287 research outputs found
On the relationship between bilevel decomposition algorithms and direct interior-point methods
Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods
Second-order optimality conditions for bilevel programs
Second-order optimality conditions of the bilevel programming problems are
dependent on the second-order directional derivatives of the value functions or
the solution mappings of the lower level problems under some regular
conditions, which can not be calculated or evaluated. To overcome this
difficulty, we propose the notion of the bi-local solution. Under the Jacobian
uniqueness conditions for the lower level problem, we prove that the bi-local
solution is a local minimizer of some one-level minimization problem. Basing on
this property, the first-order necessary optimality conditions and second-order
necessary and sufficient optimality conditions for the bi-local optimal
solution of a given bilevel program are established. The second-order
optimality conditions proposed here only involve second-order derivatives of
the defining functions of the bilevel problem. The second-order sufficient
optimality conditions are used to derive the Q-linear convergence rate of the
classical augmented Lagrangian method
A novel approach for bilevel programs based on Wolfe duality
This paper considers a bilevel program, which has many applications in
practice. To develop effective numerical algorithms, it is generally necessary
to transform the bilevel program into a single-level optimization problem. The
most popular approach is to replace the lower-level program by its KKT
conditions and then the bilevel program can be reformulated as a mathematical
program with equilibrium constraints (MPEC for short). However, since the MPEC
does not satisfy the Mangasarian-Fromovitz constraint qualification at any
feasible point, the well-developed nonlinear programming theory cannot be
applied to MPECs directly. In this paper, we apply the Wolfe duality to show
that, under very mild conditions, the bilevel program is equivalent to a new
single-level reformulation (WDP for short) in the globally and locally optimal
sense. We give an example to show that, unlike the MPEC reformulation, WDP may
satisfy the Mangasarian-Fromovitz constraint qualification at its feasible
points. We give some properties of the WDP reformulation and the relations
between the WDP and MPEC reformulations. We further propose a relaxation method
for solving WDP and investigate its limiting behavior. Comprehensive numerical
experiments indicate that, although solving WDP directly does not perform very
well in our tests, the relaxation method based on the WDP reformulation is
quite efficient
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