Abstract
AbstractRobust optimization is an effective method for dealing with the optimization problems under uncertainty. 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 programming problem with lower level second-order cone program (SOCBLP). In this paper, we present the mathematical models of the SOCBLP, and we give some basic concepts, such as constraint region, inducible region, and optimal solution. It is illustrated that the SOCBLP is generally a nonconvex and nondifferentiable optimization problem, whose feasible set may be not connected in some cases and the constraint region is generally not polyhedral. Finally under suitable conditions we propose the optimality conditions for several models of the SOCBLP in the optimistic case.MSC:90C30.
Highlights
Bilevel programming (BLP) problems are hierarchical ones-optimization problems having a second optimization problem as part of their constraints [, ]
3 Concepts and characteristics of feasible set we give some basic concepts of the SOCBLP, such as constraint region, inducible region, and optimal solution
It is illustrated that the SOCBLP is generally a nonconvex and nondifferentiable optimization problem, whose feasible set may be not connected in some cases and the constraint region is generally not polyhedral for m ≥
Summary
Bilevel programming (BLP) problems are hierarchical ones-optimization problems having a second (parametric) optimization problem as part of their constraints [ , ]. Secondorder cone programming (SOCP) problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of second-order cones [ – ]. There have been rare works about extending BLP to bilevel programming with lower level second-order cone program (SOCBLP). 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 programming problem having second-order cone programming [ ] as its lower level problem, i.e., a bilevel programming problem with lower level second-order cone programs (SOCBLP)
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