Equipped with a very robust **probabilistic** **analysis** for the Nemhauser/Ullmann algorithm, we aim in this section at analyzing more advanced algorithmic techniques for the knapsack problem. Our focus lies on the **analysis** **of** core algorithms, the predominant algorithmic concept used in practice. Despite the well known hardness **of** the knapsack problem on worst-case instances, practical studies show that knapsack core algorithms can solve large scale instances very efficiently [Pis95, KPP04, MPT99]. For example, there are algorithms that exhibit almost linear running time on purely random inputs. For comparison, the running time **of** the Nemhauser/Ullmann algorithm for this class **of** instances is about cubic in the number **of** items. Core algorithms make use **of** the fact that the optimal integral solution is usually very similar to the optimal fractional solution in the sense that only a few items need to be exchanged in order to transform one into the other. Obtaining an optimal fractional solution is computationally very inexpensive. The idea **of** the core concept is to fix most variables to the values prescribed by the optimal fractional solution and to work with only a small number **of** free variables, called the core (items). The core problem itself is again a knapsack problem with a different capacity bound and a subset **of** the original items. Intuitively, the core contains those items for which it is hard to decide whether or not they are part **of** an optimal knapsack filling. We will apply the Nemhauser/Ullmann algorithm on the core problem and exploit the remaining randomness **of** the core items to upper bound the expected number **of** enumerated Pareto points. This leads to the first theoretical result on the expected running time **of** a core algorithm that comes close to the results observed in experiments. In particular, we will prove an upper bound **of** O(npolylog (n)) on the expected running time on instances with n items whose profits and weights are drawn independently, uniformly at random. In addition, we investigate harder instances in which profits and weights are pairwise correlated. For this kind **of** instances, we can prove a tradeoff describing how the degree **of** correlation influences the running time.

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As will be shown in this paper, all numerical methods are based on one or another way **of** replacing the scattering integral with a finite sum, which replaces the desired continuous brightness distribution with **discrete** values or a set **of** coefficients for the expansion **of** this distribution over a system **of** functions. Thus, the transfer equation, its solutions, and all the corollaries from it acquire a **discrete** matrix form. In this case, the approximation is only the replacement **of** the integral by the sum, and all other conclusions can be made strictly analytically. In fact, we can talk about a **discrete** transfer theory. Kolmogorov A.N. [ 3 ] pointed out that with the development **of** modern computer technology in many cases, it is reasonable to study real phenomena, avoiding the intermediate stage **of** their stylization in the spirit **of** representations **of** mathematics **of** the infinite and continuous, moving directly to **discrete** models.

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We have developed an incremental and compositional approach for the approximation **of** the solution space **of** complex nonlinear constraints. We also presented an approach for counting the solution space **of** constraints over data structures. This allows us to extend symbolic execution to perform a **probabilistic** **analysis** - the computation **of** path condition probabilities. We also allowed adding uncertainty about the input values **of** the analyzed program and take such uncertainty into account when computing the probability **of** a path condition. Our initial experiments are promising, however our approach has same scalability issues especially when counting solutions for constraints over data structures. We plan to explore **optimization** schemes to its performance. We also plan to open source our tool.

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Abstract The shadow price **of** information has played a central role in stochastic **optimization** ever since its introduction by Rockafellar and Wets in the mid-seventies. This article studies the concept in an extended formulation **of** the problem and gives relaxed sufficient conditions for its existence. We allow for general adapted decision strategies, which enables one to establish the existence **of** solutions and the absence **of** a duality gap e.g. in various **problems** **of** financial mathematics where the usual bound- edness assumptions fail. As applications, we calculate conjugates and subdifferentials **of** integral functionals and conditional expectations **of** normal integrands. We also give a dual form **of** the general dynamic programming recursion that characterizes shadow prices **of** information.

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In order to compute solutions **of** multilevel optimal control **problems**, we are interested in computing the change **of** solutions with respect to higher level variables. Sensitivity **analysis** for ordinary differential equations (ODEs) and also differential-algebraic equations (DAEs) has been addressed by many au- thors. In [ SP02 ], sensitivity **analysis** is done for implicit ODEs with boundary conditions. In [ CLPS03 ] the case **of** general index 1 DAEs and DAEs **of** index 2 in Hessenberg form with given initial values have been treated. Adjoint equations for the tractability index have been analyzed in [ BL05 ; BM00 ]. For a comparison **of** the different index concepts we refer to, e. g., [ Meh15 ].

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Σ { e X } = E{( e X − E{ e X })( e X − E{ e X }) T }. (4.11) However, as mentioned before, this second central moment would not contain the full stochastic information in the inequality constrained case because we have to deal with truncated PDFs. There- fore, it is more conducive to compute an m-dimensional histogram **of** the parameters. This histogram can be seen as a **discrete** approximation **of** the joint PDF **of** the parameters. Approximations **of** the marginal densities can be computed the same way, adding up the particular rows **of** the hyper matrix **of** the histogram. The quality **of** approximation **of** the continuous PDF depends directly on M (cf. Alkhatib and Schuh, 2007), which therefore has to be chosen in a way that allows a satisfactory approximation while keeping the computation time at an acceptable level. In each Monte Carlo iteration a new **optimization** problem has to be solved. However, as the solution **of** the original ICLS problem can be used as feasible initial value for the parameters, convergence **of** the active-set method is usually quite fast.

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> FlowHead Conference > 28 March 2012 Starting Geometry Starting Geometry Parameterisation Parameterisation Mesh procedure Mesh procedure Flow simulation Flow simulation Objective [r]

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The liberalization **of** energy markets has induced generators, suppliers and large-scale end users to trade actively on the market. Actors like energy utilities have a variety **of** trading relations for the purchase and sale **of** electricity that are about to abandon the (still) wide-spread long-term full supply and purchase contracts. An energy utility has (or is modeled by) a portfolio **of** purchase and supply contracts for electricity. Any market movement leads to a change **of** purchase and sales possibilities and thus to a change **of** the portfolio value. In that sense portfolio **analysis** is understood as the process **of** measuring and controlling the ratio **of** risk and return **of** the portfolio. An energy utility having different fuel supply sources in contrast to a long-term full supply contract faces different types **of** risks in the liberalized energy market. The sources **of** risk are wide-ranging, just to name the market price risk, fuel price risk, risk **of** investing in production capacities or the volume risk. Thus portfolio management is closely related to risk management and the plant managers need a tool to quantify these risks. Therefore it is necessary to employ techniques that accurately incorporate the uncertain environment in the portfolio and risk management process. Uncertainty in the electricity market is additionally evoked by a number **of** factors such as political changes, weather changes or plant outages. Looking at historical electricity spot price series clearly reflects that uncertain environment and sets them apart from stock prices or equity index values. The series show sudden increases in value (known as the electricity spikes) and high levels **of** volatility. Besides that, they show a tendence to revert to a long term mean level. Such a behavior is often referred to as the mean reverting property **of** electricity prices. Moreover, one detects a seasonal pattern. The spot market is a market, where

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where G is an extended, real-valued functional on a Banach space X . The mapping G is not necessarily convex or smooth. There is no explicit assumption on G to be bounded from below. We provide sufficient condition for (P) to have a solution and give some inherently infinite dimensional examples to demonstrate their applicability. Problem (P) has been adressed in several contributions before. In [2] the authors also consider the infinite dimensional case and provide examples from linear and nonlinear elasticity theory. Our condition is somewhat weaker than the condition in [2] and we provide different examples. In [1] the finite dimensional cases are studied in much details. In [3] necessary and sufficient conditions for existence to (P) are obtained in terms **of** asymptotic behavior **of** G along sequences, which are candidates for being minimizing sequences. While this is an elegant asymptotic **analysis**, for ver- ifying existence in concrete applications the conditions given below are more direct and remain to be **of** independent importance.

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CHAPTER 2: This chapter is dedicated to cardinality constrained matroids and polymatroids. It serves, among other things, as an example for the poly- hedral **analysis** **of** the cardinality constrained version **of** a polynomial time solvable combinatorial **optimization** problem. Maurras [61] has given a com- plete linear description **of** the cardinality constrained matroid polytope. We give an elementary proof **of** this result. Moreover, we characterize the facets **of** this polytope and state a polynomial time separation procedure. Based on the results for the cardinality constrained matroid, we give a complete linear description **of** the cardinality constrained polymatroid and present a polynomial time algorithm that solves the associated separation problem. CHAPTER 3: As an example **of** NP-hard cardinality constrained combi- natorial **optimization** **problems**, we extensively study polyhedra associated with cardinality constrained versions **of** path and cycle **problems** defined on directed and undirected graphs. We show that a modification **of** forbidden cardinality inequalities leads to strong inequalities related to cardinality con- straints. Moreover, as one would expect, inequalities that define facets **of** the polytope associated with the ordinary problem usually define facets **of** the polytope associated with the cardinality constrained version.

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In general there is a certain degree **of** freedom to distribute the vertex delay to different branches **of** the tree. In this delay model we consider only binary Steiner trees where Steiner points can have the same position. By inserting a gate at a vertex **of** the tree it is possible to reduce the delay **of** one **of** the incident branches while increasing the delay on the other branch by about the same amount. As there are only a **discrete** number **of** gates with different sizes available, this effect can be modeled by so-called L 0 (k)-trees for some appropriate k ∈ N where an L 0 (k)-tree is a binary tree in which all edges have positive integral lengths and the sum **of** the lengths **of** the two edges leading from every non-leaf to its two children is k. Then the required arrival times at each sink correspond to a depth restriction for a leaf **of** the

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saddle point. This is a particularly difficult situation since WKB expan- sions starting from a minimum break down at the saddle point, and it is therefore hard to get overlapping quasimodes. Moreover, as we know from the **probabilistic** model (see in particular (0.12)), the tunneling between two minima which is responsible for the appearance **of** a given non-zero small eigenvalue may also occur through a well associated to a third minimum, which is weakly resonant in the terminology **of** [46], [47] and further compli- cates the situation. Apart from this, one has to face another complication in Step 2, related to the fact that the small eigenvalues have distinct expo- nential decay. Indeed, when diagonalizing the matrix **of** the operator, error terms propagate additively (see [48]) and therefore quantities **of** order **of** the larger exponentially small eigenvalues destroy the possibility to accurately estimate the smaller ones.

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The use **of** surrogate models is a standard method to deal with complex, real- world **optimization** **problems**. The first surrogate models were applied to con- tinuous **optimization** **problems**. In recent years, surrogate models gained impor- tance for **discrete** **optimization** **problems**. This article, which consists **of** three parts, takes care **of** this development. The first part presents a survey **of** model- based methods, focusing on continuous **optimization**. It introduces a taxonomy, which is useful as a guideline for selecting adequate model-based **optimization** tools. The second part provides details for the case **of** **discrete** **optimization** **problems**. Here, six strategies for dealing with **discrete** data structures are in- troduced. A new approach for combining surrogate information via stacking is proposed in the third part. The implementation **of** this approach will be available in the open source R package SPOT2. The article concludes with a discussion **of** recent developments and challenges in both application domains. Keywords: Surrogate, **Discrete** **Optimization**, Combinatorial **Optimization**, Metamodels, Machine learning, Expensive **optimization** **problems**, Model management, Evolutionary computation

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Chapter 4
Closer Randomized **Analysis** **of** BIT
This chapter focuses on the stochastic **analysis** **of** BIT. Request sequences considered in this chapter consist **of** requests which are i.i.d. over the set **of** list elements. In Section 4.1, uniform distribution is used to generate requests. It turns out that the cost **of** BIT can be simulated by throwing a fair die several times and counting the sum **of** the resulting points, independent **of** the initial bit setting. A further **analysis** **of** the distribution **of** bit values is performed in this section, resulting in an improved version **of** Conjecture 3.56. In Section 4.2, a formula for the expected cost **of** BIT is developed for request sequences generated by **discrete** distribution. The last section **of** this chapter provides a brief view **of** a more general case, which occurs naturally in the context **of** data compression.

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In the described experiments, we applied Bayesian **Optimization** with a standard configuration as shipped by GPyOpt, without any optimizations. However, there are well known levers for **optimization** **of** the BO algorithm: (1) Choose a more appropriate acquisition function, which in our case was expected improvement (EI), (2) optimize the acquisition function’s parameters, such as the balance between exploration and exploitation, and (3) analyze the fit **of** the surrogate model, which in our case were Gaussian processes, for the specific problem domain. For example, current research indicates that random forests might be a more suitable surrogate model for **problems** with **discrete** parameters [17].

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log-log space **of** the seismic intensity versus response, a linear relation is established using regression. The median capacity and log standard deviation can be computed from the least square linear regression, whereby, the sum **of** the squared error between the predicted values and the structural responses are minimized. The method can be seen in Ellingwood and Kinali [43], Richard and Chaudat [111], Choi et al. [22], etc. A linear relationship is assumed between the logarithm **of** the intensity and the response **of** the structure [139]. Maximum likelihood estimates (MLE), a method used by Shi- nozuka et al. [121] is yet another approach also commonly used for the computation **of** the lognormal parameters. This is a **discrete** approach as opposed to the linear regres- sion approach and the intensity is not directly influencing the lognormal parameters. The likelihood function at PGA level using the binomial probabilities are calculated, and fragility parameters are obtained by **optimization**, either by maximizing this like- lihood as shown in [7] or by minimizing the error [141], [139]. Other methods which are in practice are truncated IDA method also using a likelihood approach and multiple strip **analysis** (MSA) method. The post processing and development **of** fragility using IDA is further studied by Baker [7]. The statistical methods and concepts for fitting fragility function, and optimizing the number **of** structural **analysis** for the fragility functions fitting, etc. are explained in detail for IDA and multiple strip **analysis** meth- ods in Baker [7].

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Elevator control algorithms for elevator groups were first studied back in the 1950s, when the first automatic elevator controls were installed [Bar02]. These first algorithms were rather simple, since they had to be implemented in hardware using relays. Since then, the performance achieved by elevator control algorithms has become more important as buildings become higher and higher. In addition to algorithmic improvements one possible way to enhance this performance is to use destination call systems (sometimes also called destination hall call system). In such a system, a passenger registers his or her destination floor right at his start floor instead **of** the travel direction only. This way, the elevator control has more information as a basis for its decisions which hopefully leads to better performance. Apart from new high-rise buildings, there is another important application for destination call systems in existing buildings. It may happen that due to changing usage **of** a building the installed elevator system is not capable to cope with increased passenger demand. In this situation, changing over to a destination call system may be a relatively cheap alternative to upgrading the elevator system itself or installing additional elevators.

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Alternatively, chance-constrained optimal path design re- lies on chance-constrained **optimization** (CCO) algorithms. This type **of** algorithm allows constraint violations to be less than a user-specified risk parameter. A detailed review regarding different CCO algorithms can be found in [30] and the references therein. In [31], the authors proposed a CCO- based model predictive control scheme so as to optimize the movement **of** the ego vehicle. Considering the uncertainty in the system state as well as the constraint, a hybrid CCO method was designed in [32] and applied to solve an au- tonomous vehicle motion planning problem. Compared with RO methods, the CCO methods tend to be less conservative [30]. However, one challenge **of** the use **of** CCO methods is that the **probabilistic** functions and their derivatives cannot be calculated directly. An effective strategy to handle this issue is to replace or approximate these constraints by using deterministic functions or samples [33]–[35]. The motivation for the use **of** approximation-based strategies relies on their ability in dealing with general probability distributions for the uncertainty as well as preserving feasibility **of** approximation solutions. Until now, some approximation techniques have been proposed based on Bernstein method [24], [33], con- straint tightening approach [36], scenario approximation [37], etc. Although these strategies can be feasible for replacing the **probabilistic** constraints, there are still some open **problems**. For example, an important issue is that the conservatism is usually high and difficult to be controlled. Furthermore, the smoothness, differentiability and convergence properties **of** the approximation strategy can hardly be preserved.

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Chapter 5
**Analysis** with Simulated Data
In this chapter we generate some large scale simulated data **of** survey statistical **problems**. Simulation can enable us to work with diversely distributed variables **of** a population. Some- times it is difficult to gather exact information about the population such as the distribution **of** variables in subgroups **of** the population. In this simulation study we generate such data using R software. We generate a population **of** fixed size with variables having different distributions within the total population and also within the subgroups **of** the population. We use this simulated data to calculate robust allocations from our robust formulations. As we have already discussed in Chapter 4, Bertsimas and Sim’s approach is less conservative than Soyster’s apporach. We formulate three different robust formulations **of** the sampling allocation problem (SAP) using Bertsimas and Sim’s approach and compare the results. We perform various experiments on the robust allocations obtained for the simulated data. These experiments are helpful in explaining the benefits **of** using robust formulations. In these ex- periments we check if the uncertain parameters **of** the **optimization** **problems** can make the robust solutions infeasible.

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Recently, \average-case complexity" has received considerable attention by researchers in several elds **of** computer science. Even a problem is not (or may not be) solvable eciently in the worst-case, it may be solvable eciently on average. Indeed, several results have been obtained that show even simple algorithms work well on average (see, e.g., Joh84]). On the other hand, most **of** those results are about concrete **problems**, and not so much has been done for more general study **of** average-case complexity, though there are many interesting open questions in this area. In this paper, we consider one **of** such open questions, and improve our knowledge towards this question.

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