Abstract
A global solution strategy for multilevel optimization problems with special non-convexityformulation in the objectives of the inner level problems is presented based on branch-and-bound andmulti-parametric programming approach. An algorithm to such problems is proposed by convexifyingthe inner level problem while the variables from upper level problems are considered as parameters.The resulting convex parametric under-estimator problem is solved using multi-parametric program-ming approach. A branch-and-bound procedure is employed until a pre-specied positive tolerance issatised. Moreover, a ϵ-convergence proof is given for the algorithm.
Highlights
In many real-world problems decisions have been made in a hierarchical order where individual decision makers have no direct control upon the decisions of the others, but their actions affect all other decision makers
Even though Multi-parametric Programming (MPP) approach produces a mathematical program without equilibrium constraints depending on the upper level problems, the rational reaction set of the inner level problem may be a disconnected set
When nonconvexity appear in the inner level problem, most of the existing algorithms fail to work
Summary
In many real-world problems decisions have been made in a hierarchical order where individual decision makers have no direct control upon the decisions of the others, but their actions affect all other decision makers. By considering the upper level variables as parameters in the lower level problems Faısca et al, [10, 11] have proposed a Multi-parametric Programming (MPP) approach to solve MLOPs, when the lower level problems are convex. Many practical problems that are modeled using MLOP may contain non-convex terms in their lower level problems [8] In this case, even though MPP approach produces a mathematical program without equilibrium constraints depending on the upper level problems, the rational reaction set of the inner level problem (with non-convexity formulation) may be a disconnected set. In this paper we apply the process of convexification of the lower level problems to underestimate them by convex functions (if they are nonconvex) at each iteration and use MPP approach to propose a branch-and-bound algorithm to find a global approximate solution for multilevel problems with non-convexity at their inner levels.
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