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163
Stability, queue length and delay of deterministic and stochastic queueing networks
 IEEE Transactions on Automatic Control
, 1994
"... Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of ..."
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Cited by 192 (20 self)
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Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate `. To answer these two types of stability problems, we introduce two new notions of traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we develop a set of rules for network operations such as superposition, inputoutput relation of a single queue, and routing. Specifically, we show that (i) the MER of a superposition process is less than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, (iii) the MER of a departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to ` under a further assumption of independence. These rules provide a natural way to analyze feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the FCFS policy. Moreover, when restricting to the family of twostate Markov modulated arrival processes, the notion of MER with respect to ` is shown to be
On the Convergence of Pattern Search Algorithms
"... . We introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems. Our definition unifies an important collection of optimization methods that neither computenor explicitly approximate derivatives. We exploit our characterization of pattern sea ..."
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Cited by 172 (14 self)
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. We introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems. Our definition unifies an important collection of optimization methods that neither computenor explicitly approximate derivatives. We exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease. Our analysis is possible because the iterates of a pattern search method lie on a scaled, translated integer lattice. This allows us to relax the classical requirements on the acceptance of the step, at the expense of stronger conditions on the form of the step, and still guarantee global convergence. Key words. unconstrained optimization, convergence analysis, direct search methods, globalization strategies, alternating variable search, axial relaxation, local variation, coordinate search, evolutionary operation, pattern search, multidirectional search, downhill simplex search AMS(M...
Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods
 SIAM REVIEW VOL. 45, NO. 3, PP. 385–482
, 2003
"... Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked ..."
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Cited by 143 (14 self)
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Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.
On the Use of NonStationary Penalty Functions to Solve Nonlinear Constrained Optimization Problems with GA's
 In
, 1994
"... In this paper we discuss the use of nonstationary penalty functions to solve general nonlinear programming problems (NP ) using realvalued GAs. The nonstationary penalty is a function of the generation number; as the number of generations increases so does the penalty. Therefore, as the penalty i ..."
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Cited by 109 (7 self)
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In this paper we discuss the use of nonstationary penalty functions to solve general nonlinear programming problems (NP ) using realvalued GAs. The nonstationary penalty is a function of the generation number; as the number of generations increases so does the penalty. Therefore, as the penalty increases it puts more and more selective pressure on the GA to find a feasible solution. The ideas presented in this paper come from two basic areas: calculusbased nonlinear programming and simulated annealing. The nonstationary penalty methods are tested on four NP test cases and the effectiveness of these methods are reported.. 1 Introduction Constrained function optimization is an extremely important tool used in almost every facet of engineering, operations research, mathematics, and etc. Constrained optimization can be represented as a nonlinear programming problem. The general nonlinear programming problem is defined as follows: (NP ) minimize f(X) subject to (nonlinear and linear)...
Mesh adaptive direct search algorithms for constrained optimization
 SIAM J. Optim
, 2004
"... Abstract. This paper introduces the Mesh Adaptive Direct Search (MADS) class of algorithms for nonlinear optimization. MADS extends the Generalized Pattern Search (GPS) class by allowing local exploration, called polling, in a dense set of directions in the space of optimization variables. This mean ..."
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Cited by 91 (14 self)
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Abstract. This paper introduces the Mesh Adaptive Direct Search (MADS) class of algorithms for nonlinear optimization. MADS extends the Generalized Pattern Search (GPS) class by allowing local exploration, called polling, in a dense set of directions in the space of optimization variables. This means that under certain hypotheses, including a weak constraint qualification due to Rockafellar, MADS can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained. The main GPS convergence result is to identify limit points where the Clarke generalized derivatives are nonnegative in a finite set of directions, called refining directions. Although in the unconstrained case, nonnegative combinations of these directions spans the whole space, the fact that there can only be finitely many GPS refining directions limits rigorous justification of the barrier approach to finitely many constraints for GPS. The MADS class of algorithms extend this result; the set of refining directions may even be dense in R n, although we give an example where it is not. We present an implementable instance of MADS, and we illustrate and compare it with GPS on some test problems. We also illustrate the limitation of our results with examples. Key words. Mesh adaptive direct search algorithms (MADS), convergence analysis, constrained optimization, nonsmooth analysis, Clarke derivatives, hypertangent, contingent cone.
The kserver problem
 Computer Science Review
"... The kserver problem is perhaps the most influential online problem: natural, crisp, with a surprising technical depth that manifests the richness of competitive analysis. The kserver conjecture, which was posed more that two decades ago when the problem was first studied within the competitive ana ..."
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Cited by 69 (5 self)
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The kserver problem is perhaps the most influential online problem: natural, crisp, with a surprising technical depth that manifests the richness of competitive analysis. The kserver conjecture, which was posed more that two decades ago when the problem was first studied within the competitive analysis framework, is still open and has been a major driving force for the development of the area online algorithms. This article surveys some major results for the kserver. 1
On The Convergence Of The Multidirectional Search Algorithm
, 1991
"... . This paper presents the convergence analysis for the multidirectional search algorithm, a direct search method for unconstrained minimization. The analysis follows the classic lines of proofs of convergence for gradientrelated methods. The novelty of the argument lies in the fact that explicit ca ..."
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Cited by 67 (8 self)
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. This paper presents the convergence analysis for the multidirectional search algorithm, a direct search method for unconstrained minimization. The analysis follows the classic lines of proofs of convergence for gradientrelated methods. The novelty of the argument lies in the fact that explicit calculation of the gradient is unnecessary, although it is assumed that the function is continuously differentiable over some subset of the domain. The proof can be extended to treat most nonsmooth cases of interest; the argument breaks down only at points where the derivative exists but is not continuous. Finally, it is shown how a general convergence theory can be developed for an entire class of direct search methodswhich includes such methods as the factorial design algorithm and the pattern search algorithmthat share a key feature of the multidirectional search algorithm. Key words. unconstrained optimization, convergence analysis, direct search methods, parallel optimization, mult...
A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs : II. Application of Theory and Test Problems
 Engng
, 1990
"... In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until th ..."
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Cited by 54 (22 self)
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In Part I (Floudas and Visweswaran, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an fflglobal optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include : (i) the general quadratic programming problem, (ii) quadratic programming problems with quadratic constraints, (iii) pooling and blending problems, and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach. Keywords : Global Optimization, Quadratic Programming, Quadratic Constraints, Polynomial functions, Pooling and Blending Problems. Author to whom...
A Microeconomic View of Data Mining
, 1998
"... We present a rigorous framework, based on optimization, for evaluating data mining operations such as associations and clustering, in terms of their utility in decisionmaking. This framework leads quickly to some interesting computational problems related to sensitivity analysis, segmentation and th ..."
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Cited by 45 (2 self)
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We present a rigorous framework, based on optimization, for evaluating data mining operations such as associations and clustering, in terms of their utility in decisionmaking. This framework leads quickly to some interesting computational problems related to sensitivity analysis, segmentation and the theory of games. Department of Computer Science, Cornell University, Ithaca NY 14853. Email: kleinber@cs.cornell.edu. Supported in part by an Alfred P. Sloan Research Fellowship and by NSF Faculty Early Career Development Award CCR9701399. y Computer Science Division, Soda Hall, UC Berkeley, CA 94720. christos@cs.berkeley.edu z IBM Almaden Research Center, 650 Harry Road, San Jose CA 95120. pragh@almaden.ibm.com 1 Introduction Data mining is about extracting interesting patterns from raw data. There is some agreement in the literature on what qualifies as a "pattern" (association rules and correlations [1, 2, 3, 5, 6, 12, 20, 21] as well as clustering of the data points [9], are ...
Sample Path Large Deviations and Intree Networks
 Queueing Systems
, 1994
"... Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distri ..."
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Cited by 41 (8 self)
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Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distributions in an intree network with routing can be derived by a set of recursive equations. The solution of this set of equations is related to the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. We also prove a conditional limit theorem that illustrates how a queue builds up in an intree network.