On the local representation of piecewise smooth equations as a Lipschitz manifold

We study systems of equations, F(x)=0$F(x)=0$, given by piecewise differentiable functions F:Rⁿ→R^k$F:{\mathbb{R}}^n\to {\mathbb{R}}^k$, k?n$k?n$. The focus is on the representability of the solution set locally as an (n−k)$(n-k)$-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke?s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke?s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke?s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k=3$k=3$.     

Bilevel optimization: on the structure of the feasible set

We consider bilevel optimization from the optimistic point of view. Let the pair (x, y) denote the variables. The main difficulty in studying such problems lies in the fact that the lower level contains a global constraint. In fact, a point (x, y) is feasible if y solves a parametric optimization problem L(x). In this paper we restrict ourselves to the special case that the variable x is one-dimensional. We describe the generic structure of the feasible set M. Moreover, we discuss local reductions of the bilevel problem as well as corresponding optimality criteria. Finally, we point out typical problems that appear when trying to extend the ideas to higher dimensional x-dimensions. This will clarify the high intrinsic complexity of the general generic structure of the feasible set M and corresponding optimality conditions for the bilevel problem U.     

On the closure of the feasible set in generalized semi-infinite programming

In generalized semi-infinite programming the feasible set is known to be not closed in general. In this paper, under natural and generic assumptions, the closure of the feasible set is described in explicit terms. Oliver Stein gratefully acknowledges support through a Heisenberg grant of the Deutsche Forschungsgemeinschaft.     

One-Parameter Families of Feasible Sets in Semi-infinite Optimization

Feasible sets in semi-infinite optimization are basically defined by means of infinitely many inequality constraints. We consider one-parameter families of such sets. In particular, all defin-ing functions - including those defining the index set of the inequality constraints - will depend on a parameter. We note that a semi-infinite problem is a two-level problem in the sense that a point is feasible if and only if all global minimizers of a corresponding marginal function are nonnegative. For a quite natural class of mappings we characterize changes in the global topological structure of the corresponding feasible set as the parameter varies. As long as the index set (-mapping) of the inequality constraints is lower semicontinuous, all changes in topology are those which generically appear in one-parameter sets defined by finitely many constraints. In the case, however, that some component of the mentioned index set is born (or vanishes), the topological change is of global nature and is not controllable. In fact, the change might be as drastic as that when adding or deleting an (arbitrary) inequality constraint.     

Parallel Dynamics of Extremely Diluted Symmetric Q-Ising Neural Networks

The parallel dynamics of extremely diluted symmetric Q-Ising neural networks is studied for arbitrary Q using a probabilistic approach. In spite of the extremely diluted architecture the feedback correlations arising from the symmetry prevent a closed-form solution in contrast with the extremely diluted asymmetric model. A recursive scheme is found determining the complete time evolution of the order parameters taking into account all feedback. It is based upon the evolution of the distribution of the local field, as in the fully connected model. As an illustrative example an explicit analysis is carried out for the Q=2 and Q=3 model. These results agree with and extend the partial results existing for Q=2. For Q>2 the analysis is entirely new. Finally, equilibrium fixed-point equations are derived and a capacity-gain function diagram is obtained.     

Generalized semi-infinite optimization: A first order optimality condition and examples

We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x)‖xεM}, where M={x∈ℝⁿ|h_i(x)=0i=l,...m, G(x,y)⩾0, y∈Y(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the setY(x) is compact for allx under consideration and the set-valued mappingY(.) is upper semi-continuous. The difference with a standard semi-infinite problem lies in thex-dependence of the index setY. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible setM. This work was partially supported by the “Deutsche Forschungsgemeinschaft” through the Graduiertenkolleg “Mathematische Optimierung” at the University of Trier.     

On the Complexity of Equalizing Inequalities

We study two approaches to replace a finite mathematical programming problem with inequality constraints by a problem that contains only equality constraints. The first approach lifts the feasible set into a high-dimensional space by the introduction of quadratic slack variables. We show that then not only the number of critical points but also the topological complexity of the feasible set grow exponentially. On the other hand, the second approach bases on an interior point technique and lifts an approximation of the feasible set into a space with only one additional dimension. Here only Karush–Kuhn–Tucker points with respect to the positive and negative objective function in the original problem give rise to critical points of the smoothed problem, so that the number of critical points as well as the topological complexity can at most double.     

Development of radiation medicine at DLNP, JINR

The Dzhelepov Laboratory of Nuclear Problems’ activity is aimed at developing three directions in radiation medicine: 3D conformal proton therapy, accelerator techniques for proton and carbon treatment of tumors, and new types of detector systems for spectrometric computed tomography (CT) and positron emission tomography (PET). JINR and IBA have developed and constructed the medical proton cyclotron C235-V3. At present, all basic cyclotron systems have been built. We plan to assemble this cyclotron at JINR in 2011 and perform tests with the extracted proton beam in 2012. A superconducting isochronous cyclotron C400 has been designed by the IBA-JINR collaboration. This cyclotron will be used for radiotherapy with proton, helium and carbon ions. The ¹²C⁶⁺ and ⁴He²⁺ ions will be accelerated to an energy of 400 MeV/amu, the protons will be extracted at the energy 265 MeV. The construction of the C400 cyclotron was started in 2010 within the framework of the Archarde project (France). Development of spectrometric CT tomographs may allow one to determine the chemical composition of a substance together with the density, measured using traditional CT. This may advance modern diagnostic methods significantly. JINR develops fundamentally new pixel detector systems for spectrometric CT. The time-of-flight (TOF) system installed in the positron emission tomograph (PET) permits essential reduction in the detector noise from occasional events of different positron annihilations. The micropixel avalanche photodiodes (MAPDs) developed at JINR allow a factor of 1.5 reduction in the resolution time for the PET TOF system and suppression of the noise level as compared to commercial PET. The development of a combined PET/MRI is of considerable medical interest, but it cannot be made with the existing PET tomographs based on detectors of compact photomultipliers due to strong alternating magnetic field of MRI. Change-over to detectors of micropixel avalanche photodiodes permits making a combined PET/MRI.