57 research outputs found
Towards Supremum-Sum Subdifferential Calculus Free of Qualification Conditions
We give a formula for the subdifferential of the sum of two convex functions where one of them is the supremum of an arbitrary family of convex functions. This is carried out under a weak assumption expressing a natural relationship between the lower semicontinuous envelopes of the data functions in the domain of the sum function. We also provide a new rule for the subdifferential of the sum of two convex functions, which uses a strategy of augmenting the involved functions. The main feature of our analysis is that no continuity-type condition is required. Our approach allows us to unify, recover, and extend different results in the recent literature.Research of the first and the second authors is supported by CONICYT grants, Fondecyt 1150909 and 1151003, Basal PFB-03, and Basal FB003. Research of the second and third authors is supported by MINECO of Spain and FEDER of EU, grant MTM2014-59179-C2-1-P. Research of the third author is also supported by the Australian Research Council: Project DP160100854
Valadier-like formulas for the supremum function II: The compactly indexed case
We generalize and improve the original characterization given by Valadier
[20, Theorem 1] of the subdifferential of the pointwise supremum of convex
functions, involving the subdifferentials of the data functions at nearby
points. We remove the continuity assumption made in that work and obtain a
general formula for such a subdifferential. In particular, when the supremum is
continuous at some point of its domain, but not necessarily at the reference
point, we get a simpler version which gives rise to Valadier formula. Our
starting result is the characterization given in [10, Theorem 4], which uses
the epsilon-subdiferential at the reference point.Comment: 23 page
A non-convex relaxed version of minimax theorems
Given a subset of a locally convex space (with
compact) and a function such that
are concave and upper semicontinuous, the minimax
inequality is shown to hold provided that be the set of
such that is proper, convex and lower semi-contiuous. Moreover, if
in addition , then we can take as
the set of such that is convex. The relation to Moreau's
biconjugate representation theorem is discussed, and some applications to\
convex duality are provided.
Key words. Minimax theorem, Moreau theorem, conjugate function, convex
optimization
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
Contributions to complementarity and bilevel programming in Banach spaces
In this thesis, we derive necessary optimality conditions for bilevel programming problems (BPPs for short) in Banach spaces. This rather abstract setting reflects our desire to characterize the local optimal solutions of hierarchical optimization problems in function spaces arising from several applications.
Since our considerations are based on the tools of variational analysis introduced by Boris Mordukhovich, we study related properties of pointwise defined sets in function spaces. The presence of sequential normal compactness for such sets in Lebesgue and Sobolev spaces as well as the variational geometry of decomposable sets in Lebesgue spaces is discussed.
Afterwards, we investigate mathematical problems with complementarity constraints (MPCCs for short) in Banach spaces which are closely related to BPPs. We introduce reasonable stationarity concepts and constraint qualifications which can be used to handle MPCCs. The relations between the mentioned stationarity notions are studied in the setting where the underlying complementarity cone is polyhedric. The results are applied to the situations where the complementarity cone equals the nonnegative cone in a Lebesgue space or is polyhedral.
Next, we use the three main approaches of transforming a BPP into a single-level program (namely the presence of a unique lower level solution, the KKT approach, and the optimal value approach) to derive necessary optimality conditions for BPPs. Furthermore, we comment on the relation between the original BPP and the respective surrogate problem.
We apply our findings to formulate necessary optimality conditions for three different classes of BPPs. First, we study a BPP with semidefinite lower level problem possessing a unique solution. Afterwards, we deal with bilevel optimal control problems with dynamical systems of ordinary differential equations at both decision levels. Finally, an optimal control problem of ordinary or partial differential equations with implicitly given pointwise state constraints is investigated
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