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 nonlinearoptimization .
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. ). 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 nonlinearmultiobjectiveoptimization 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.
There has been a lot of research about robust model predictive control. Mayne et al.  give a good overview about the beginnings of the research about model predictive control until the year 2000. They consider works about constrained linear andnonlinear 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 , Kothare et al. ,Lee and Yu  or Scokaert and Mayne . Bemporad et al. and Magni et al.  describe methods in their work that combine dynamic and parametric programming approachesfor 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  or in Scokaert and Mayne , where single finite di- mensional optimization techniques are used. Lee and Yu  use dynamic programming by discretizing the state space to solve the predictive optimal control problem in a state feedback form.
Applications The design optimization examples from electrical engineering have shown that the presented approach for solving mixed-integer nonlinearproblems 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 nonlinearproblems, 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 ), Sacks and his colleagues pointed out that first, simulation problems often do not behave like stochastic problemsand 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 optimizationproblems based on a subsurface flow simulation. They are also established as a benchmark set of problemsfor 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.
There is a considerable body of literature on separable convex programming (integer or con- tinuous) with linear constraints, providing efficient algorithms forsolution, (cf. Hochbaum and Shantikumar (1990)). These results are still to be exploited for (bi)proportional rounding pur- poses. More general nonlinear integer optimizationproblems 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.
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.
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 optimizationproblems. 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 optimizationproblems if start from line A and 14 problems if we start from line B.
The motivation for the use of evolutionary optimization algorithms relies on their ability in dealing with local optimal solutionand 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 , wherein a Genetic Algorithm (GA) was applied to generate the optimal reentry trajectories. Similarly, in  a low-thrust interplanetary trajectory problem was formulated and solved via a modified GA. Pontani and Conway  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.
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 optimizationproblems 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  ; b) dynamic penalty functions based on adaptive
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 solutionand 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 solutionand determine the uniqueness of the solution by checking for active parallel constraints.
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 . The existence of multiple solutions and sign-changing solutions for zero Neumann boundary values has been proven in [88, 107, 108, 122] and , 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 . 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
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.
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.
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.
Documents in EconStor may be saved and copied for your personal and scholarly purposes.
You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public.
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.
Multiobjective mixed integer convexoptimization refers to mathematical pro- gramming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. We present a branch-and-bound method based on the use of properly de- fined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. We are able to guaran- tee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convexproblems according to a prescribed precision. As far as we know, the procedure we present is the first deterministic algorithm devised to handle this class of problems. Our numerical experiments show results on biobjective and triobjective mixed integer convex instances.
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 problemsand 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 problemsfor 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 fornonlinear inverse problems [ 74 , 8 ]. Their eﬃciency in practice is also illustrated by some numerical examples in two above problems.