Shape optimization is indispensable for designing and constructing industrial components.
Many problems that arise in application, particularly in structural mechanics and in the optimal
control of distributed parameter systems, can be formulated as the minimization of functionals
defined over a class of admissible domains.
The present talk aims at surveying on parametric shape optimization with elliptic or parabolic
state equation. Especially, the following items will be addressed:
This talk is devoted to discussing optimistic bilevel programs based on the so-called value/marginal function approach. This approach allows us reducing bilevel programs to single-value problems of mathematical programming with unavoidably nondifferentiable (and often nonconvex) data. Applying generalized differentiation theory of variational analysis in convex and nonconvex settings together with methods of nondifferentiable programming, we derive verifiable optimality conditions for optimistic bilevel programs, which give us the potential for numerical implementations. We also discuss a relatively new class of seminfinite bilevel programs and formulate several open questions of the further research.
In this talk we will present an approach to abstract convex analysis and nonconvex duality theory based on Fenchel-Moreau conjugations; in particular, we will discuss quasiconvex conjugation and duality in detail.
References:
Some newly emerging imaging methods lead to the inverse problem of determining one or several coefficient function(s) in an elliptic partial differential equation from (partial) knowledge of its solutions. In particular let us mention electrical impedance tomography (EIT), where electrical currents are driven through a patient to image its interior. Often the main interest is on the detection of anomalous regions in which the conductivity differs from a known normal background value. In this talk, we will describe recent advances on such shape reconstruction problems that are based on monotonicity relations with respect to matrix definiteness and the concept of localized potentials.
Many set-valued integrators for parametric ODEs fail to stabilize the computed enclosures on infinite horizons, even though the ODE trajectories themselves may be asymptotically stable. This talk starts by describing a new discretized set-valued integration algorithm that uses a predictor-validation approach to propagate generic affine set-parameterizations, whose images are guaranteed to enclose the ODE solution set. Sufficient conditions are then derived for this algorithm to be locally asymptotically stable, in the sense that the computed enclosures are guaranteed to remain stable on infinite time horizons when applied to a dynamic system in the neighborhood of a locally asymptotically stable periodic orbit (or equilibrium point). The key requirement here is quadratic Hausdorff convergence of function extensions in the chosen affine set-parameterization. We illustrate these stability properties with several numerical case studies.
Topic of the talk are (first) ideas to solve optimistic bilevel optimization problems. Bilevel optimization problems are nonconvex problems the feasible set of which is (in part) given by the graph of a second parametric optimization problem. The transformation of it into an optimization problem with a nonconvex, nonsmooth constraint is discussed. To solve this problem the feasible set of it can be approximated with increasing accuracy or algorithms for nonsmooth optimization can be used.
In this talk I will discuss recent results concerning spectral optimization problems for the eigenvalues of the Laplace operator with different boundary conditions (Dirichlet and Robin). From a theoretical point of view, these problems are seen in the free boundary/free discontinuity frameworks. The question of existence of an optimal shape is crucial for both theoretical and numerical purposes. In case of non-existence, a relaxed form of the problem should be searched in a Gamma-convergence framework providing as well in this way a useful tool for numerical approximations.
During the last ten years or so level set techniques have become popular for solving specific inverse problems where the main information sought from indirect data can be described by a collection of shapes, possibly in combination with internal and external parameter profiles. We will present some of these applications in this talk, together with some tailor-made level set reconstruction techniques. Numerical experiments will be presented which show how level set techniques perform in these applications.
This introductory talk is a motivation for the workshop in its present form. I will define and illustrate the concept of generalized convexity, which induces in a very natural way classes of sets with properties depending on the coupling function. We will see that some of the representations of sets used today are particular cases of the approach via generalized convexity, and I will explain which tasks lie ahead if this approach will be pursued with nonstandard coupling functions.
Over the last two decades, robust optimization has emerged as a computationally attractive approach to formulate and solve min-max decision problems. More recently, robust optimization has been successfully applied to min-max-min problems with continuous second-stage decisions. This talk takes a step towards extending the robust optimization methodology to problems with integer second-stage decisions, which have largely resisted solution so far. To this end, we approximate these problems by their corresponding K-adaptability problems, in which the decision maker pre-commits to K second-stage policies in the first stage and implements the best of these policies in the second stage. We study the approximation quality and the computational complexity of the K-adaptability problem, and we propose two mixed-integer linear programming reformulations that can be solved with off-the-shelf software. We demonstrate the effectiveness of our reformulations for stylized instances of supply chain design and capital budgeting problems.