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The generalized bilevel programming problem (GBLP) is a bilevel mathematical program where the lower level is a variational inequality. In this paper we prove that if the objective function of a GBLP is uniformly Lipschitz continuous in the lower level decision variable with respect to the upper level decision variable, then using certain uniform parametric error bounds as penalty functions gives single level problems equivalent to the GBLP. Several local and global uniform parametric error bounds are presented, and assumptions guaranteeing that they apply are discussed. We then derive Kuhn-Tucker-type necessary optimality conditions by using exact penalty formulations and nonsmooth analysis.

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The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush-Kuhn-Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.

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This paper focuses on the adaptive trajectory tracking control for a remotely operated vehicle (ROV) with an unknown dynamic model and the unmeasured states. Unlike most previous trajectory tracking control approaches, in this paper, the velocity states and the angular velocity states in the body-fixed frame are unmeasured, and the thrust model is inaccurate. Obviously, it is more in line with the actual ROV systems. Since the dynamic model is unknown, a new local recurrent neural network (local RNN) structure with fast learning speed is proposed for online identification. To estimate the unmeasured states, an adaptive terminal sliding-mode state observer based on the local RNN is proposed, so that the finite-time convergence of the trajectory tracking error can be guaranteed. Considering the problem of inaccurate thrust model, an adaptive scale factor is introduced into thrust model, and the thruster control signal is considered as the input of the trajectory tracking system directly. Based on the local RNN output, the adaptive scale factor, and the state estimation values, an adaptive trajectory tracking control law is constructed. The stability of the trajectory tracking control system is analyzed by the Lyapunov theorem. The effectiveness of the proposed control scheme is illustrated by simulations.

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