Abstract: Dijkstra’s Algorithm is one of the most popular algorithms in computer science. It is also popular in operations research. It is generally viewed and presented as a greedy algorithm. In this paper we attempt to change this perception by providing a dynamic programming perspective on the algorithm. In particular, we are reminded that this famous algorithm is strongly inspired by Bellman’s Principle of Optimality and that both conceptually and technically it constitutes a dynamic programming successive approximation procedure par excellence. One of the immediate implications of this perspective is that this popular algorithm can be incorporated in the dynamic programming syllabus and in turn dynamic programming should be (at least) alluded to in a proper exposition/teaching of the algorithm.

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Abstract: We consider a discounted Markov Decision Process (MDP) supplemented with the requirement that another discounted loss must not exceed a specified value, almost surely. We show that the problem can be reformulated as a standard MDP and solved using the Dynamic Programming approach. An example on a con- trolled queue is presented. In the last section, we briefly reinforce the connection of the Dynamic Programming approach to another close problem statement and present the corresponding example. Several other types of constraints are discussed, as well.

Abstract: In Knopik (2005) the ageing class MTFR (Mean Time to Failure or Repair) of lifetime distribution was introduced. In this paper, we show that the family MTFR is closed under weak convergence of distribution and convolution. We prove that the dual family MTFR D (in a particular case) is closed under mixtures.

Abstract: Tabu search has proven highly successful in solv- ing hard combinatorial optimization problems. In this paper, we propose a hybrid method that combines adaptive memory, sparse dynamic programming, and reduction techniques to reduce and ex- plore the search space. Our approach starts with a bi-partition of the variables, involving a small core problem, which never exceeds 15 variables, solved using the "forward" phase of the dynamic program- ming procedure. Then, the remaining subspace is explored using tabu search, and each partial solution is completed with the informa- tion stored during the forward phase of dynamic programming. Our approach can be seen as a global intensification mechanism, since at each iteration, the move evaluations involve solving a reduced prob- lem implicitly. The proposed specialized tabu search approach was tested in the context of the multidimensional 0-1 knapsack prob- lem. Our approach was compared to ILOG's commercial product CPLEX and to the corresponding "pure" tabu search (i.e., without a core problem) for various sets of test problems available in OR- libraries. The results are encouraging. In particular, this enhances the robustness of the approach, given that it performs better than the corresponding pure tabu search most of the time. Moreover, our approach compares well with CPLEX when the number of variables is large; it is able to provide elite feasible solutions in a very reason- able amount of computational time.

Abstract: We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term The symmetries are then used to obtain families of Noether’s first integrals, possibly in the presence of nonconservative external forces As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8) Mathematics Subject Classification 2000: 49K15; 49-04; 49S05

Abstract: Controlling entropy of a system with unknown pa- rameters is treated here as an adaptive control problem Necessary conditions for optimality and an algorithm for computing extremal controls in the spirit of R Rishel are obtained

Abstract: Infinite-dimensional linear dynamic systems describ- ed by infinite matrices are studied. Approximate controllability for systems with lower-diagonal matrices is investigated, whereas ob- servability is studied for systems with row-finite and upper-diagonal matrices. Different necessary or sufficient conditions of approximate controllability and observability of such systems are given. They are used to show dualities between these properties. The theorems on dualities extend the results known for finite-dimensional systems.

Abstract: In the real estate sector, especially in construction or purchasing of commercial buildings, adequate evaluation of market development and property management is of paramount importance. In this paper, application of mathematical modelling to evaluating the efficiency and risk of investment projects is discussed. Most of the microeconomic models are discrete, implying that the initial data and the results obtained are discrete values. In the suggested model, the most likely variability intervals of the parameters are taken as the basis of modelling. The models suggested in the present pa- per deal with local investment problems, which should be promptly solved in the presence of a great number of alternative investment possibilities. The modelling is aimed at determining zones related to the quality of decisions in the area of investment. The principles of mathematical modelling and determination of various financial risk zones are described. An example of determination of risk zones of investments in Vilnius are presented.

Abstract: In this paper a parametric description for the state space of an arbitrary TPN is given. An enumerative procedure for reducing the state space is introduced. The reduction is defined as a truncated multistage decision problem and solved recursively. A reachability graph is defined in a discrete way by using the reachable integer-states of the TPN.

Abstract: We model an environment where orders arrive proba- bilistically over time, with their revenues and capacity requirements becoming known upon arrival. The decision is whether to accept an order, receiving a reward and reserving capacity, or reject an order, freeing capacity for possible future arrivals. We model the dynamic, stochastic multiple knapsack problem (DSMKP) with stochastic dy- namic programming (SDP). Multiple knapsacks are used as orders may stay in the system for multiple periods. As the state space grows exponentially in the number of knapsacks and the number of possible orders per period, we utilize linear programming and duality to quickly approximate the end-of-horizon values for the SDP. This helps mitigate end-of-study effects when solving the SDP directly, allowing for the solution of larger problems and leading to increased quality in solutions.

Abstract: Evaluation of efficiency of each of the DMUs (Deci- sion Making Units) in a company is a very important task. Thus, the studies of evaluation of efficiency are being actively carried out, based on production function. Until quite recently, the loglinear production function (the Cobb-Douglas function) has been used for evaluation purposes. The loglinear model evaluates the DMUs by measuring the average efficiency. Of late, the DEA (Data Envel- opment Analysis) focussed the interest as the available method, in the form of either the CCR (Charnes-Cooper-Rhodes) or the BCC (Banker-Charnes-Cooper) model. However, the DEA approach does not provide for the lower limit of the production set, but only for the upper one. Hence, considering the fact that in the real-life problems the production set ranges between the lower and the upper limit, it is proposed that the possibility production function be constructed by introducing fuzziness into the loglinear production function. When we try to evaluate efficiency with the help of this possibility func- tion, we can obtain from it two efficiency ratings, corresponding to the upper and lower limits. The DEA and the fuzzy loglinear models perform evaluation in the sense of inclusion of all the DMU data and provide a dual possibility image of efficiency in the sense that the DEA assesses the lower limit of inputs for the given output, while the fuzzy loglinear model assesses the maximum output for the given inputs. Hence, by making full use of this duality, we try to fuse the DEA and the fuzzy loglinear model in the evaluation of DMU ef- ficiency by introducing a fuzzy goal. We propose to construct the fuzzy goal by evaluating the ratings for individual outputs with the help of fuzzy loglinear analysis, and introduce this fuzzy goal into the DEA. This approach can yield both efficiency and ability as obtained from the comparison of the CCR-based efficiencies.

Abstract: We obtain a method to compute effective first integrals by combining Noether’s principle with the Kozlov-Kolesnikov integrability theorem A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained We illustrate our approach on some problems taken from the literature An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2, 3, 5) is also given

Abstract: In multiobjective (vector) optimization problems, among the given objective functions there exist some, which do not influence the set of efficient solutions. These objective functions are said to be nonessential. In this paper we present a new method to decide if a given linear objective function is nonessential or not. Keywords: multiple criteria decision making, multiobjective (vector) optimization, efficient solutions, nonessential objectives.

Abstract: Newton's problem of minimal resistance is one of the first problems of optimal control: it was proposed, and its solution given, by Isaac Newton in his masterful Principia Mathematica, in 1686. The problem consists of determining, in dimension three, the shape of an axis-symmetric body, with assigned radius and height, which offers minimum resistance when it is moving in a resistant medium. The problem has a very rich history and is well docu- mented in the literature. Of course, at a first glance, one suspects that the two dimensional case should be well known. Nevertheless, we have looked into numerous references and asked at least as many experts on the problem, and we have not been able to identify a sin- gle source. Solution was always plausible to everyone who thought about the problem, and writing it down was always thought not to be worthwhile. Here we show that this is not the case: the two- dimensional problem is richer than the classical one, being, in some sense, more interesting. Novelties include: (i) while in the classi- cal three-dimensional problem only the restricted case makes sense (without restriction on the monotonicity of admissible functions the problem does not admit a local minimum), we prove that in dimen- sion two the unrestricted problem is also well-posed when the ratio of height versus radius of base is greater than a given quantity; (ii) while in three dimensions the (restricted) problem has a unique so- lution, we show that in the restricted two-dimensional problem the minimizer is not always unique - when the height of the body is less or equal than its base radius, there exists infinitely many minimizing functions.

Abstract: The paper addresses the problem of real-time emis- sion control in a given set of air pollution sources. The approach applied utilizes the optimal control technique for distributed para- meter systems. A set of pointwise emission sources with a predefined location and emission characteristics is considered as the controlled object. The problem is formulated as on-line minimization of an en- vironmental cost function, by the respective modification of emission level in the controlled sources, according to the changing meteoro- logical conditions (e.g. wind direction and velocity). Dispersion of atmospheric pollution is governed by a multi-layer, dynamic model of SO x transport, which is the main forecasting tool used in the optimization algorithm. The objective function includes the envi- ronmental damage related to air quality as well as the cost of the controlling action. The environmental cost index depends on the current level of SO x concentration and on the sensitivity of the area to this type of air pollution. The adjoint variable, related to the main transport equation of the forecasting model, is applied to cal- culate the gradient of the objective function in the main optimiza- tion procedure. The test computations have been performed for a set of major power plants in the industrial region of Upper Silesia (Poland).

Abstract: The statistical procedure for determination of the type of relation – equivalence or tolerance – in a finite set of elements, estimated on the basis of pairwise comparisons with random errors, is presented. The procedure consists of two tests based on Chebyshev’s inequality for variance of a random variable; the test statistic is a mixture of some random variables. An example of application of the procedure – determination of relation type in the set of functions expressing profitability of treasury securities sold at auctions in Poland – is presented, too.

Abstract: The paper studies the realization problem for series and parallel connections of nonlinear single-input single-output sys- tems, described by higher order differential equations. Necessary and sufficient conditions are given for the existence of the classical state space realization in both cases. It is proved that post- and parallel compensators are of no help in overcoming non-realizability. Results are illustrated by an example.

Abstract: Necessary and sufficient conditions for a set-valued map K : R ։ R n to be GDQ-differentiable are given. It is shown that K is GDQ differentiable at t0 if and only if it has a local mul- tiselection that is Cellina continuously approximable and Lipschitz at t0. It is also shown that any minimal GDQ of K at (t0, y0) is a subset of the contingent derivative of K at (t0, y0), evaluated at 1. Then this fact is used to prove a viability theorem that asserts existence of a solution to the initial value problem ˙ y(t) ∈ F (t, y(t)), with y(t0) = y0, where F : Gr(K) ։ R n is an orientor field (i.e. multivalued vector field) defined only on GrK and K : T ։ R n is a time-varying constraint multifunction. One of the assumptions is GDQ differentiability of K.

Abstract: The optimal control problems and a weight mini- mization problem are considered for elastic three-layered plate with inner obstacle and friction condition on a part of the boundary. The state problem is represented by a variational inequality and the de- sign variables influence both the coefficients and the set of admissible state functions. We prove the existence of a solution to the above- mentioned problem on the basis of a general theorem on the control of variational inequalities. Next, the approximate optimization prob- lem is proved on the basis of the general theorem for the continuous problem. When the mesh/size tends to zero, then any sequence of appropriate solutions converges uniformly to a solution of the con- tinuous problem. Finally, the application to the optimal design of unilaterally supported of rotational symmetrical load elastic annular plate is presented. Keywords: control of variational inequalities, elasto-orthotropic plate, optimal design, weight minimization, approximate optimiza- tion problem.

Abstract: This paper addresses the asymptotic stability analy- sis problem for a class of linear large-scale systems with time delay in the state of each subsystem as well as in the interconnections. Based on the Lyapunov stability theory, a delay-dependent criterion for stability analysis of the systems is derived in terms of a linear matrix inequality (LMI). Finally, a numerical example is given to demonstrate the validity of the proposed result.

Abstract: We consider two semidynamical systems, ( e Tt)t>0 and (Tt)t>0, generated by different partial differential equations of von Foerster-Lasota type. We investigate some of their common prop- erties in the integrable space with the p-exponent. We show that their chaotic or stable behaviour depends on the common value of the parameter γ = λ(0).

Abstract: As a result of the severe practical and ethical con- straints imposed on medical measurements, the parameter estima- tion procedure designed for diagnosis and therapy is often a difficult problem. When blood sampling provides the data, the number of samples and the observation interval should be minimized. Design- ing an experiment for parameter estimation requires a step known as quantitative experiment design, usually preceded by a step called qualitative experiment design. The latter answers if a model is identifiable under particular experimental conditions. The former is suitable for the purpose of obtaining the maximum information from the data to be collected. An experiment design is based on the optimization of a suitable criterion formulated with respect to the analyzed variables of the experiment (input shape, sampling sched- ule). This paper considers sampling schedule design. New criteria for new optimal sampling schedules (OSS) have been formulated on the basis of sensitivity function. These are referred to as S-OSS and RS-OSS designs. The results of optimization for both criteria are compared with the result obtained with a reputable established D- optimal design based on the Fisher information matrix. By showing the results of S-OSS and RS-OSS design we can present the reliabil- ity and efficiency of the new criteria in comparison to D-OSS design. Illustrative examples are presented.

Abstract: In a previous work, Prieur, Trelat (2006), we derived a result of semi-global minimal time robust stabilization for analytic control systems with controls entering linearly, by means of a hybrid state feedback law, under the main assumption of the absence of minimal time singular trajectories. In this paper, we investigate the Martinet case, which is a model case in IR 3 , where singular mini- mizers appear, and show that such a stabilization result still holds. Namely, we prove that the solutions of the closed-loop system con- verge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small mea- surement noise, external disturbances and actuator errors.

Abstract: The aim of this paper is to present bond portfolio immunization strategies in the case of multiple liabilities, based on single-risk or multiple-risk measure models under the assumption of multiple shocks in the term structure of interest rates referring, in particular, to Fong and Vasicek (1984), Nawalkha and Chambers (1996), Balbas and Ibanez (1998) and Hurlimann (2002). Immu- nization problem is formulated as a constrained optimization prob- lem under a fixed open loop strategy. New risk measures associated with changes of the term structure are also defined.

Abstract: We prove an existence and uniqueness result for the solutions to the Skorokhod problem on uniformly prox-regular sets through a deterministic approach. This result can be applied in order to investigate some regularity properties of the value function for differential games with reflection on the boundary.

Abstract: IRIS (increasing reward with increasing service) real- time scheduling appears frequently in real-time control applications such as heuristic control. IRIS requires not only meeting deadlines, but also finding the schedule with the best result (highest reward). In this paper, a framework is presented that uses Timed Petri nets (TPN) to transform an IRIS problem into a dynamic programming (DP) problem, allowing the application of known TPN and DP tech- niques. In the presented approach, an IRIS problem with tasks having discrete-time optimal parts is transformed into a (possibly unbounded) TPN. Then, the critical path problem of the TPN state graph can be tackled with DP. This approach allows for the IRIS problem multiple constraints and negative rewards.

Abstract: In this paper we describe an approach to the ex- tension of geographic information systems to take advantage of the continuing development of capabilities of the Semantic Web. This is presented in the context of a portal based Geospatial Information Database (GIDB TM ), an object-oriented spatial database capable of storing multiple data types from multiple sources. We have devel- oped our approach for a specific domain, spatially oriented, meteo- rological and oceanographic, but this can clearly be applied to other spatial data domains. Finally we illustrate the use of the ontology development system based on Generative Sublanguage Ontologies (GSO), a type of linguistic ontology inspired by the Generative Lex- icon Theory, to develop effective domain ontologies.

Abstract: In this paper we investigate observability of small solutions of linear differential-algebraic systems with delays (DAD), i.e. solutions that vanish after some finite time. In particular, we prove existence of two kinds of small solutions of DAD systems. The main result are the rank type conditions for observability of three kinds of small solutions of linear differential-algebraic systems with delay.