Class Of Bilevel Programs Research Articles

Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem

When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel program with lower level second-order cone programming problem (SOCBLP). In this paper, we show that the Lagrange multiplier set mapping of the lower level problem of a class of the SOCBLPs is upper semicontinuous under suitable assumptions. Based on this fact, we detect the similarities and relationships between the SOCBLP and its KKT reformulation. Then we derive the specific expression of the critical cone at a feasible point, and show that the second order sufficient conditions are sufficient for the second order growth at an M-stationary point of the SOCBLP under suitable conditions.

Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities

We propose algorithms for finding the projection of a given point onto the solution set of the pseudomonotone variational inequality problem. This problem arises in the Tikhonov regularization method for pseudomonotone variational inequality. Since the solution set of the variational inequality is not given explicitly, the available methods of mathematical programming and variational inequalities cannot be applied directly.

A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints

We present a branch-and-bound algorithm for discretely-constrained mathematical programs with equilibrium constraints (DC-MPEC). This is a class of bilevel programs with an integer program in the upper-level and a complementarity problem in the lower-level. The algorithm builds on the work by Gabriel et al. (Journal of the Operational Research Society 61(9):1404–1419, 2010) and uses Benders decomposition to form a master problem and a subproblem. The new dynamic partition scheme that we present ensures that the algorithm converges to the global optimum. Partitioning is done to overcome the non-convexity of the Benders subproblem. In addition Lagrangean relaxation provides bounds that enable fathoming in the branching tree and warm-starting the Benders algorithm. Numerical tests show significantly reduced solution times compared to the original algorithm. When the lower level problem is stochastic our algorithm can easily be further decomposed using scenario decomposition. This is demonstrated on a realistic case.

An Evolutionary Algorithm for Solving Bilevel Programming Problems Using Duality Conditions

Bilevel programming is characterized by two optimization problems located at different levels, in which the constraint region of the upper level problem is implicitly determined by the lower level problem. This paper is focused on a class of bilevel programming with a linear lower level problem and presents a new algorithm for solving this kind of problems by combining an evolutionary algorithm with the duality principle. First, by using the prime‐dual conditions of the lower level problem, the original problem is transformed into a single‐level nonlinear programming problem. In addition, for the dual problem of the lower level, the feasible bases are taken as individuals in population. For each individual, the values of dual variables can be obtained by taking the dual problem into account, thus simplifying the single‐level problem. Finally, the simplified problem is solved, and the objective value is taken as the fitness of the individual. Besides, when nonconvex functions are involved in the upper level, a coevolutionary scheme is incorporated to obtain global optima. In the computational experiment, 10 problems, smaller or larger‐scale, are solved, and the results show that the proposed algorithm is efficient and robust.

Genetic algorithm for solving quadratic bilevel programming problem

By applying Kuhn-Tucker condition the quadratic bilevel programming, a class of bilevel programming, is transformed into a single level programming problem, which can be simplified by some rule. So we can search the optimal solution in the feasible region, hence reduce greatly the searching space. Numerical experiments on several literature problems show that the new algorithm is both feasible and effective in practice.