: 10.1080/02331930701617551 · Zbl 1172.90481
[6] Dempe, S. and Gadhi, N. (2007). Necessary optimality conditions for bilevel set optimization problems.Journal of Global Optimization, 39(4), 529-542.doi: 10.1007/s10898-007-9154-0 · Zbl 1190.90178
[7] Demyanov, V. F. and Jeyakumar, V. (1997). Hunting for a Smaller Convex Subdifferential.Journal of Global Optimization, 10(3), 305-326.doi: 10.1023/A:1008246130864 · Zbl 0872.90083
[8] Dutta,J. and Chandra,S. (2002). Convexifactors,Generalized Convexity,and Optimality Conditions.Journal of Optimization Theory and Applications, 113(1), 41-64.doi: 10.1023/a:1014853129484 · Zbl 1172.90500
[9] G¨opfert, A., Riahi, H., Tammer, C. and Z‘alinescu, C. (2003). Variational Methods in Partially Ordered Spaces. Springer-Verlag: Canadian Mathematical Society.doi: /10.1007/b97568 · Zbl 1140.90007
[10] Jeyakumar, V. and Luc, D. T. (1999). Nonsmooth Calculus, Minimality, and Monotonicity of Convexifactors.Journal of Optimization Theory and Applications, 101(3), 599-621.doi: 10.1023/a:1021790120780 · Zbl 0956.90033
[11] Kohli, B. (2012). Optimality Conditions for Optimistic Bilevel Programming Problem Using Convexifactors.Journal of Optimization Theory and Applications. 152(3), 632-651.doi: 10.1007/s10957-011-9941-0 · Zbl 1262.90137
[12] Luderer, B. (1983). ¨Uber der ¨Aquivalenz nichtlinearer Optimierungsaufgaben. Technical Report, Technische Universit¨at Karl-Marx-Stadt. · Zbl 0556.90081
[13] Michel, P., Penot, J. P. and Brezis, H. (1992). A generalized derivative for calm and stable functions.Differential and Integral Equations, 5(2), 433-454.https://projecteuclid.org/euclid. die/1371043981 · Zbl 0787.49007
[14] Mordukhovich, B. S. and Nam, N. M. (2005). Variational Stability and Marginal Functions via Generalized Differentiation.Mathematics of Operations Research, 30(4), 800-816.doi: 10.1287/moor.1050.0147 · Zbl 1284.90083
[15] Mordukhovich, B. S., Nam, N. M., and Yen, N. D. (2009). Subgradients of marginal functions in parametric mathematical programming.Mathematical Programming, 116(1-2), 369-396.doi: 10.1007/s10107-007-0120-x · Zbl 1177.90377
[16] Mordukhovich, B. S. and Shao, Y. (1995). On Nonconvex Subdifferential Calculus in Banach Spaces.Journal of Convex Analysis, 2(1-2), 211-227.http://www.heldermann.de/JCA/jca02.htm · Zbl 0838.49013
[17] Ren, A. (2018). Solving the General Fuzzy Random Bilevel Programming Problem ThroughM e Measure-Based Approach.IEEE Access, 6, 25610-25620.doi: 10.1109/access.2018.2828706
[18] Singh, V. P. and Chakraborty, D. (2017). Solving bi-level programming problem with fuzzy random variable coefficients.Journal of Intelligent and Fuzzy Systems, 32(1), 521-528.doi: 10.3233/jifs152354 · Zbl 1366.90228
[19] Ye, J. J. and Zhu, D. L. (1995). Optimality conditions for bilevel programming problems.Optimization, 33(1), 9-27.doi: 10.1080/02331939508844060 · Zbl 0820.65032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.