Nonsmooth Optimization; Proceedings of an IIASA Workshop, March 28 - April 8, 1977

Optimization, a central methodological tool of systems analysis, is used in many of IIASA's research areas, including the Energy Systems and Food and Agriculture Programs. IIASA's activity in the field of optimization is strongly connected with nonsmooth or nondifferentiable extreme problems, which consist of searching for conditional or unconditional minima of functions that, due to their complicated internal structure, have no continuous derivatives. Particularly significant for these kinds of extreme problems in systems analysis is the strong link between nonsmooth or nondifferentiable optimization and the decomposition approach to large-scale programming. This volume contains the report of the IIASA workshop held from March 28 to April 8, 1977, entitled Nondifferentiable Optimization. However, the title was changed to Nonsmooth Optimization for publication of this volume as we are concerned not only with optimization without derivatives, but also with problems having functions for which gradients exist almost everywhere but are not continous, so that the usual gradient-based methods fail. Because of the small number of participants and the unusual length of the workshop, a substantial exchange of information was possible. As a result, details of the main developments in nonsmooth optimization are summarized in this volume, which might also be considered a guide for inexperienced users. Eight papers are presented: three on subgradient optimization, four on descent methods, and one on applicability. The report also includes a set of nonsmooth optimization test problems and a comprehensive bibliography

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

This paper pursues a twofold goal. First to derive new results on generalized differentiation in variational analysis focusing mainly on a broad class of intrinsically nondifferentiable marginal/value functions. Then the results established in this direction apply to deriving necessary optimality conditions for the optimistic version of bilevel programs that occupy a remarkable place in optimization theory and its various applications. We obtain new sets of optimality conditions in both smooth and smooth settings of finite-dimensional and infinite-dimensional spaces

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