Guide Duality in Global Optimization: Optimality Conditions and Algorithmical Aspects

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A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated.

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If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case.

We also give a numerical solution method for the case of a finite set of alternatives. This paper aims at combining variable ordering structures with set relations in set optimization, which have been dened using the constant ordering cone before. Since the purpose is to connect these two important approaches in set optimization, we do not restrict our considerations to one certain relation. Conversely, we provide the reader with many new variable set relations generalizing the relations from [16, 25] and discuss their usefulness.

After analyzing the properties of the introduced relations, we dene new solution notions for set-valued optimization problems equipped with variable ordering structures and compare them with other concepts from the literature. In order to characterize the introduced solutions a nonlinear scalarization approach is used.

These variational principles are useful for deriving optimality conditions, epsilon-Kolmogorov conditions in approximation theory, and epsilon-maximum principles in optimal control. In this paper we consider a generalized k-server problem with parallel requests where several servers can also be located on one point which was initiated by an operations research problem. In section 4 the ''compound Harmonic algorithm'' for the generalized k-server problem is presented. Certain multi-step transition probabilities and absorbing probabilities are used by the compound Harmonic algorithm.

For their computation one step of the generalized k-server problem is replaced by a number of steps of other generalized specific k-server problems. We show that this algorithm is competitive against an adaptive online adversary.

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In the case of unit distances the Harmonic algorithm and the compound Harmonic algorithm are identical. Copositivity tests are presented based on new necessary and suffcient conditions requiring the solution of linear complementarity problems LCP. Methodologies involving Lemke's method, an enumerative algorithm and a linear mixed-integer programming formulation are proposed to solve the required LCPs. A new necessary condition for strict copositivity based on solving a Linear Program LP is also discussed, which can be used as a preprocessing step.

The algorithms with these three different variants are thoroughly applied to test matrices from the literature and to max-clique instances with matrices up to dimension x As the solution set of a multiobjective problem is often rather large and contains points of no interest to the decision-maker, strategies are sought that reduce the size of the solution set. This approach can be used to reduce the dimensionality of the solution set as well as to discarde certain unwanted solutions, especially the 'extreme' ones found by minimizing just one of the objectives given in the classical sense while disregarding all others.

In der Praxis werden viele Prozesse durch Unsicherheiten beeinflusst. Ein Ansatz dazu ist die Nutzung der wahrscheinlichkeitsrestringierten Optimierung. In dieser Arbeit werden entsprechende Methoden zur Berechnung solcher Integrale vorgestellt. Eine weitere Verringerung der Rechenzeit wird durch die effiziente Approximierung der unterliegenden Modellgleichungen erreicht. We will distinguish the surplus-situation where the request can be completely fulfilled by means of the k servers and the scarcity-situation where the request cannot be completely met.

We use the method of the potential function by Bartal and Grove in order to prove that a corresponding Harmonic algorithm is competitive for the more general k-server problem in the case of unit distances. For this purpose we partition the set of points in relation to the online and offline servers' positions and then use detailed considerations related to sets of certain partitions. In vector optimization with a variable ordering structure the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space.

As these vector optimization problems are not only of interest in applications but also mathematical challenging, in recent publications it was started to develop a comprehensive theory. In doing that also notions of proper efficiency where generalized to variable ordering structures. In this paper we study the relations between several types of proper optimality notions, among others based on local and global approximations of the considered sets. We give scalarization results based on new functionals defined by elements from the dual cones which allow characterizations also in the nonconvex case.

Multiobjective optimization problems with a variable ordering structure, instead of a partial ordering, have recently gained interest due to several applications. In the previous years, a basic theory has been developed for such problems. The binary relations of a variable ordering structure are defined by a cone-valued map that associates, with each element of the linear space R m, a pointed convex cone of dominated or preferred directions. The difficulty in the study of the variable ordering structures arises from the fact that the binary relations are in general not transitive.

In this paper, we propose numerical approaches for solving such optimization problems. For continuous problems a method is presented using scalarization functionals, which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method, known from multiobjective optimization with a partial ordering, is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison.

The SDDP problem is based on an industrial problem, which contains an optimal conversion of machines. Partitions of integers as states of these stochastic dynamic programming problems involves combinatorial aspects of SDDP problems.

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Under the assumption of identical "basic costs" in other words of "unit distances" and independent and identically distributed requirements we will show in many cases by means of combinatorial ideas that decisions for feasible states with least square sums of their parts are optimal solutions. Optimal decisions of such problems can be used as approximate solutions of corresponding SDDP problems, in which the basic costs differ only slightly from each other or as starting decisions if corresponding SDDP problems are solved by iterative methods, such as the Howard algorithm.

In this paper, an erratum is provided to the article "On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets", published in Optim Lett, Due to precise observation of the first author, it has been found that the proof of Lemma 9 has a nontrivial gap, and consequently the main result Theorem 10 is incorrect. In this erratum, we prove that Corollary 14 is still correct in the original setting while to fix the proof of Theorem 10 we need additional assumptions.

We provide a list of different commonly used assumptions making this theorem to be true, and a new version of this theorem, which is now Theorem We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actually a semidefinite programming problem.


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Our main concern in this article are concepts of nondominatedness w. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Theory Appl. Our studies are motivated by some recent applications e. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop-Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.

In this paper, proper optimality concepts in vector optimization with variable ordering structures are introduced for the first time and characterization results via scalarizations are given. New type of scalarizing functionals are presented and their properties are discussed. The scalarization approach suggested in the paper does not require convexity and boundedness conditions. We present three new copositivity tests based upon difference-of-convex d.

The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d. We report on first numerical experience with this procedure which are very promising. This manuscript is on the theory and numerical procedures of vector optimization w.

That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Zu den Verfahren werden Experimente beschrieben, die anhand der von R. Reinhardt entwickelten und von A. So kann der Leser die Verfahren der Optimierung selbst erleben. Automatic differentiation is an often superior alternative to numerical differentiation that is yet unregarded for calculating derivatives in the optimization of imaging optical systems. Freeform optical surfaces offer additional degrees of freedom for designing imaging systems without rotational symmetry.

This allows for a reduction in the number of optical elements, leading to more compact and lightweight systems, while at the same time improving the image quality. This also enables new areas of application. Radial basis functions RBF have been used for many years e. In this contribution we investigate properties specific to RBF-based optical surfaces and compare the performance of RBF-based surfaces to other representations in selected optical imaging systems.

Interesting aspects include the dependency on the number of RBF that are summed to form the surface, the locality structure and its effects on optimization. Cone-valued maps are special set-valued maps where the image sets are cones. Such maps play an important role in optimization, for instance in optimality conditions or in the context of Bishop-Phelps cones.

In vector optimization with variable ordering structures, they have recently attracted even more interest. We show that classical concepts for set-valued maps as cone-convexity or monotonicity are not appropriate for characterizing cone-valued maps. For instance, every convex or monotone cone-valued map is a constant map. Similar results hold for cone-convexity, sublinearity, upper semicontinuity or the local Lipschitz property.

Therefore, we also propose new concepts for cone-valued maps. Multiobjective optimization problems with a variable ordering structure instead of a partial ordering have recently gained interest due to several applications. In the last years a basic theory has been developed for such problems. The difficulty in their study arises from the fact that the binary relations of the variable ordering structure, which are defined by a cone-valued map which associates to each element of the image space a pointed convex cone of dominated or preferred directions, are in general not transitive.

For continuous problems a method is presented using scalarization functionals which allows the determination of an approximation of the infinite optimal solution set. For discrete problems the Jahn-Graef-Younes method known from multiobjective optimization with a partial ordering is adapted to allow the determination of all optimal elements with a reduced effort compared to a pairwise comparison. In the paper a k-server problem with parallel requests where several servers can also be located on one point is considered.

In this paper, a chance constrained nonlinear dynamic optimization problem is considered, which will be investigated by using a moving horizon scheme. In each horizon, the chance constraints will be written transformed in terms of those input random variables with known probability distributions by using monotonicity relations. Some definitions and properties related to the required monotonicity properties are introduced.

For the application problem considered these monotonicity properties hold automatically true. The chance constraints and their gradients are evaluated by computing multivariate normal integrals using direct numerical integration. Numerical experimentation results will also be reported.

Der 2. Hence, we have attempted here to extend the robust analysis of Zheng et al. We are also of strong conviction that, the results of our investigation could open a way to apply the IGOM for the numerical treatment of some class of parametric optimization problems, when global optima are required. Diese Arbeit versucht erstmals, ohne Unterhalbstetigkeit der Index-Abbildung auszukommen.

Im zweiten Zugang werden zwei Straffunktionen vorgestellt. Es wird gezeigt, dass die entstehende Straffunktionen oberrobust i. A Sigma-Delta Modulator of order 2 is approximated by a differential equation of first order. A filter with this error bound related to the moving mean value of the input signal is proposed. In this paper an exact model for a basic, first order sigma-delta-modulator is derived by means of a difference equation with discontinuous nonlinearity. An explicit solution is given in terms of the greatest integer function under certain boundedness and initial conditions of the input signal.

Assumptions are made under which the explicit formula remains a solution of the difference equation although the suppositions of the main theorem are violated. The problem of finding the nearest in the Hausdorff metric circle to a non-empty convex compact set T in the plane is considered from geometrical point of view.

It can be characterized by a geometric Chebyshevian alternance. As a consequence in the particular case of a polygon the centre of the circle is described as an intersection of a midline between some two vertices and a bisectrix of some two sides. In the general case geometrical algorithms corresponding to the one and the four point exchange Remez algorithms are described.

They assure correspondingly linear and superlinear convergence. Following the idea, in the case of a polygon to get the exact solution in finite number of steps, a modified two point exchange algorithm is suggested and illustrated by a numerical example. An application is given to estimate the Hausdorff distance between an arbitrary convex set and its Hausdorff nearest circle. The considered problem arises as a practical problem by measuring and pattern recognition in the production of circular machine parts. We give an equivalence between the tasks of computing the essential supremum of a summable function and of finding a certain zero of a one-dimensional convex function.

Publications

Interpreting the integral method as Newton-type method we show that in the case of objective functions with an essential supremum that is not spread the algorithm can work very slowly. Let M be a Hausdorff compact topological space, let C M be the Banach space of the continuous on M functions supplied with the supremum norm and let V be a finite dimensional subspace of C M.

The problem of the Chebyshev approximation of a function f of C M by functions from V is generalized to two Chebyshev kind approximations of a point-to set mapping by a single valued continuous function from V using suitable distances between a point and a set. The first problem occur e. The second problem is useful for calculating continuous selections with special uniform distance properties.

Then the function F of a, defined by the integral of f - a over all x of D with f x larger as or equal to a, is continuous, non-negative, non-increasing, convex, and has almost everywhere the measure of the level set as derivative F' a. These properties can be used for computing the essential supremum of f. As example, two algorithms are stated. If the function f is dense, or lower semicontinuous, or if -f is robust, then supremum and the essential supremum of f coincide. In this case, the algorithms mentioned can be applied for determining the supremum of f, i.

We consider mixed suffucient conditions of optimality for abstract optimal control problems being local w. Using a special kind of augmented Lagrangian we don't need the kernel of the first derivative of the state equation or inequality for the formulation of the sufficient conditions. An example of its application to control problems with integral operators is given. We consider the approximation of the characteristics of thermocouples by polynomials of given degree on the largest possible range such that an apriori given error for the temperature is not violated.

Using a nonsymmetric duality for abstract continuous convex control problems optimality conditions are derived for calculating the primal and dual solutions in the case of linear on state depending dual operators. Functional and pointwise conditions are considered.

Using the distances of a point x to two convex sets we obtain an upper estimation of the distance of x to the intersection of these two sets. Applications to the intersection of point-to-set mappings are given. Reinhardt, A. Hoffmann, T. Springer, , p. Leipzig: Ed. ISBN TU Ilmenau Homelink. Karriere Weiterbildung Alumni International Aktuelles. Studieninteressierte Studierende Mitarbeiter Journalisten Wirtschaft. V, ISSN , , first online: 29 July , 23 Seiten In real life applications, optimization problems with more than one objective function are often of interest.

Thomann, Jana; Eichfelder, Gabriele; A trust-region algorithm for heterogeneous multiobjective optimization. Niebling, Julia; Eichfelder, Gabriele; A branch-and-bound-based algorithm for nonconvex multiobjective optimization. Eichfelder, Gabriele; Pilecka, Maria; Ordering structures and their applications.

Hildenbrandt, Regina; The k-server problem with parallel requests and the corresponding generalized paging problem. Boeck, Thomas; Terzijska, Dzulia; Eichfelder, Gabriele; Maximum electromagnetic drag configurations for a translating conducting cylinder with distant magnetic dipoles. Bao, Truong Quang; Eichfelder, Gabriele; Soleimani, Behnam; Tammer, Christiane; Ekeland's variational principle for vector optimization with variable ordering structure.

Niebling, Julia; Eichfelder, Gabriele; A branch-and-bound algorithm for bi-objective problems. Eichfelder, Gabriele; Pilecka, Maria; Set approach for set optimization with variable ordering structures : part II: scalarization approaches. Eichfelder, Gabriele; Pilecka, Maria; Set approach for set optimization with variable ordering structures : part I: set relations and relationship to vector approach. Hildenbrandt, Regina; The k-server problem with parallel requests and the compound Harmonic algorithm. Eichfelder, Gabriele; Jahn, Johannes; Vector and set optimization. Eichfelder, Gabriele; Gerlach, Tobias; Sumi, Susanne; A modification of the [alpha]BB method for box-constrained optimization and an application to inverse kinematics.

Eichfelder, Gabriele; Gerlach, Tobias; Characterization of properly optimal elements with variable ordering structures. Eichfelder, Gabriele; Variable ordering structures - what can be assumed? Eichfelder, Gabriele; Vector optimization in medical engineering. Eichfelder, Gabriele; Pilecka, Maria Set approach for set optimization with variable ordering structures - Ilmenau : Techn.

Eichfelder, Gabriele; Kasimbeyli, Refail Properly optimal elements in vector optimization with variable ordering structures.

Veröffentlichungen

Bao, Truong Q. Hildenbrandt, Regina; The k-server problem with parallel requests and the compound Harmonic algorithm - Ilmenau : Techn. Eichfelder, Gabriele; Variable ordering structures in vector optimization - Berlin : Springer, - xiii, Seiten. Hildenbrandt, Regina; A k-server problem with parallel requests and unit distances. Eichfelder, Gabriele; Gerlach, Tobias Characterization of proper optimal elements with variable ordering structures - Ilmenau : Techn.

Eichfelder, Gabriele; Numerical procedures in multiobjective optimization with variable ordering structures. Hildenbrandt, Regina; Partitions-requirements-matrices as optimal Markov kernels of special stochastic dynamic distance optimal partitioning problems.

Mod-09 Lec-33 Optimality Conditions and Simplex Tableau

Dickinson, Peter J. Recalls Convex analysis : asymptotic function Linearization methods The semismooth Newton algorithm for nonlinear equations : Clarke differential, semismoothness, local convergence Interior point methods Semidefinite optimization : problem definition and examples, existence of solution, optimality conditions. Examination written examination, made of "exercises", 3 hours, with documents distributed in the course.

Second part The course is scheduled on November: 20, 27; Decembre: 4, 11, It is composed of 5 sessions of 4h each on Monday 14hh , which makes it 20h long. Examination: 8th and 15th of January , 14h by appointment. The goal of this course is to guide the student in the implementation of some well known optimization algorithms and in its use to solve a concrete problem.


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