Nach oben pdf Vector and matrix apportionment problems and separable convex integer optimization

Vector and matrix apportionment problems and separable convex integer optimization

Vector and matrix apportionment problems and separable convex integer optimization

Algorithms for the proportional rounding of a nonnegative vector, and for the bipro- portional rounding of a nonnegative matrix are discussed. Here we view vector and matrix rounding as special instances of a generic optimization problem that employs an additive version of the objective function of Gaffke and Pukelsheim (2007). The generic problem turns out to be a separable convex integer optimization problem, in which the linear equal- ity constraints are given by a totally unimodular coefficient matrix. So, despite the integer restrictions of the variables, Fenchel duality applies. Our chief goal is to study the implied algorithmic consequences. We establish a general algorithm based on the primal optimiza- tion problem. Furthermore we show that the biproportional algorithm of Balinski and Demange (1989), when suitably generalized, derives from the dual optimization problem. Finally we comment on the shortcomings of the alternating scaling algorithm, a discrete variant of the well-known Iterative Proportional Fitting procedure.
Mehr anzeigen

24 Mehr lesen

Convex Optimization for Inequality Constrained Adjustment Problems

Convex Optimization for Inequality Constrained Adjustment Problems

It is important to notice that the direction of the manifold cannot be changed through the intro- duction of inequality constraints. More specifically, a translation (case 2b), a general restriction (case 1b) or a dimension reduction (case 1c and 2a) of the manifold are possible, but never a rota- tion. This leaves us in the comfortable situation that it is possible to determine the homogeneous solution of an ICLS problem by determining the homogeneous solution of the corresponding uncon- strained WLS problem and reformulate the constraints in relation to this manifold. Therefore, our framework consists of the following major parts that will be explained in detail in the next sections: To compute a general solution of an ICLS problem (3.8), we compute a general solution of the unconstrained WLS problem and perform a change of variables to reformulate the constraints in terms of the free variables of the homogeneous solution. Next, we determine if there is an intersection between the manifold of solutions and the feasible region. In case of an intersection, we determine the shortest solution vector in the nullspace of the design matrix with respect to the inequality constraints and reformulate the homogeneous solution and the inequalities accordingly. If there is no intersection, we use the modified active-set method described in Sect. 5.2.2 to compute a particular solution and determine the uniqueness of the solution by checking for active parallel constraints.
Mehr anzeigen

140 Mehr lesen

Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems

Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems

Impressum: Herausgeber: Institut f¨ur Mathematik Universit¨at Augsburg 86135 Augsburg http://www.math.uni-augsburg.de/forschung/preprint/ ViSdP: Friedrich Pukelsheim Institut f¨ur Mathem[r]

26 Mehr lesen

Generalized Functional Matrix Learning Vector Quantization (GFMLVQ)

Generalized Functional Matrix Learning Vector Quantization (GFMLVQ)

The basis of the success of all these LVQ algorithms was the simple heuristic learning strategy of LVQ1 only by moving prototypes in a simple way (see Section 2.3). The major problem is that a stable behaviour and convergence is not guaranteed during the learning phase, but in most cases the simple LVQ1-algorithm leads to a reasonable classifier in an appropriate time and the solution is understandable for all users. Improvements were made to diminish the bad attributes of the basic LVQ1 approach. The most important and revolutionary improvement was the introduction of a differentiable cost-function in GLVQ by Sato and Yamada (see Section 2.4) - a quantum leap for those classification methods. On the one hand, due to a differentiable cost-function, a stable behaviour and convergence is satisfied and on the other hand more and more ideas can be realized (see Figure 2.29), like learning a problem-specific metric during the learning-phase. Both GRLVQ and GMLVQ have remarkable properties and enabled the user to understand why the classification is realized in such a good way or not - additional informations about the data can be found. Another advantage is that the concepts can be fitted to specific data. The GFRLVQ algorithm is the best example how to modify the GRLVQ algorithm to take the functional context of data into account.
Mehr anzeigen

123 Mehr lesen

On core stability and apportionment methods

On core stability and apportionment methods

The realization of this thesis would not have been possible if it were not for the assistance of numerous people. The author must therefore thank, first and foremost, his supervisors Professor emeritus Joachim Rosenmüller and Associate Professor Peter Sudhölter. Both have contributed enormously not only to the theoretical results but also to the presenta- tion, the questions which were investigated, to my financial, psychological and emotional support as well as contributing significantly to the enrichment of my stay in Germany in many other ways. I thank Claus-Jochen Haake for his assistance with many of my ques- tions concerning all sorts of aspects of game theory. I must also thank Matthias Schleef for his unrelenting assistance with my Latex problems, Yaron Azrieli for reading and cor- recting some of the section on fuzzy games and the members of EBIM and BIGSEM for their numerous discussions and exchanges of opinions.
Mehr anzeigen

77 Mehr lesen

Approximation and online algorithms for selected network optimization problems

Approximation and online algorithms for selected network optimization problems

Though overcoming this obstacle can be done in a surprisingly simple way, this approach can be easily overlooked. When designing the first PTAS for the Santa Claus scheduling problem on identical machines Woeginger [Woe97] stated: “Rounding large jobs to a constant number of distinct job sizes of course simplifies the problem, but there seems to be no way of integrating the small jobs into a dynamic program.” The idea to circumvent this problem is to abstract from specific small rate allocations and only consider the required total volume. As already exploited in the PTAS in Section 4.1.3.1, Lemma 4.9 does not impose any assumption about the structure of the set of small channels other than that their individual sizes are bounded by a given threshold  ∗ := λd. This even allows us to replace them by a set of q := bV / ∗ c many channels of size  ∗ and one channel of size V −  ∗ q without affecting the approximation guarantee. We may further omit this last channel and lose at most an additive term of  ∗ in the total data rate of at most one terminal.
Mehr anzeigen

135 Mehr lesen

Exploiting structure in non-convex quadratic optimization and gas network planning under uncertainty

Exploiting structure in non-convex quadratic optimization and gas network planning under uncertainty

In the first part of the thesis, we concentrate on quadratic programs as an impor- tant subclass of MINLP. In this problem class, the multiplication of two variables is the only source of nonlinearity and McCormick [McC76] already in 1976 provided a tight relaxation for the product of two bounded variables. This termwise relaxation, however, is weak when terms interact or when more constraints are present in the model. The so-called Reformulation-Linearization-Technique (RLT) [SA92] is well known to strengthen the relaxation of quadratic programs by capturing structure of the constraints. In a nutshell, the ideas is to multiply a linear constraint by a variable which yields product terms. Auxiliary variables representing the products of two variables are used to reformulate the constraint making it linear again. Different implementations for this method have different ways to handle product terms that appear only in constraints generated by this approach. We propose to project them out by replacing them with appropriate over- and underestimators, an approach that has not been described in the literature to the best of our knowledge. Within the work on this thesis, the author implemented the projected RLT cuts in the commercial solver CPLEX [IBMb], where they are enabled in the default settings for optimization problems with a non-convex quadratic objective in version 12.7.0. An overview about reformulations and relaxations for quadratic programs together with a presentation of (projected) RLT is given in Chapter 3.
Mehr anzeigen

411 Mehr lesen

Existence and approximation results for shape optimization problems in rotordynamics

Existence and approximation results for shape optimization problems in rotordynamics

in rotordynamics Frank Strauß ∗ Vincent Heuveline † Ben Schweizer ‡ Abstract We consider a shape optimization problem in rotordynamics where the mass of a rotor is min- imized subject to constraints on the natural frequencies. Our analyis is based on a class of rotors described by a Rayleigh beam model including effects of rotary inertia and gyroscopic moments. The solution of the equation of motion leads to a generalized eigenvalue problem. The governing operators are non-symmetric due to the gyroscopic terms. We prove the existence of solutions for the optimization problem by using the theory of compact operators. For the numerical treatment of the problem a finite element discretization based on a variational formulation is considered. Applying results on spectral approximation of linear operators we prove that the solution of the discretized optimization problem converges towards the solution of the continuous problem if the discretization parameter tends to zero. Finally, a priori estimates for the convergence order of the eigenvalues are presented and illustrated by a numerical example.
Mehr anzeigen

19 Mehr lesen

Domain-independent local search for linear integer optimization

Domain-independent local search for linear integer optimization

Within the empirical study, our original results include – evidence that large and tightly constrained problems beyond the limitations of ILP branch-and-bound can be solved directly from[r]

151 Mehr lesen

Geostationary Satellite Station-Keeping Using Convex Optimization

Geostationary Satellite Station-Keeping Using Convex Optimization

angle γ ∈ [0, 90 ◦ ], both rotations are toward the radial direction. A second rotation is applied as follows; thrusters T1 and T2 are rotated by an angle β ∈ [0, 90 ◦ ] about the North-axis (towards the East) and T3 and T4 at an angle β ∈ [0, 90 ◦ ] about the South-axis (towards the West). One configuration (A) used in this paper is obtained for γ = 45 ◦ and β = 90 ◦ , shown in Fig. 2. The arrows point in the direction of the acceleration that is exerted on the spacecraft by each thruster (which is opposite to the exhaust plume). With this choice for β and γ the thrust vector lies completely in the tangential, normal plane, which is a common choice for geostationary satellites. This configuration is similar as implemented on the Hispasat Advanced Generation 1 mission [13]. Another configuration (B) that is analyzed in this paper is obtained by choosing γ = 45 ◦ and β = 10 ◦ ,. Fig. 3 shows the projected thrust force vectors in the TN, RT and RN planes. Note that for this configuration the thrusters are all pointing away from the solar panels as well as away from the Earth-facing panel. We also use a reference configuration (REF) with four thrusters pointing respectively North, East, South and West (Fig. 1). Note that additionally the location of the thruster could be optimized, for example to support a combined orbit control and attitude control or momentum management strategy. The configurations used in this work assume each thrust force vector to pass through the satellite’s center of mass and attitude dynamics are ignored.
Mehr anzeigen

36 Mehr lesen

Projected Push-Sum Gradient Descent-Ascent for Convex Optimization with Application to Economic Dispatch Problems

Projected Push-Sum Gradient Descent-Ascent for Convex Optimization with Application to Economic Dispatch Problems

One of the first projection-based, distributed gradient meth- ods was published in [8]. However, the proposed method relies on double-stochastic matrix-updates, which restrict the communication to undirected graphs. The contributions in [4] and [15] rely on row-stochastic communication matrices for diffusing the projected gradient information. Finally, [14] employs the push-sum consensus combined with a projection that uses a convex proximal function. The major drawback of the mentioned projection-based methods in relation to the specific structure of the DEDP under consideration is the assumption that all constraints of the distributed optimization problem are known by every agent and therefore global. This restricts the privacy of the agents with local constraints. Compared to the projection-free method in [13], they have the advantage that no penalization parameter sequence needs to be chosen.
Mehr anzeigen

8 Mehr lesen

Collocation of geostationary satellites using convex optimization

Collocation of geostationary satellites using convex optimization

Syncom II was the first satellite to arrive in a geosyn- chronous orbit. Since that time many satellites followed, and in particular the geostationary orbit has become in- creasingly populated. Driven by the need to avoid radio- frequency interference between different satellites, the geo- stationary orbit was divided into slots, which are allocated by the International Telecommunication Union. The lim- ited availability and difficulty of obtaining these slots, es- pecially at key locations above highly populated areas, together with the ever increasing need for geostationary satellite services lead several organizations to collocate multiple satellites within a geostationary slot, see e.g. [1]. Collocation strategies used to control more than two satellites in one slot generally rely on a coordinated ap- proach to specifying desired states. The satellites’ desired mean eccentricity and inclination vectors are defined in a configuration that is passively safe. Each satellite in the fleet is then controlled individually to stay close to this de- sired state [2]. The idea is to maintain relative eccentricity and inclination vectors (anti-)parallel, to ensure that ra- dial separation is maximum when normal separation van-
Mehr anzeigen

11 Mehr lesen

Integer programming methods for special college admissions problems

Integer programming methods for special college admissions problems

The first special feature of the application is the presence of ties, and the solution concept of stable score-limits. According to the Hungarian admission policy, when two applicants have the same score at a programme then they should either both be accepted or rejected by that programme. The solution of stable score-limits ensures that no quota is violated, hence the last group of students with the same score that would cause a quota violation is always rejected. A set of stable score-limits always exists, and a student- optimal solution can be found efficiently by an extension of the Gale-Shapley algorithm, as shown in [9]. This method is the basis of the heuristic used in the Hungarian application. The second and third special features studied in this paper are the lower and common quotas. A university may set not just an upper quota for the number of admissible students for a programme, but also a lower quota. A violation of this lower quota would imply the cancellation of the programme. Furthermore, a common upper quota may be also introduced for a set of programmes, to limit the number of students admitted to a faculty, at a university or nationwide with regard to the state-financed seats in a particular subject. These concepts were studied in [7], where the authors showed that each of these special features makes the college admission problem NP-hard, even in the form that is present in the Hungarian application. Finally, students can apply for pairs of programmes in case of teacher education programmes. This possibility was reintroduced in the scheme in 2010. This problem is closely related to the Hospitals / Residents problem with Couples, where couples may apply for pairs of positions. The latter problem is also known to be NP-hard [24], even for unit-capacity hospitals, for so-called consistent preferences [21], and also for a specific setting present in Scotland [8] where hospitals have common rankings. The fact that the unit-capacity case is also NP-hard implies the NP-hardness of the paired application problem as well.
Mehr anzeigen

33 Mehr lesen

On the use of some tangent cones and sets in vector optimization

On the use of some tangent cones and sets in vector optimization

(8), with D = R p + :  For a more general approach to sucient rst-order optimality conditions see Giorgi, Jimenez and Novo (2008) . We have to note that condition (ii) is not very useful, as it is the same both for local ecient minimum points and for local ecient maximum points. In order to obtain useful conditions one has to impose some kind of generalized convexity (or concavity, in case of a maximum problem) on the objective function f (see, e. g., Cambini and Martein (1993, 1994) ).

25 Mehr lesen

Formula apportionment: Factor allocation and tax avoidance

Formula apportionment: Factor allocation and tax avoidance

Compared to other firm panels like Compustat or AMADEUS, AFiD holds a number of major advantages for our analysis. Unlike public accounting data, the Investment Survey and the Monthly Report provide very detailed information on the volume and composition of payrolls, investments, and sales revenues, and these data are collected at the establishment level. Since we complement the data with information on local tax rates, we are not only able to analyze correlations between payroll expense and tax rates, but also correlations with number of employees, number of working hours per employee, sales revenue, gross investment, and measures for tax avoidance (payroll per number of hours worked, payroll per unit of sales revenue) on the establishment and firm levels. To our knowledge, this is a unique feature, allowing us a more detailed analysis than in previous research. Both surveys are conducted as a mandatory census for all domestic establishments in the manufacturing and mining industries with at least 20 employees; therefore, non-response and sample selection are not issues. An additional advantage stems from the fact that the data are anonymized and available only for political and scientific use. Hence, there should be a smaller incentive for survey participants to “brighten the numbers” as in balance sheet information.
Mehr anzeigen

36 Mehr lesen

Formula apportionment: Factor allocation and tax avoidance

Formula apportionment: Factor allocation and tax avoidance

The German practitioner literature discusses a range of tax avoidance strategies to manipulate payroll expense as an FA factor (Dietrich & Krakowiak, 2009; Scheffler, 2011). To obtain a better understanding of tax avoidance opportunities in the German FA system, we rely on a series of qualitative interviews with a focus on tax-planning and audit procedures. Overall, we use 19 interviews with tax advisers, business taxpayers, staff members of tax authorities and staff members of municipal authorities. 7 Importantly, this qualitative research confirms anecdotal evidence of a weak tax audit system for FA purposes. Hence, the fiscal administration and the municipal authorities have only very limited opportunities to check the validity of the payroll distribution reported in a firm’s partition statement. This is partially driven by the fact that firms’ central offices administer employee contracts, while the establishment is not a legally distinct entity. As a result, there exist no official documents on the workplaces of employees and no official accounts on the distribution of payroll among establishments. Using the tax practitioner literature, as well as the findings of our own qualitative research, we were able to identify tax-planning strategies for FA factor manipulation. One example for such tax planning is that an establishment in a low-tax municipality leases employees to other establishments or subsidiaries of the same firm group that are located in high-tax municipalities. The other business units pay a leasing fee for the employees. However, this leasing fee is not considered a payroll expense under German FA rules. As a result, the FA-relevant payroll expense is paid in a low-tax municipality, while the hours worked are performed in a high-tax municipality. This increases (reduces) the weight of low-tax (high-tax) municipalities in the FA formula.
Mehr anzeigen

53 Mehr lesen

Separable graphs, minors and the reconstruction conjecture

Separable graphs, minors and the reconstruction conjecture

The third chapter deals with minors in the reconstruction conjecture and the edge- reconstruction conjecture. We pose the question whether containing a specific mi- nor or not is reconstructible or edge-reconstructible for certain graphs. The classes of graphs we investigate are based upon their connectness. We give a bound for 2-connected graphs and connected minors as well as we show that the problem is edge-reconstructible for a range of other graph classes. The chapter closes with an application of the results. In particular we were able to show that the Hadwiger num- ber and the treewidth is edge-reconstructible for a wide range of cases. We also give a bound for the edge-reconstruction of the Hadwiger number and the treewidth based upon the ratio between the order of the graph and the order of the minor. Further- more these results can easily be applied to various exluded minor theorems (a graph has a specific property if and only if it has no minors isomorphic to a specific set of graphs).
Mehr anzeigen

122 Mehr lesen

Formula apportionment: Factor allocation and tax avoidance

Formula apportionment: Factor allocation and tax avoidance

The negative and significant results for the Payroll per hour ratio and the Payroll per revenue ratio document that the impact of TaxD on the FA factor payroll expense is stronger than FA impacts on input and output measures. This constitutes evidence for the existence of more or less artificial tax avoidance strategies that do not result from a reallocation of real labor input (see also Section 2). Our regression results imply that tax avoidance may be an important element of the overall FA effect on payroll. Comparing the range of coefficients of tax avoidance measures (-0.545 to -0.929) with the effect on working hours (-0.633), tax avoidance may be responsible for a significant part of the overall impact of TaxD on Payroll share. 13 Considering that our tax avoidance measures are probably not appropriate to identify all tax avoidance strategies, tax avoidance may even be responsible for a major part of the FA impact. However, as it is very hard to measure tax avoidance under an FA system directly and consequently our proxies are indirect measures for tax avoidance, the results have to be interpreted with due caution. An alternative explanation for a tax-driven reduction of payroll expense without a corresponding change in the underlying real input measure (number of working hours) or output measure (sales revenues) might be tax incidence. It has been argued that businesses may impose the local tax burden on their employees by reducing gross wages (Fuest et al., 2013). However, taking into account the binding force of labor market contracts and the tariff commitment of most German industries (especially in the manufacturing sector), this does not seem to be a likely explanation for the strong immediate tax effects on the Payroll per hour ratio found in our paper. While employment and incidence effects of taxes should have delayed effects (Fuest et al., 2013, Siegloch, 2014), our regression results rather imply a rapid effect from the tax rate differential on Payroll share, the Payroll per hour ratio and the Payroll per
Mehr anzeigen

38 Mehr lesen

A convex analysis approach to multi-material topology optimization.

A convex analysis approach to multi-material topology optimization.

2006 ] are amongst the earliest and most frequently used techniques. A standard approach for the two-material case consists in setting u(x) = u 1 w(x) + u 2 (1 − w(x)) and minimizing over the set of all characteristic functions w(x) ∈ {0, 1}. This problem is non-convex, but its convex relaxation – minimizing over all w(x) ∈ [0, 1] – often has a bang-bang solution, i.e., w(x) ∈ {0, 1} almost everywhere. For multi-material optimization, this approach can be extended by introducing multiple characteristic functions; non-overlapping materials can be enforced by considering the third domain as an intersection of two (possibly overlapping) domains, e.g., u(x) = u 1 w 1 (x) + u 2 (1 − w 1 (x))w 2 (x) + u 3 (1 − w 1 (x))(1 − w 2 (x)) for w 1 (x), w 2 (x) ∈ [0, 1]. For an increasing number d of materials, this approach has obvious drawbacks due to the combinatorial nature and increasing non-linearity. Shape calculus techniques [ Pironneau 1984 ; Sokołowski and Zolésio 1992 ] focus on the effect of smooth perturbations of the interfaces on the cost functional and have reached a high level of sophistication. From the point of view of numerical optimization, they are first-order methods and stable, with the drawback that they mostly allow only smooth variations of the reference geometry. When combined with level-set techniques [ Allaire, Jouve, and Toader 2004 ; Ito, Kunisch, and Li 2001 ], they are flexible enough to allow vanishing and merging of connected components, but they do not allow the creation of holes. This is allowed in the context of topological sensitivity analysis [ Garreau, Guillaume, and Masmoudi 2001 ; Sokołowski and Żochowski 1999 ], which investigates the effect of the creation of holes on the cost. Let us point out that in our work we do not rely in any explicit manner on knowledge of the shape or the topological derivatives. Moreover, the numerical technique that we propose is of second order rather than of gradient nature. Second-order shape or topological derivative analysis is available, but it is involved when it comes to numerical realization. Multi-material optimization for elasticity problems are further investigated in [ Haslinger et al. 2010 ] by means of H-convergence methods and by phase-field methods in [ Blank et al. 2014 ]. The work which in part is most closely related to ours is [ Amstutz 2011 ], see also [ Amstutz and Andrä 2006 ; Amstutz 2010 ], where for the case of linear solution operators and two materials, the set of coefficients is expressed in terms of characteristic functions, and the resulting problem is considered in function spaces rather than in terms of subdomains and their boundaries. The first order-optimality condition is derived and formulated as a nonlinear equation for which a semi-smooth Newton method is applicable.
Mehr anzeigen

23 Mehr lesen

Synthesis of plantwide control strategies using mixed integer optimization

Synthesis of plantwide control strategies using mixed integer optimization

In case 3, the vapor boil-up rate VB, the reflux ratio set-point RR and purity of the fresh feeds (i.e., presence of other reactant in the fresh feed) are four uncertain disturbances acting on the column. The magnitude of these step disturbances follows the joint normal PDF with mean and covariance according to Table 3.3. The GBD based sequential solution approach demands 75 minutes of computation time using the sigma point method, since the complexity of the resulting MIDO problem is increased with the number of uncertain disturbances. The optimal control structure and controller parameters are summarized in Table 3.5 which are different from the other two cases as well as the heuristic method and the nominal case. The expectation and the variance of the performance index at the optimal decentralized control system is compared with specialized cubature formula [57] and verified with random samples which are given in Table 3.9. Here, we use the specialized cubature formula for comparison purpose in stead of the specialized product Gauss formula due to the fact that it is the choice of a suitable numerical integration method for 3 ≤ n ≤ 7 [57]. However, it requires 24 number of grid points which is higher than that of the sigma point method. Further, the sigma point method estimates the expectation within 0.8% error around the random samples with 10000 observations which is shown in Fig 3.7. Like the previous cases, the optimal decentralized control system from the stochastic approach has superior performance compared to the heuristic method and the nominal case which is shown in Table 3.10 through the statistical objective function value. Further, we observed in this case also that not much improvement was found for a multivariable controller compared to the decentralized control system.
Mehr anzeigen

114 Mehr lesen

Show all 10000 documents...