Two-level value function approach to non-smooth optimistic and pessimistic bilevel programs

ABSTRACT

The authors' paper in Dempe et al. [Necessary optimality conditions in pessimistic bilevel programming. Optimization. 2014;63:505–533], was the first one to provide detailed optimality conditions for pessimistic bilevel optimization. The results there were based on the concept of the two-level optimal value function introduced and analysed in Dempe et al. [Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22 (2012), 1309–1343], for the case of optimistic bilevel programs. One of the basic assumptions in both of these papers is that the functions involved in the problems are at least continuously differentiable. Motivated by the fact that many real-world applications of optimization involve functions that are non-differentiable at some points of their domain, the main goal of the current paper is to extend the two-level value function approach by deriving new necessary optimality conditions for both optimistic and pessimistic versions in bilevel programming with non-smooth data.

KEYWORDS:

• Optimistic and pessimistic bilevel programming
• two-level value functions
• variational analysis
• generalized differentiation
• optimality conditions

1991 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 49K35
• 49K40
• 90C26
• 90C30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the first author has been supported by Deutsche Forschungsgemeinschaft, Project DE 650/10-1. The research of the second author was partially supported by the USA National Science Foundation [grant number DMS-1512846], by the USA Air Force Office of Scientific Research [grant number 15RT0462], and by the RUDN University Program 5-100. The research of the third author was partially supported by the Engineering and Physical Sciences Research Council (EPSRC) [grant EP/P022553/1].