Journal of Convex Analysis 14 (2007), No. 2, 227--237
Copyright Heldermann Verlag 2007
An Explicit Descent Method for Bilevel Convex Optimization
Mikhail Solodov
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil
solodov@impa.br
We consider the problem of minimizing a smooth convex function over the set of constrained minimizers of another smooth convex function. We show that this problem can be solved by a simple and explicit gradient descent type method. Standard constrained optimization is a particular case in this framework, corresponding to taking the lower level function as a penalty of the feasible set. We note that in the case of standard constrained optimization, the method does not require solving any penalization (or other optimization) subproblems, not even approximately, and does not perform projections (although explicit projections onto simple sets can be incorporated).
Keywords: Convex minimization, bilevel optimization, penalty methods, descent methods.
MSC: 90C30, 65K05
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