19 research outputs found
Solving ill-posed bilevel programs
This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem
Set optimization - a rather short introduction
Recent developments in set optimization are surveyed and extended including
various set relations as well as fundamental constructions of a convex analysis
for set- and vector-valued functions, and duality for set optimization
problems. Extensive sections with bibliographical comments summarize the state
of the art. Applications to vector optimization and financial risk measures are
discussed along with algorithmic approaches to set optimization problems
A Mountain Pass-type Theorem for Vector-valued Functions
The mountain pass theorem for scalar functionals is a fundamental result
of the minimax methods in variational analysis. In this work we extend this theorem
to the class of C^1 functions f : R^n \u2192 R^m, where the image space is ordered by the
nonnegative orthant R^m_+. Under suitable geometrical assumptions, we prove the
existence of a critical point of f and we localize this point as a solution of a minimax
problem. We remark that the considered minimax problem consists of an inner
vector maximization problem and of an outer set-valued minimization problem. To
deal with the outer set-valued problem we use an ordering relation among subsets of
R^m introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type
principle for set-valued maps and we extensively use the notion of vector
pseudogradient