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Keywords: **Shortest** Path, Dijkstra Algorithm, Floyd-Warshall Algorithm, Road Networks 1. Introduction
It is, perhaps, by nature that human beings mostly emphasize optimization in real life phenomena. Finding the shorter times **and** the lower costs are not only looked upon by individuals but also by multinational companies. For an individual, it is often only a matter of convenience, but for a corporation it is of strategic importance when direct monetary cost is involved. The **shortest** path problem is one of the most important optimization **problems** in such fields as operations research / management science, computer sciences **and** artificial intelligence. One of the reasons is that essentially any combinatorial optimization problem can be formulated as a **shortest** path problem (Sniedovich, 2005). The development of algorithms for this **shortest** path finding problem, their computational testing **and** efficient implementation have remained important research topics within **related** disciplines (Dijkstra (1959), Dial et al. (1979), Glover et al. (1985), Ahuja et al. (1990), Goldberg **and** Radzik (1993)).

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Based on existence **and** uniqueness results for the solution to a SPDE, one can often reformulate the control problem as a minimization problem on a set of admissible controls given by a suitable Hilbert space or a suitable subset of this Hilbert space. For that reason, the main idea of the stochastic maximum principle is to state necessary **and** sufficient optimality conditions the optimal control has to satisfy. In general, the necessary optimality condition can be derived using the Gˆ ateaux derivative of the cost functional. Using this necessary optimality condition, one can derive an explicit formula of the optimal control based on the adjoint equation, which is given by a backward stochastic partial differential equation (BSPDE). Sufficient optimality conditions are often stated based on the second order Fr´ echet derivative of the cost functional. If the control problem is additionally convex, then the necessary optimality condition is also sufficient. For general concepts of optimization **problems** on Hilbert spaces, we refer to [57, 93]. Closely **related** is Pontryagin’s maximum principle, where one minimizes the Hamiltonian instead of the original control problem. However, one still obtains an explicit formula of the optimal control based on the adjoint equation. As a consequence, it remains to solve the so called Hamiltonian system. For applications, we refer to [14, 36, 47, 67]. In this context, we may also note the general theory for finite dimensional control **problems** presented in [91]. In contrast to these methods, the dynamic programming principle considers the control problem at different initial times **and** initial states through the so called value function. This value function is the solution of a nonlinear partial differential equation given by the Hamilton-Jacobi-Bellman equation. If the equation is solvable, then one can obtain a feedback law of the optimal control, see [33]. For applications, we refer to [22, 26, 32, 61, 83, 92]. We also note the finite dimensional approach presented in [35, 91].

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From this brief description it may already become clear that the latter of our two rout- ing **problems** constitutes a generalization of the former, which is why similar ideas **and** concepts can be adopted for both. Another linkage between these topics concerns the ap- plication of our planar-separator algorithms to compute so-called selected vertices needed for the multi-level technique: although several other ways of doing so have been considered as well, using planar separators is one of two methods performing best for all instances tested. **Shortest**-path computation is one of the fields in computer science that has for the last couple of years been evolving at a bewildering speed, which is equally true for the size of many kinds of real-world graphs (especially in the realm of traffic-planning). It therefore hardly surprises to notice that the multi-level technique as developed some few years ago cannot quite live up to today’s standards any more. However, the systematic investigation conducted, involving a large number of parameter settings **and** graphs, has delivered some valuable insights into the behavior of this technique in realistic scenarios. Such findings strongly helped spark development of two of the presently quickest **shortest**-path algo- rithms for road networks, the high-performance multi-level technique **and** transit node routing, both closely **related** to our approach.

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The focus of this thesis is the investigation of the facial structure of the cardinality **constrained** matroid, path **and** cycle polytopes. As it might be expected **and** is exemplarily shown for path, cycle, **and** matroid polytopes, an inequality that induces a facet of the polytope associated with the ordinary problem usually induces a facet of the polytope associated with the cardinality re- stricted version. However, we are in particular interested in inequalities that cut off the incidence vectors of solutions that are feasible for the ordinary problem but infeasible for its cardinality restricted version. In this context, the most important class of inequalities for this thesis are the so-called forbidden cardinality inequalities. These inequalities are valid for a polytope associated with a cardinality **constrained** combinatorial optimization problem independent of its specific combinatorial structure. Using these inequalities as prototype for inequalities incorporating com- binatorial structures of a problem, we derive facet defining inequalities for polytopes associated with several cardinality **constrained** combinatorial optimization **problems**, in particular, for the above mentioned polytopes. Moreover, for cardinality **constrained** path **and** cycle polytopes we derive further classes of facet defining inequalities **related** to cardinality restrictions, also those inequalities specific to odd/even path/cycles **and** hop **constrained** **paths**.

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Abstract—**Constrained** trajectory optimization has been a critical component in the development of advanced guidance **and** control systems. An improperly planned reference trajectory can be a main cause of poor online control performance. Due to the existence of various mission-**related** constraints, the feasible solution space of a trajectory optimization model may be restricted to a relatively narrow corridor, thereby easily resulting in local minimum or infeasible solution detection. In this work, we are interested in making an attempt to handle the **constrained** trajectory design problem using a biased particle swarm op- timization approach. The proposed approach reformulates the original problem to an unconstrained multi-criterion version by introducing an additional normalized objective reflecting the total amount of constraint violation. Besides, to enhance the progress during the evolutionary process, the algorithm is equipped with a local exploration operation, a novel 𝜀-bias selection method, **and** an evolution restart strategy. Numerical simulation experi- ments, obtained from a **constrained** atmospheric entry trajectory optimization example, are provided to verify the effectiveness of the proposed optimization strategy. Main advantages associated with the proposed method are also highlighted by executing a number of comparative case studies.

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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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The multigrid approach to optimal control **problems** is, apart from a few contributions, rather recent. Our framework is the one-shot multigrid strategy as first pro- posed in [1]. **Related** approaches can be found in [9] within the successive quadratic programming method, **and** in [11] where the optimality conditions are refor- mulated as fixed-point problem **and** solved by multigrid methods. A one-shot multigrid algorithm means solv- ing the optimality system for the state, the adjoint, **and** the control variables in parallel in the multigrid process. This is in contrast to solving sequentially for the state **and** the adjoint equations **and** then updating the control variables along with the gradients provided by the op- timality condition. Notice that this ‘gradient’ approach was also used in [1] to design the smoothers. This strat- egy has disadvantages: it cannot be applied to singu- lar optimal control **problems** **and**, in general, it results in less robust multigrid schemes. The present approach of collectively solve for all three optimization variables overcomes these difficulties.

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In his landmark paper [55, p.220], Lyons gave a long **and** visionary list of advantages (to a probabilist) of constructing stochastic objects in a pathwise fashion: stochastic flows, differential equations with boundary conditions, Stroock–Varadhan support theo- rem, stochastic anlysis for non-semimartingales, numerical algorithms for SDEs, robust stochastic filtering, stochastic PDE with spatial roughness. Many other applications have been added to this list since. (We do not attempt to give references; an up-to-date bib- liography with many applications of the (continuous) rough path theory can be found e.g. in [17].) The present work lays in particular the foundation to revisit many of these **problems**, but not allowing for systematic treatment of jumps. We also note that inte- gration against general rough **paths** can be considered as a generalization of the F¨ ollmer integral [14] **and**, to some extent, Karandikar [38], (see also Soner et al. [77] 1 ), but free of implicit semimartingale features.

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1 Introduction
Algorithm engineering exhibited an impressive surge of interest during the last years, spearheaded by one of the showpieces of algorithm engineering: computation of **shortest** **paths**. In this field, many speed- up techniques for D IJKSTRA ’s algorithm have been developed (see [1] for an overview). Recent re- search [2, 3] even made the calculation of the distance between two points in road networks of the size of Europe a matter of microseconds. One problem arising for algorithm engineering in general, **and** **shortest** **paths** in particular, is the following: Performance of algorithms highly depends on the used input, e.g. [2, 3] were developed for road networks **and** use properties of those networks in order to gain their enormous speed-up. Due to the availability of huge road networks, recent research in short- est **paths** solely concentrated on those networks [4]. However, fast algorithms are also needed for other applications, e.g. timetable information or routing in sensor networks.

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An extension of the time-expanded approach able to count train transfers is presented by M¨ uller-Hannemann **and** Weihe along with an experimental study fo- cused on multi-criteria **problems** in [MHW01]. The results of this study are quite promising: they show that in practice (among other data also the time-expanded graph was considered) the number of Pareto-optimal **paths** is often very small, **and** labelling approaches are feasible. In [MHSW02], the same authors together with Schnee investigate the issue of space consumption when more complex real-world scenarios shall be modelled. In a subsequent study, M¨ uller-Hannemann **and** Schnee extensively investigate multi-criteria optimisation in the time-expanded graph by a labelling approach [MHS]; they relax the notion of Pareto-optimal connections in order to find all attractive train connections. M¨ohring suggests the time-expanded model as graph-theoretic concept for timetable information in [M¨oh99]. He further discusses algorithms for solving multi-criteria **problems**, **and** focuses on a distributed approach for timetable information, which is also the topic of the recent projects DELFI [DEL] **and** EU-Spirit [EUS]: the railway network is considered as consist- ing of several (overlapping) subnetworks (e.g., each subnetwork is operated by a different company or institution), **and** a global solution is constructed from several subqueries to the conventional timetable information systems operated on the re- spective subnetworks. In a sense such new systems operate like meta search engines for the web.

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Additionally, an enormous potential exists to enhance the algorithmic formulations proposed in this work. The flexibility of the FEM with respect to, for example, geometric adaptability **and** rigorous error control, should be further exploited in future approaches. The FEM offers the particular advantage of providing methods that allow for spatial discretization in such a way that a given functional of interest, namely, a figure that is determined from the solution of the PDE, can be approximated as well as possible with the numerical resources at hand. An efficient way to achieve this goal is mesh adaptation based on a posteriori error control, for example, with the method of dual weighted residuals or so-called goal-oriented error estimators. The constraints to be evaluated for the approximating MILPs refer to just a small number of degrees of freedom of the FEM problem defining the continuous state variables. Only these must be known **and** computed accurately in the context of an active-set strategy. The results presented in this work further suggest that a general framework for a highly efficient treatment of MIPDECO **problems** could be based on a time-space-Galerkin method relying on sets of tensor-product basis functions with a discontinuous Galerkin method in time. Also of interest is how far the utilized FEM basis functions **and** meshes can be adapted to guarantee a fast computation of all states **related** to a suitable basis of the control space, so that the discretization of the PDE-defined constraints can be turned into discrete versions in such a way that IBM ILOG CPLEX (or another MILP solver) can deal with them in the most efficient way.

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For adopted children, who generally do not know either of their genetic parents, this process of searching has been recognised for some time (Brodzinsky & Schechter, 1990; Hibbs, 1991; Howe & Feast, 2000). More recently, a similar problem was detected among another group: children conceived by donor insemi- nation (Blyth, 1998; Hunter, Salter-Ling & Glover, 2000; Meerum Terwogt, 1993; Shenfield & Steele, 1997). In both groups, the apparent need for information about one’s genetic background is explicitly linked to information about one’s identity (Golombok & Murray, 1999; Haines, 1987). In addition, the need for information appears to exist independent of the quality of the relationship with the social par- ents. The children are not searching for new parents, but want to gain a greater un- derstanding of themselves (McWhinnie, 2000). Nonetheless, the child’s interest in his or her genetic parents is usually experienced by both the parents **and** the child as an assault on the mutual loyalty within the existing family (Hunter et al., 2000; Triseliotis, 1973; 1991). Therefore, the children often dare not come out into the open with their desires. When they do, it frequently appears that their fears were justified as the relationships within the family come under strain. Sometimes the child then backs down **and** suppresses or represses his or her desires.

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Abstract Nonlinear **constrained** optimization problem (NCOP) has been arisen in a diverse range of sciences such as portfolio, economic management, airspace engineering **and** intelligence system etc. In this paper, a new multiobjective imperialist competitive algorithm for solving NCOP is proposed. First, we review some existing excellent algorithms for solving NOCP; then, the nonlinear **constrained** optimization problem is transformed into a biobjective optimization problem. Second, in order to improve the diversity of evolution country swarm, **and** help the evolution country swarm to approach or land into the feasible region of the search space, three kinds of different methods of colony moving toward their relevant imperialist are given. Thirdly, the new operator for exchanging position of the imperialist **and** colony is given similar as a recombination operator in genetic algorithm to enrich the exploration **and** exploitation abilities of the proposed algorithm. Fourth, a local search method is also presented in order to accelerate the convergence speed. At last, the new approach is tested on thirteen well-known NP-hard nonlinear **constrained** optimization functions, **and** the experiment evidences suggest that the proposed method is robust, efficient, **and** generic when solving nonlinear **constrained** optimization problem. Compared with some other state-of-the-art algorithms, the proposed algorithm has remarkable advantages in terms of the best, mean, **and** worst objective function value **and** the standard deviations.

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Both in the stationary **and** the instationary case each surface is approximated by a triangula- tion Γ h on which a finite element scheme for the state equation is formulated along the lines of [Dzi88] **and** [DE07], respectively. Here we assume n = 1, 2, 3 in order that the interpolation be well defined. The approximation error of this discretization of the state equation decomposes into a finite element error, arising from the projection onto a finite dimensional Ansatz space, **and** a geometrical part which is due to the approximation of Γ by Γ h . We prove convergence

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Table 3.8: Output functional error: maximum relative error, primal-dual error bound, **and** associated eﬀectivity for varying N **and** M. For the deﬁnition of these quantities see Appendix A.1. error bound ∆ u,∗ N,max,rel over N; we further show the maximum relative primal-dual control error bound ∆ u,∗ N,M,max,rel over N for various values of M. The corresponding plots for the cost **and** output functional errors are shown respectively in Figures 3.4 **and** 3.5. We observe that the error bounds decrease for ﬁxed M as N increases **and** vice versa. A speciﬁc desired accuracy of the bound can thus be achieved for diﬀerent combinations of N **and** M. Furthermore, we note that the dual approach clearly improves the convergence rates of the control **and** output error bounds (Figures 3.3 **and** 3.5), whereas the convergence rate of the cost functional error bound is fairly insensitive to M (Figure 3.4). Regarding the cost functional, a larger M simply allows to “shift” the convergence curve of the error bound closer to the actual error resulting in a sharper bound. We may thus select values of N **and** M, so as to (say) minimize the computational cost required for a desired accuracy, or to minimize the eﬀectivity of the error bound. If the main interest is in sharp bounds (**and** hence small eﬀectivities), for example, we need to choose M larger than N for the problem at hand. We present results for diﬀerent combinations of N **and** M in Table 3.7 for the control **and** cost, **and** in Table 3.8 for the output. Here, we choose N vs. M based on Figures 3.3 to 3.5: for a given N we select the smallest possible M so as to minimize the error bound, **and** thus the eﬀectivity. We note that the actual values of N **and** M to achieve this goal are strongly problem-dependent.

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Elliptic Optimal Control **Problems** with Distributed Control
In this chapter, we extend the framework from the previous two chapters into several directions. First, we consider optimal control **problems** involving distributed controls. Compared to scalar controls, distributed controls pose an additional challenge, because the control space is high-dimensional. To this end, we follow the approach originally proposed in [35], **and** introduce reduced basis spaces not only for the state **and** adjoint variables, but also a separate reduced basis (control) space for the distributed control. We thus considerably reduce the dimension of the first-order optimality system. Second, we propose two different rigorous a posteriori error bounds for the optimal control **and** associated cost functional. The first proposed bound is an extension of the bound from the previous two chapters to distributed controls. The second bound is directly derived from the error residual equations of the optimality system. Third, we compare our bounds with a previously proposed bound [44] based on the Banach-Nečas-Babuška (BNB) theory. We then show that the reduced order optimal control problem **and** associated a posteriori error bounds can be efficiently evaluated in an offline-online computational procedure. Finally, we present numerical results for two model **problems**: a Graetz flow problem **and** a heat transfer problem.

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In contrast to the mixed-integer reformulation in [10], problem (1.2) has only continuous variables. This potentially allows the application of methods from nonlinear optimization **and** therefore covering nonlinear **problems**. However, due to the complementarity constraint, most conditions that ensure that optimality conditions hold in a local minimum cannot be expected to be satisfied, see [15]. The complementarity formulation has a very similar struc- ture to a mathematical program with complementarity constraints (MPCC). In that setting the additional constraint x ≥ 0 is present. These **problems** also violate most standard con- straint qualifications. Therefore custom constraint qualifications, stationary conditions **and** numerical methods for MPCCs were introduced. Yet, even if the additional constraint x ≥ 0 is present, the results for MPCCs are not readily applicable, since the feasible set of (1.2) violates most of the custom constraint qualifications for MPCCs, see [15]. In [14, 15] concepts from the theory on MPCCs were transferred to the complementarity formulation. Some of the results for the complementarity formulation are even stronger than the corresponding results for MPCCs. In this thesis we further follow this path. For an overview of the sub- ject of MPCCs, see [59, 69] (whenever we resort to particular concepts, we will give further references).

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