Showing papers in "Annals of Operations Research in 1992"

Direct and indirect methods for trajectory optimization

Abstract: This paper gives a brief list of commonly used direct and indirect efficient methods for the numerical solution of optimal control problems. To improve the low accuracy of the direct methods and to increase the convergence areas of the indirect methods we suggest a hybrid approach. For this a special direct collocation method is presented. In a hybrid approach this direct method can be used in combination with multiple shooting. Numerical examples illustrate the direct method and the hybrid approach.

The dynamics of speculative behaviour

Abstract: A number of recent empirical studies cast some doubt on the random walk theory of asset prices and suggest these display significant transitory components and complex chaotic motion. This paper analyses a model of fundamentalists and chartists which can generate a number of dynamic regimes which are compatible with the recent empirical evidence.

Hierarchical optimization: an introduction

Abstract: Decision problems involving multiple agents invariably lead to conflict and gaming. In recent years, multi-agent systems have been analyzed using approaches that explicitly assign to each agent a unique objective function and set of decision variables; the system is defined by a set of common constraints that affect all agents. The decisions made by each agent in these approaches affect the decisions made by the others and their objectives. When strategies are selected simultaneously, in a noncooperative manner, solutions are defined as equilibrium points [13,51] so that at optimality no player can do better by unilaterally altering his choice. There are other types of noncooperative decision problems, though, where there is a hierarchical ordering of the agents, and one set has the authority to strongly influence the preferences of the other agents. Such situations are analyzed using a concept known as a Stackelberg strategy [13, 14,46]. The hierarchical optimization problem [11, 16, 23] conceptually extends the open-loop Stackelberg model toK players. In this paper, we provide a brief introduction and survey of recent work in the literature, and summarize the contributions of this volume. It should be noted that the survey is not meant to be exhaustive, but rather to place recent papers in context.

A sequential LCP method for bilevel linear programming

Abstract: In this paper, we discuss an SLCP algorithm for the solution of Bilevel Linear Programs (BLP) which consists of solving a sequence of Linear Complementarity Problems (LCP) by using a hybrid enumerative method. This latter algorithm incorporates a number of procedures that reduce substantially the search for a solution of the LCP or for showing that the LCP has no solution. Computational experience with the SLCP algorithm shows that it performs quite well for the solution of small- and medium-scale BLPs with sparse structure. Furthermore, the algorithm is shown to be more efficient than a branch-and-bound method for solving the same problems.

Pollution control: a differential game approach

Abstract: Transnational pollution is formulated as a differential game between two sovereign governments. The symmetric open loop Nash equilibrium is shown to yield more pollution than in a cooperative solution. A model of Stackelberg leadership in pollution control is also investigated. The possibility of limit cycles is illustrated, using bifurcation theory.

Cyclic reservation schemes for efficient operation of multiple-queue single-server systems

Abstract: We study two new cyclic reservation schemes for the efficient operation of systems consisting of a single server and multiple queues. The schemes are the Globally Gated regime and the Cyclic-Reservation Multiple-Access (CRMA). Both procedures possess mechanisms for prioritizing the queues and lend themselves to a closed-form analysis. The combination of these two properties allows for effective and efficient operation of the systems, for which we provide a thorough delay analysis and derive simple rules for optimal operation.

Double penalty method for bilevel optimization problems

Abstract: A penalty function method approach for solving a constrained bilevel optimization problem is proposed. In the algorithm, both the upper level and the lower level problems are approximated by minimization problems of augmented objective functions. A convergence theorem is presented. The method is applicable to the non-singleton lower-level reaction set case. Constraint qualifications which imply the assumptions of the general convergence theorem are given.

Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming

Abstract: When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

Solving nonlinear bilevel programming models of the equilibrium network design problem: a comparative review

Abstract: Nonlinear bilevel programming problems, of which the equilibrium network design problem is one, are extremely difficult to solve. Even if an optimum solution is obtained, there is no sure way of knowing whether the solution is the global optimum or not, due to the nonconvexity of the bilevel programming problem. This paper reviews and discusses recent developments in solution methodologies for nonlinear programming models of the equilibrium network design problem. In particular, it provides a primer for descent-type algorithms reported in the technical literature and proposes certain enhancements thereof.

Efficient implementation of heuristics for the continuous network design problem

Abstract: In this paper, we present an efficient implementation of heuristic procedures for solving the continuous network design problem where network users behave according to Wardrop's first principle of traffic equilibrium. Numerical results involving a “standard” benchmark problem are given. Also, it is shown that the cost mapping arising in the Iterative-Optimization-Assignment algorithm is integrable if and only if the volume-delay function is of either the BPR or some logarithmic form.

Heuristic algorithms for delivered price spatially competitive network facility location problems

Abstract: We review previous formulations of models for locating a firm's production facilities while simultaneously determining production levels at those facilities so as to maximize the firm's profit. We enhance these formulations by adding explicit variables to represent the firm's shipping activities and discuss the implications of this revised approach. In these formulations, existing firms, as well as new entrants, are assumed to act in accordance with an appropriate model of spatial equilibrium. The firm locating new production facilities is assumed to be a large manufacturer entering an industry composed of a large number of small firms. Our previously reported proof of existence of a solution to the combined location-equilibrium problem is briefly reviewed. A heuristic algorithm based on sensitivity analysis methods which presume the existence of a solution and which locally approximate price changes as linear functions of production perturbations resulting from newly established facilities is presented. We provide several numerical tests to illustrate the contrasting locational solutions which this paper's revised delivered price formulation generates relative to those of previous formulations. An exact, although computationally burdensome, method is also presented and employed to check the reliability of the heuristic algorithm.

A nonlinear bilevel model for analysis of electric utility demand-side planning issues

Abstract: An application of bilevel programming in the electric utility industry is presented. The model is nonlinear and is used to analyze various economic issues that affect electric utility planning. The electric utility at the upper level of the model seeks to minimize costs or maximize benefits while controlling electric rates and subsidizing energy conservation programs. Customers at the lower level attempt to maximize their net benefit by consuming electricity and investing in conservation. This model considers factors such as free riders and the rebound effect which affect the net benefits of utility resource plans but are ignored by most planning models. The model's solutions shed light on utility issues including whether there can be a practical difference between various objectives, including minimizing cost (“least cost” planning) and maximizing net social welfare (“value based” planning).

An algorithm for the mixed-integer nonlinear bilevel programming problem

Abstract: The bilevel programming problem (BLPP) is a two-person nonzero sum game in which play is sequential and cooperation is not permitted. In this paper, we examine a class of BLPPs where the leader controls a set of continuous and discrete variables and tries to minimize a convex nonlinear objective function. The follower's objective function is a convex quadratic in a continuous decision space. All constraints are assumed to be linear. A branch and bound algorithm is developed that finds global optima. The main purpose of this paper is to identify efficient branching rules, and to determine the computational burden of the numeric procedures. Extensive test results are reported. We close by showing that it is not readily possible to extend the algorithm to the more general case involving integer follower variables.

Construction of a real-world bilevel linear programming model of the highway network design problem

Abstract: The formulation of the highway network design problem (NDP) as a bilevel linear program (BLP) allows more realistic solutions taking into account the reaction of users to improvements made by the system. In this paper, a conceptual framework for the optimization of investments in inter-regional highway networks in developing countries is proposed. The model is applied to the Tunisian network using actual data. Considerable effort was expended to make the implementation as realistic as possible, taking into consideration travel times, operating costs, accident costs, improvement costs, conservation laws, and effects of intra-regional flows. A new formulation allowing the incorporation of any improvement cost functions, including non-convex and non-concave functions, is introduced.

Separable routing: a scheme for state-dependent routing of circuit switched telephone traffic

Abstract: Separable routing is the first of a number of routing schemes for circuit switched telephone traffic invented at Bellcore. These routing schemes are state dependent, in the sense that, for each call attempt, a routing decision is made on the basis of the state of the network (defined in terms of the numbers of busy and idle trunks in the various trunk groups at the moment of the call attempt). In this paper, we describe separable routing and its mathematical background. Simulation results we have presented elsewhere show that the family of state-dependent routing schemes, of which separable routing is a member, is very attractive in terms of blocking rate, built-in network management features, and behavior in the presence of traffic forecast error.

OPTCON: an algorithm for the optimal control of nonlinear stochastic models

Abstract: In this paper we describe the algorithm OPTCON which has been developed for the optimal control of nonlinear stochastic models. It can be applied to obtain approximate numerical solutions of control problems where the objective function is quadratic and the dynamic system is nonlinear. In addition to the usual additive uncertainty, some or all of the parameters of the model may be stochastic variables. The optimal values of the control variables are computed in an iterative fashion: First, the time-invariant nonlinear system is linearized around a reference path and approximated by a time-varying linear system. Second, this new problem is solved by applying Bellman's principle of optimality. The resulting feedback equations are used to project expected optimal state and control variables. These projections then serve as a new reference path, and the two steps are repeated until convergence is reached. The algorithm has been implemented in the statistical programming system GAUSS. We derive some mathematical results needed for the algorithm and give an overview of the structure of OPTCON. Moreover, we report on some tentative applications of OPTCON to two small macroeconometric models for Austria.

Locating concentrators in centralized computer networks

Abstract: Topological design of centralized computer communication networks is a complex problem that is generally solved in two phases. The first phase of the design process involves dividing network nodes (terminals or clusters of terminals) into groups, and selecting a concentrator location for each group so that all the nodes in a group are assigned to the same concentrator. The next phase determines topology of links that connect network nodes to concentrators and concentrators to each other and to the central computer. The design problem studied in this paper contains some aspects of both phases. In this problem locations of concentrators, assignments of user nodes to concentrators and the topology of the links connecting concentrators to the central computer are jointly determined. The proposed design method is built around the well known sweep heuristic which is used to partition the node space into sectors. Each of these sectors contain a “backbone” path connecting concentrators to the central computer.

Attributed graphs, graph-grammars, and structured modeling

Abstract: Structured Modeling, proposed by Geoffrion [5, 6], aspires to provide a rich, computer-based modeling environment. Structured Modeling relies on a particular type of attributed graph to represent models. Jones [17] has proposed the use of a graph-based modeling system (GBMS) based on graph-grammars to facilitate the creation of modeling environments for attributed graphs. Since Structured Modeling can be viewed as a type of attributed graph, and graph-grammars support attributed graphs, this paper explores how graph-grammars can support Structured Modeling. As will be seen, graph-grammars have the potential to provide a graphical, syntax-directed editing environment for Structured Modeling. A prototype implementation, Networks/SM, based on Jones's prototype GBMS, Networks, is also discussed.

The computational complexity of multi-level linear programs

Abstract: We show that (L+1)-level linear programs are as difficult as levelL of the polynomial-time hierarchy, even if one only considers problems with unique optimal solutions.

State-dependent routing on symmetric loss networks with truck reservations. II: asymptotics, optimal design

Abstract: We investigate a distributed, state-dependent, dynamic routing strategy for circuit-switched loss networks which we have called Aggregated-Least-Busy-Alternative (ALBA). The networks considered are symmetric and fully connected, the offered calls form Poisson streams and routes have at most two links. In ALBA(K), the states of each link are lumped intoK (K ≥ 2) aggregates and the route of each call is determined by local information on the aggregate-states of the links of the alternate routes. The last aggregate is always the set of states reserved for direct traffic and defined by the trunk reservation parameter. The particular case of ALBA in which there is no aggregation is Least-Busy-Alternative (LBA); ALBA(2) represents the other extreme of aggregation. We consider two separate asymptotic scalings based on Fixed Point Models for ALBA(K) which were obtained and investigated in an earlier paper. In the first, it is assumed that the number of network nodes, the offered traffic and trunk group size are all large; their ratios have been chosen to reflect practical interest. The results show that there exists a threshold which delineates fundamentally different behavior: for offered traffic below the threshold, the network loss probability decreases exponentially with increasing network size, while above the threshold the decrease is only polynomial. In the related second asymptotic scaling, the asymptotically optimum trunk reservation parameter is obtained as the solution of a simple equation. Such asymptotically optimal designs are compared to the outputs of exhaustive numerical searches for some realistically sized networks and found to perform very well.

Model management systems: a survey

Abstract: This paper provides a survey of model management literature within the mathematical modeling domain. The first part of the survey is a review and a summary of the literature. After giving some basic definitions of modeling, modeling life cycle, and model management, two representative algebraic modeling languages followed by three approaches to modeling are introduced. These approaches are database, graph-based, and knowledge-based. The discussion is followed by a review of two specialized model management systems. The second part of the survey is a categorization of various modeling systems based on the modeling functions they provide and some of their features. These functions include life cycle support and model base administration. The degree of model independence provided by model management systems and the implemented environment systems is also summarized. The last part of the paper provides directions for future research.

Computational algorithms for blocking probabilities in circuit-switched networks

Abstract: In this paper, we develop two new general purpose recursive algorithms for the exact computation of blocking probabilities in multi-rate product-form circuit-switched networks with fixed routing. The first algorithm is a normalization constant approach based on the partition function of the state distribution. The second is a mean-value type of algorithm with a recursion cast in terms of blocking probabilities and conditional probabilities. The mean value recursion is derived from the normalization constant recursion. Both recursions are general purpose ones that do not depend on any specific network topology. The relative advantage of the mean-value algorithm is numerical stability, but this is obtained at the expense of an increase in computational costs.

A dynamical characterization of evolutionarily stable states

Abstract: Evolutionary stability, the central solution concept in evolutionary game theory, is closely related to local asymptotic stability in a certain nonlinear dynamical system operating on the state space, the so-called "replicator dynamics". However, a purely dynamical characterization of evolutionary stability is not available in an elementary manner. This characterization can be achieved by investigating so-called "derived games" which consist of mixed strategies corresponding to successful states in the original game. Using well-known facts, several characterization results are obtained within this context. These also may shed light on the extremality properties of evolutionary stability.

Cyclical strategies in two-dimensional optimal control models: necessary conditions and existence

Abstract: This paper derives necessary conditions such that cyclical policies may be optimal in concave, two state variable (economic) control problems. These conditions identify four different routes. One major implication is that two of these four conditions may be met by separable models. This possibility has been overlooked so far. Therefore, even separable and structurally very simple models may be characterized by optimal cyclical policies. Indeed, it will be shown that stable limit cycles exist for concave and separable control problems.

Nonlinear mode-interaction in the macroeconomy

Abstract: A central problem for a dynamic formulation of macroeconomic theory is how to explain the occurrence of different, relatively well-defined economic modes such as the short term business cycle, the construction (or Kuznets) cycle, and the economic long wave (or Kondratiev cycle). Equally important is a description of the various phenomena that can arise through interaction between these cycles. Modern nonlinear theory suggests that different cyclical modes may be entrained through the process of mode-locking, where the periods of the interacting modes adjust to one another, so as to attain a rational ratio. This type of interaction is well documented in physical and biological systems. However, despite the importance of the problem and abundant evidence for nonlinearity in the economy, modern concepts of nonlinear mode-interaction have not yet been applied to the problem of entrainment between economic cycles. We show how mode-locking and other highly nonlinear dynamic phenomena arise in a model of the economic long wave. The behavior of the model is mapped as a function of the frequency and amplitude of an external forcing, producing both a devil's staircase and a detailed Arnol'd tongue diagram. Two different routes to chaos are identified. The Lyapunov exponents are calculated, allowing the strength of the chaos to be assessed, and the fractal nature of the basins of attraction for two simultaneously existing periodic solutions is illustrated. The paper concludes with a discussion of the implications for economic theory.

Empirical chaotic dynamics in economics

Abstract: Barnett and Chen [4–6] have displayed evidence of chaos in certain monetary aggregates, but the tests have unknown statistical sampling properties. Using monthly growth rates in monetary aggregates, we conduct bispectral tests for nonlinearity. Our tests have known sampling properties, and we find deep nonlinearity in some monetary aggregate series.

Supplier-customer interaction in quality control

Abstract: We develop a game theoretic model of joint quality control in a single sourcing environment which integrates supplier and customer decisions. In this model, both parties behave strategically and take each other's incentives into account when deciding on their respective sampling plans. The specific sampling plans considered are of the “single sample fraction defective with rectifying inspection” type. A method to find the optimal sampling plans for both supplier and customer in two different informational setups is constructed. The resulting models lead to examination of Stackelberg and Nash equilibria. Numerical examples of such equilibria are presented, indicating how optimal strategies depend on the parameters of the problem.

A comparison of mathematical programming modeling systems

Abstract: We compare three mathematical programming modeling languages, GAMS, OMNI and MathPro. To understand the properties of these languages, we formulate four linear programs in each language. The formulations are representative of the kinds of model structures one encounters in practice. Each of the languages focuses on a different view of linear programs. GAMS approximates algebra, OMNI uses the activity view and MathPro uses a block schematic. We summarize our experiences with the languages and suggest areas for further enhancement.

Uniqueness results and algorithm for Stackelberg-Cournot-Nash equilibria

Abstract: This paper uses recently developed theory of sensitivity analysis to explore the reaction of a Cournot-Nash equilibrium to a Stackelberg firm, and to analyze the effect of this reaction on the uniqueness of the Stackelberg-Cournot-Nash equilibrium. Some of the results presented are not new, but the methods used provide simpler proofs and a different perspective. More importantly, the methods used here allow the development of new conditions for a unique Stackelberg-Cournot-Nash equilibrium that extends those previously known. The methods used also provide for the development of an efficient algorithm for finding the equilibrium.

Chaotic dynamics in agricultural markets

Abstract: In a recent article, Chiarella [7] used a nonlinear supply curve with exactly one inflection point in the context of a cobweb model in order to make plausible that chaotic behaviour may result in such a model. There is, however, no exact proof under what conditions chaos can actually show up since Chiarella confines the analysis to a second order approximation to his difference equation. In a somewhat different model (a linear supply and a nonlinear demand curve which can be given a microeconomic foundation) we show that, under the formation of adaptive price expectations, the resulting adjustment mechanism can generate a wide range of dynamic behaviour (depending on the prevailing parameter constellations) such as stability, bifurcations with stable cycles of period 2, 4, 8, ... and, finally, aperiodic time paths, (i.e. we can show that period-3 cycles exist). In a second step, long historical time series of weekly price observations in German agricultural markets are scrutinized with regard to the hypothesis of nonlinearities. We study their correlation dimensions only recently employed by economists as a characteristic measure which allows, under certain conditions, to distinguish between deterministic chaos and random noise. Our results do not provide evidence to reject the hypothesis, although noise infection of the series cannot be ruled out.