Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Solving nonconvex planar location problems by nite dominating sets

It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by nding a fi nite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location. In this paper it is fi rst established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems. Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal ob jective value. For the approximation problem two di erent approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed e - to polynomial approximation algorithms with accuracy for solving the general model considered in this paper.Dirección General de Enseñanza Superio

A finite dominating set of cardinality O(k) and a witness set of cardinality O(n) for 1.5D terrain guarding problem

1.5 dimensional (1.5D) terrain is characterized by a piecewise linear curve. Locating minimum number of guards on the terrain (T) to cover/guard the whole terrain is known as 1.5D terrain guarding problem. Approximation algorithms and a polynomial-time approximation scheme have been presented for the problem. The problem has been shown to be NP-Hard. In the problem, the set of possible guard locations and the set of points to be guarded are uncountable. To solve the problem to optimality, a finite dominating set (FDS) of size O (n2) and a witness set of size O (n3) have been presented, where n is the number of vertices on T. We show that there exists an even smaller FDS of cardinality O (k) and a witness set of cardinality O(n), where k is the number of convex points. Convex points are vertices with the additional property that between any two convex points the piecewise linear curve representing the terrain is convex. Since it is always true that k≤ n for n≥ 2 and since it is possible to construct terrains such that n= 2 k, the existence of an FDS with cardinality O(k) and a witness set of cardinality of O (n) leads to the reduction of decision variables and constraints respectively in the zero-one integer programming formulation of the problem. © 2017, Springer Science+Business Media New York

Applying multiobjective evolutionary algorithms in industrial projects

During the recent years, multiobjective evolutionary algorithms have matured as a flexible optimization tool which can be used in various areas of reallife applications. Practical experiences showed that typically the algorithms need an essential adaptation to the specific problem for a successful application. Considering these requirements, we discuss various issues of the design and application of multiobjective evolutionary algorithms to real-life optimization problems. In particular, questions on problem-specific data structures and evolutionary operators and the determination of method parameters are treated. As a major issue, the handling of infeasible intermediate solutions is pointed out. Three application examples in the areas of constrained global optimization (electronic circuit design), semi-infinite programming (design centering problems), and discrete optimization (project scheduling) are discussed

Parallel software tool for decomposing and meshing of 3d structures

An algorithm for automatic parallel generation of three-dimensional unstructured computational meshes based on geometrical domain decomposition is proposed in this paper. Software package build upon proposed algorithm is described. Several practical examples of mesh generation on multiprocessor computational systems are given. It is shown that developed parallel algorithm enables us to reduce mesh generation time significantly (dozens of times). Moreover, it easily produces meshes with number of elements of order 5 · 107, construction of those on a single CPU is problematic. Questions of time consumption, efficiency of computations and quality of generated meshes are also considered

Structure and pressure drop of real and virtual metal wire meshes

An efficient mathematical model to virtually generate woven metal wire meshes is presented. The accuracy of this model is verified by the comparison of virtual structures with three-dimensional images of real meshes, which are produced via computer tomography. Virtual structures are generated for three types of metal wire meshes using only easy to measure parameters. For these geometries the velocity-dependent pressure drop is simulated and compared with measurements performed by the GKD - Gebr. Kufferath AG. The simulation results lie within the tolerances of the measurements. The generation of the structures and the numerical simulations were done at GKD using the Fraunhofer GeoDict software

Review of Optimization Problems in Wireless Sensor Networks

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Using the Sharp Operator for edge detection and nonlinear diffusion

In this paper we investigate the use of the sharp function known from functional analysis in image processing. The sharp function gives a measure of the variations of a function and can be used as an edge detector. We extend the classical notion of the sharp function for measuring anisotropic behaviour and give a fast anisotropic edge detection variant inspired by the sharp function. We show that these edge detection results are useful to steer isotropic and anisotropic nonlinear diffusion filters for image enhancement

A proof of convergence of a finite volume scheme for modified steady Richards’ equation describing transport processes in the pressing section of a paper machine

A number of water flow problems in porous media are modelled by Richards’ equation [1]. There exist a lot of different applications of this model. We are concerned with the simulation of the pressing section of a paper machine. This part of the industrial process provides the dewatering of the paper layer by the use of clothings, i.e. press felts, which absorb the water during pressing [2]. A system of nips are formed in the simplest case by rolls, which increase sheet dryness by pressing against each other (see Figure 1). A lot of theoretical studies were done for Richards’ equation (see [3], [4] and references therein). Most articles consider the case of x-independent coefficients. This simplifies the system considerably since, after Kirchhoff’s transformation of the problem, the elliptic operator becomes linear. In our case this condition is not satisfied and we have to consider nonlinear operator of second order. Moreover, all these articles are concerned with the nonstationary problem, while we are interested in the stationary case. Due to complexity of the physical process our problem has a specific feature. An additional convective term appears in our model because the porous media moves with the constant velocity through the pressing rolls. This term is zero in immobile porous media. We are not aware of papers, which deal with such kind of modified steady Richards’ problem. The goal of this paper is to obtain the stability results, to show the existence of a solution to the discrete problem, to prove the convergence of the approximate solution to the weak solution of the modified steady Richards’ equation, which describes the transport processes in the pressing section. In Section 2 we present the model which we consider. In Section 3 a numerical scheme obtained by the finite volume method is given. The main part of this paper is theoretical studies, which are given in Section 4. Section 5 presents a numerical experiment. The conclusion of this work is given in Section 6