by (−2, . . . , −2) **and** (2, . . . , 2) was randomly chosen. All methods were tested starting from this set of initial points. The Algorithms were implemented in Matlab 2018a using the included solvers fmincon **and** quadprog to solve the search direction **problems**. The computations were executed on an Intel Core i5 processor with 3.2 GHz **and** 8 GB of ram. Table 3.1 shows the maximum **and** average num- ber of iterations (which also coincides with the number of computations of the Hessian matrix **for** the newton method), the average time **for** reaching the final iterate **and** the average number of function evaluations (where one evaluation is an evaluation of the m-dimensional function f (x)). We observe that the bulk of the function evaluations occurs during the computation of the Armijo step sizes. The average runtimes depend highly on the performance of the solvers used **and** should only be un- derstood in this context. We can observe a distinct reduction in the number of iterations when using Newton **and** BFGS instead of the steepest descent method. This is expected **and** in accordance with results from scalar **nonlinear** **optimization** [36].

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work are subsystems of the RailCab vehicle which is a novel linear motor driven railway system including the latest technologies **and** innovative pro- totypes, developed by the project RailCab (“Neue Bahntechnik Paderborn”, cf. [73]). In particular, we have considered a hybrid energy storage system, the guidance module **and** the doubly-fed linear drive. The new aspect of our study of these applications is the numerical approximation of entire Pareto sets. Based on the resulting global picture on the **solution** set, the engineers were now able to choose suitable Pareto optimal solutions. In the case of the linear drive we even developed a situation-dependent heuristic which allows the automatic choice of suitable Pareto optimal solutions during operation time. In the case of the guidance module certain reference trajectories are de- sired to be optimized with respect to several objectives. This would directly lead to an optimal control problem with multiple objectives. As the guidance module is modeled as a differentially flat system, we were able to reformu- late the optimal control problem into a **nonlinear** **multiobjective** **optimization** problem. This type of reformulation has been introduced by Murray et al (cf. [72, 104, 70, 31]) **for** the case of one single objective function. In this work it is shown that the same idea works **for** the **multiobjective** case.

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There has been a lot of research about robust model predictive control. Mayne et al. [63] give a good overview about the beginnings of the research about model predictive control until the year 2000. They consider works about constrained linear **and** **nonlinear** dynamic systems **and** about model predictive control of **problems** that are difficult to solve, e.g. control of unconstrained **nonlinear** systems **and** time varying systems. Good **and** general introductions to feedback, closed-loop **and** control policies can be found, e.g., Mayne [62], Kothare et al. [49],Lee **and** Yu [55] or Scokaert **and** Mayne [74]. Bemporad et al.[7] **and** Magni et al. [59] describe methods in their work that combine dynamic **and** parametric programming **approaches** **for** solving discrete min-max optimal control **problems** under the assumption that the perturbations take values in a poly- tope. Another approach **for** solving a predictive optimal control problem can be found in Kerrigan **and** Maciejowski [42] or in Scokaert **and** Mayne [74], where single finite di- mensional **optimization** techniques are used. Lee **and** Yu [55] use dynamic programming by discretizing the state space to solve the predictive optimal control problem in a state feedback form.

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Applications The design **optimization** examples from electrical engineering have shown that the presented approach **for** solving mixed-integer **nonlinear** **problems** determines so- lutions of high quality **for** the system design. The benefit of a direct handling of implicit constraints has been demonstrated **for** the design of a magnetic bearing. **For** the design of a superconductive synchrotron magnet all earlier applied methods have not been able to deal with the integer-valued variables that define the devices’ structure. With the amount of simulation evaluations that other methods needed to solve the inherent continuous-valued **nonlinear** **problems**, a competitive design close to the theoretical minimum was identified. The second type of applications discussed is taken from environmental engineering. In a reply to Lucas (both in [180]), Sacks **and** his colleagues pointed out that first, simulation **problems** often do not behave like stochastic **problems** **and** second, that finally, even the test **problems** should be some kind of simulation application. The presented applications, called Community **Problems**, consist not only of a number of **optimization** **problems** based on a subsurface flow simulation. They are also established as a benchmark set of **problems** **for** any derivative free **optimization** method, as was claimed by Sacks. It has been shown earlier, **for** the Community **Problems**, that deterministic sampling methods can outperform metaheuristic search methods [93–95]. The presented surrogate **optimization** approach, combined with a newly introduced mixed-integer **nonlinear** problem formulation, resulted in an even better **solution** obtained by only a small fraction of black box evaluations needed by other **approaches**.

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There is a considerable body of literature on separable **convex** programming (integer or con- tinuous) with linear constraints, providing efficient algorithms **for** **solution**, (cf. Hochbaum **and** Shantikumar (1990)). These results are still to be exploited **for** (bi)proportional rounding pur- poses. More general **nonlinear** integer **optimization** **problems** are considered in Murota, Saito, Weismantel (2004) **and** in Hemmecke (2003). We will concentrate on separable **convex** integer programming **problems** under totally unimodular linear equations.

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In practice, the input data of the problem is seldom known exactly. The reasons **for** uncertainty in the input data are manifold **and** include measurement errors, model simplifications, **and** uncertain forecasts. Investments into the gas network infrastructure are extremely costly **and** impact the network performance **for** decades to come. Therefore, small changes in the quality of the decisions can result in substantial financial gains or losses. At the same time the future transport patterns in Europe are highly uncertain as the EU is increasingly dependent on gas imports [Com]. While deterministic planning **approaches** focus on one bottleneck scenario, the long planning horizon paired with high uncertainty ask **for** planning methods that take several scenarios **for** future demand into account in order to prepare the network **for** future challenges. We use the framework of Robust **Optimization** [BGN09] where instead of assuming that the data that describes the objective **and** the constraints is known, the input data assumed to realize itself within an uncertainty set. The decisions that are to be determined then must be robust, i.e., they need to be feasible no matter how the data manifests itself with the uncertainty set. The concept of robust **optimization** is reviewed in Section 2.3. **For** linear mixed-integer **problems**, tractable robust counterparts can be derived **for** several classes of uncertainty sets, such as conic or polyhedral sets. As we are facing a complex MINLP, much less is known about tractable robust counterparts. We therefore consider a discrete uncertainty set that consists of a finite number of scenarios **for** the uncertain data. This reflects the situation in which different scenarios are collected from historical data, future forecasts, or domain experts. One typical property of these models is that only very few variables **and** constraints couple the different scenarios **and** thus decomposition methods are a common weapon of choice. We also follow this route **and** design a decomposition algorithm that takes the particular structure of our model into account.

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min
x y x . In our definition we have only two anchor points in total: (0,1) **and** (1,0).
In the original approach the displacement along the utopia line does not provide any **solution**, except the utopia points because box D is in the unfeasible space Y \ Y * . Therefore, the off-set strategy is required in this example. If the displacement of point M is performed along the line parallel to the line A as shown in Figure 9a, a complete Pareto frontier can be obtained by the original method. Yet, this Pareto set is not evenly distributed. It is important to note here that only five different Pareto points are obtained in spite of solving eleven local **optimization** **problems**. This result appears because some locations of point M lead to the same Pareto **solution**. It is to be noted that we chose line A utilizing our a priori knowledge of the Pareto **solution**. If we consider line B then we obtain only 4 Pareto solutions marked by the circles. In combination, these two computations altogether provide us an evenly distributed Pareto set. However, in order to obtain 7 Pareto solutions we have to consider 13 single-objective **optimization** **problems** if start from line A **and** 14 **problems** if we start from line B.

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The motivation **for** the use of evolutionary **optimization** algorithms relies on their ability in dealing with local optimal **solution** **and** control constraints, that naturally arise in **nonlinear** optimal control **problems** [1, 4, 5]. Contributions made to apply evolutionary **optimization** techniques can be found in literatures. **For** instance, a constrained space plane reentry problem was solved in [13], wherein a Genetic Algorithm (GA) was applied to generate the optimal reentry trajectories. Similarly, in [14] a low-thrust interplanetary trajectory problem was formulated **and** solved via a modified GA. Pontani **and** Conway [15] investigated an optimal finite-thrust rendezvous trajectory problem. In their work, a Particle Swarm **Optimization** (PSO) algorithm was applied to solve the rendezvous optimal control problem. The main advantage with evolutionary **optimization** methods is that it is simple to understand **and** easy to apply. Besides, it is more likely than traditional gradient- based methods to locate the global optimum **solution**. Therefore, in this study, an enhanced GA is introduced to optimize the transcribed **optimization** model. Compared with traditional GA, it uses a hybrid evolutionary strategy **and** tends to have better local searching ability.

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As far as generic methods are concerned, since these algorithms are generic, some per- formances of them in some case can’t be fully satisfied. However, these special methods are applicable either to these **optimization** **problems** having **convex** search region only or to these op- timization problem whose objective **and** constraint functions are differentiable. In fact, among the generic methods, the most popular approach in real **optimization** fields to deal with the constraint of an **optimization** problem is the penalty function method, which involves a num- ber of penalty parameters **and** we must to set right in any algorithms in order to obtain the optimal **solution**, **and** this performance on penalty parameter has led many researches to de- vise the sophisticated penalty function method. These methods mainly can be divided three categories: a) multi-level penalty functions [16] ; b) dynamic penalty functions based on adaptive

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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.

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when p < r < p ∗ **and** infinitely many solutions when 1 < r < p by using the Mountain-Pass Theorem **and** the ”concentration-compactness principle”, respectively. A similar result of the same authors is also developed in [124]. The existence of multiple solutions **and** sign-changing solutions **for** zero Neumann boundary values has been proven in [88, 107, 108, 122] **and** [125], respectively. Analogous results **for** the Dirichlet problem have been recently obtained in [35, 36, 37, 41, 57, 99, 101]. An interesting problem about the existence of multiple solutions **for** both, the Dirichlet problem **and** the Neumann problem, can be found in [44]. The authors study the existence of multiple solutions to the abstract equation J p u = N f u, where J p is the duality mapping on a real reflexive **and** smooth Banach space X , corresponding to the gauge function ϕ(t) = t p−1 , 1 < p < ∞ **and** N

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In the descent phase the guidance subsystem has to generate a trajectory from the last position of the map- ping phase to the selected landing site. It also needs to maintain its attitude within certain bounds to make sure, that the target body is in sight. The HDA system is still active in this phase to make sure in case of con- tingency, that a retargeting is initiated **for** the guidance algorithm to reevaluate **for** a new optimal trajectory. It would generate the required acceleration profiles **for** this **and** provide it to the control subsystem to execute them. While generating the acceleration profiles it also needs to take care of the control actuator limitations. The TAG descent requires a combination of long ascent thrusts while preserving a desired safe attitude to prevent contact with the surface of the target body. These require a tight coupling between the trajectory **and** attitude control **and** a six DOF guidance **and** control (3 **for** the position, 3 **for** the attitude) (Xinfu et al. (2017)). As discussed in the optimisation algorithm heritage, we would approach this problem with successive **convex**- ification. Advantages of this method are, that it guarantees convergence **for** a well-posed **convex** problem, the **solution** is the global optimum **and** is a **solution** to the original problem, a number of efficient solvers have been developed **for** this kind of problem **and** constraints **and** penalties can be imposed. The novel method of this thesis is the combination of the descent problem in the form of a **convex** problem using dual quaternions with relative states.

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Chapter 7
NRW Income **and** Taxation Data
Microcensus in Germany is being carried out since 1957. The German microcensus survey is integrated into the Labour Force Survey of the European Union (EU Labour Force Survey). The microcensus aims to collect official statistical figures about the population. It is helpful to have an inference about the labour market, economic **and** social activity of the population, education **and** training situations **and** on health **and** housing situations. Stratified sampling techniques are used to select samples **and** the sample size is taken to be 1% of the people **and** households in Germany. The microcensus provides very important data not just **for** administrative purposes but also **for** research purposes. We have already seen the robust allocation approach **for** simulated datasets in Chapter 5 **and** Chapter 6. In this chapter, we focus on the real dataset on income **and** taxation of North Rhine-Westphalia (NRW), Germany. This data is available **for** research purposes from (Forschungsdatenzentrum, 2001). Income **and** taxes are two important factors that have a direct impact on the financial situation of a country (Alesina **and** Perotti, 1997). Income **and** taxation data are collected from different states of a country. This data helps the government in taking financial decisions. **For** our survey statistical problem we take the income **and** taxation data of North Rhine- Westphalia. This data includes information about a population of size 274, 743 with their age, sex, income (yearly **and** monthly), amount of tax paid in a year, taxation level **and** social structure etc.

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Third, the computational efficiency of the EP algorithm requires careful implementation, e.g., computing the diagonal elements of the inverse of large (**and** possibly dense) inverse covariance matrix **and** accurate numerical quadrature of low-dimensional but nonsmooth integrals. This necessitates the study of relevant numerical issues, e.g., semi-analytic formulas **and** error estimates. It may be viable to first compute a MAP estimate by an **optimization** strategy **and** then use that to precondition the EP method. Alternatively, one may modify the EP method to only supplement the MAP estimate with the variance. Fourth, to combine the rapid approximation of Expectation Propagation with the accuracy of a long-running MCMC chain, preconditioning methods **for** MCMC should be studied. There are multiple ways on how to incorporate the results from Expectation Propagation to accelerate a subsequent MCMC algorithm. **For** example, in random walk MCMC methods, a proposal distribution with a covariance computed by EP will yield a higher acceptance rate. Independence sampler, i.e. MCMC methods with a fixed proposal, can be enhanced by using the EP approximation to build the proposal. In preconditioned MCMC with multiple levels, the Gaussian approximation by EP can be employed on the first level to cheaply reject proposals that would probably be rejected by the true posterior.

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Andreas Rieder: Inexact Newton Regularization Using Conjugate Gradients as Inner IteractionMichael Jan Mayer: The ILUCP preconditioner Andreas Rieder: Runge-Kutta Integrators Yield Optim[r]

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Integrated Order Batching **and** Vehicle Routing Operations in Grocery Re- tail – A General Adaptive Large Neighborhood Search Algorithm.
Kuhn, H., Schubert, D. und Holzapfel, A. (2020)
Abstract In recent years, established **and** well-known grocery retailers have increasingly been investing in the business of micro stores **and** petrol station shops. Supplying these stores with perishable **and** durable goods leads to noticeable logistics challenges **for** the retailers. Since the total sales volumes of these shops are typically low **and** the respective sales areas are very limited, highly frequent deliveries of small sizes are required. These noticeably affect a number of operational planning **problems**. In the warehouse, the items requested have to be collected in small order sizes. In order to achieve efficient picking operations, orders are therefore combined into larger picking orders, i.e., batches. After- wards the orders have to be delivered to the stores at high frequency. In practice, all the planning **problems** mentioned are heavily interconnected due to the short planning horizon.

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There are a lot of practical important **problems** falling into the category of inverse **problems**. In [ 30 , 53 ], the authors have listed some inverse **problems** that have several applications **and** have attracted a lot of researchers. There, parameter identiﬁcation **problems** have been pointed out to be one of the most interested ﬁelds because of their applications in many practical situations. One of those is the diﬀusion coeﬃcient identiﬁcation problem, which describes the ﬂow of a ﬂuid (e.g. groundwater) through some medium with permeability. **For** a good review, we refer to the books by Cannon [ 14 ], Banks **and** Kunisch [ 5 ], **and** Engl et al. [ 30 ]. Another is electrical impedance tomography, which is an imaging tool with important applications in ﬁelds such as medicine, geophysics, environmental sciences **and** nondestructive testing of materials. We refer to Borcea’s paper [ 9 ] **and** the references therein **for** a good review. Although, many researchers have examined these **problems** **and** some regularization methods have been applied, there have been few proposed results of the convergence **and** convergence rates of regularization methods as well as eﬃcient numerical algorithms **for** reconstructing the parameters. In this work, we ﬁrst investigate sparsity regularization **for** the diﬀusion coeﬃcient identiﬁcation problem **and** electrical impedance tomography. These **problems** will be later used as model **problems** **for** the algorithms studied in the thesis. Second, we propose several numerical algorithms in order to solve minimization **problems** arising from sparsity regularization. Our algorithms are more eﬃcient than the others proposed in sparsity regularization **for** **nonlinear** inverse **problems** [ 74 , 8 ]. Their eﬃciency in practice is also illustrated by some numerical examples in two above **problems**.

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Our experiments show that the ANS heuristic is capable of nding a larger number of feasible delivery slots than the Simple Insertion heuristic, requiring run times that are suited for A[r]

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