In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n(-1)Q(nt), where Q(t) represents the joint number of clients at time t in a polling system...
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In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n(-1)Q(nt), where Q(t) represents the joint number of clients at time t in a polling system with N nodes, one server and Markovian routing. By the way, the large deviation principle is proved and the rate function is shown to have the form conjectured by Dupuis and Ellis. We introduce a so called empirical generator consisting of Q(t) and of two empirical measures associated with S-t, the position of the server at time t. One of the main step is to derive large deviations bounds for a localized version of the empirical generator. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program. The method seems to have a wide range of application including the famous Jackson networks, as shown at the end of this study. An example illustrates how this technique can be used to estimate stationary probability decay rate.
A bound is obtained in this note for the distance between the integer and real solutions to convex quadratic programs. This bound is a function of the condition number of the Hessian matrix. We further extend this pro...
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A bound is obtained in this note for the distance between the integer and real solutions to convex quadratic programs. This bound is a function of the condition number of the Hessian matrix. We further extend this proximity result to convex programs and mixed-integer convex programs. We also show that this bound is achievable in certain situations and the distance between the integer and continuous minimizers may tend to infinity. (C) 2001 Elsevier Science B.V. All rights reserved.
This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint func...
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This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint functions of the problem are analytic, we also characterize the limiting behavior of these paths as they approach the set of optimal solutions. A duality relationship between a certain pair of logarithmic barrier problems is also discussed.
This note derives bounds on the length of the primal-dual affine scaling directions associated with a linearly constrained convex program satisfying the following conditions: 1) the problem has a solution satisfying s...
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This note derives bounds on the length of the primal-dual affine scaling directions associated with a linearly constrained convex program satisfying the following conditions: 1) the problem has a solution satisfying strict complementarity, 2) the Hessian of the objective function satisfies a certain invariance property. We illustrate the usefulness of these bounds by establishing the superlinear convergence of the algorithm presented in Wright and Ralph [22] for solving the optimality conditions associated with a linearly constrained convex program satisfying the above conditions.
The authors present a primal interior-point algorithm for solving convex programs with nonlinear constraints. The algorithm uses a predictor-corrector strategy to follow a smooth path that leads from a given starting ...
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The authors present a primal interior-point algorithm for solving convex programs with nonlinear constraints. The algorithm uses a predictor-corrector strategy to follow a smooth path that leads from a given starting point to an optimal solution. A convergence analysis is given showing that under mild assumptions the algorithm simultaneously iterates towards feasibility and optimality. The matrices involved can be kept sparse if the nonlinear functions are separable or depend on only a few variables. A preliminary implementation has been developed. Some promising numerical results indicate that the algorithm may be efficient in practice, and that it can deal in a single phase with infeasible starting points without relying on some ''big M'' parameter.
A differential inclusion is designed for solving cone-constrained convex programs. The method is of subgradient-projection type. It involves projection, penalties and Lagrangian relaxation. Non-smooth data can be acco...
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A differential inclusion is designed for solving cone-constrained convex programs. The method is of subgradient-projection type. It involves projection, penalties and Lagrangian relaxation. Non-smooth data can be accommodated. A novelty is that multipliers converge monotonically upwards to equilibrium levels. An application to stochastic programming is considered.
In this paper, we investigate the impact of the locations of the gateways on the performance of the internet. We consider the problem of determining i) the routing assignments for the intranet and internet traffic and...
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In this paper, we investigate the impact of the locations of the gateways on the performance of the internet. We consider the problem of determining i) the routing assignments for the intranet and internet traffic and ii) the number of gateways and their locations to interconnect existing data networks to minimize a linear combination of the average internet and intranet packet delays subject to a cost constraint on the amount to be spent to establish the gateways. This joint routing and topological design problem is important in the design of internets and should be solved before networks are actually interconnected. This problem is formulated as a nonlinear combinatorial optimization problem. When the gateway locations are fixed, the resulting routing problem is not a convex programming problem. This is unexpected since the routing problem in datagram networks is usually formulated as a convex program, We develop an algorithm based upon Lagrangian relaxation to solve this problem. In the computational experiments, the algorithm was shown to be effective in interconnecting i) two WAN's and ii) two grid networks. The experiments also showed that the algorithm finds better feasible solutions than an exchange heuristic.
We introduce and characterize a class of differentiable convex functions for which the Karush-Kuhn-Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterizati...
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We introduce and characterize a class of differentiable convex functions for which the Karush-Kuhn-Tucker condition is necessary for optimality. If some constraints do not belong to this class, then the characterization of optimality generally assumes an asymptotic form. We also show that for the functions that belong to this class in multi-objective optimization, Pareto solutions coincide with strong Pareto solutions,. This extends a result, well known for the linear case.
An algorithm for minimizing the largest eigenvalue of an affine combination of symmetric matrices is presented. The nonsmooth problem is transformed into an equivalent smooth constrained problem, which is solved by a ...
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An algorithm for minimizing the largest eigenvalue of an affine combination of symmetric matrices is presented. The nonsmooth problem is transformed into an equivalent smooth constrained problem, which is solved by a predictor-corrector interior-point method taking full advantage of the differentiability and convexity. Some promising numerical results obtained from a preliminary implementation are included.
When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an art...
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When we apply interior point algorithms to various problems including linear programs, convex quadratic programs, convex programs and complementarity problems, we often embed an original problem to be solved in an artificial problem having a known interior feasible solution from which we start the algorithm. The artificial problem involves a constant M (or constants) which we need to choose large enough to ensure the equivalence between the artificial problem and the original problem. Theoretically, we can always assign a positive number of the order O(2L) to M in linear cases, where L denotes the input size of the problem. Practically, however, such a large number is impossible to implement on computers. If we choose too large M, we may have numerical instability and/or computational inefficiency, while the artificial problem with M not large enough will never lead to any solution of the original problem. To solve this difficulty, this paper presents ''a little theorem of the big M'', which will enable us to find whether M is not large enough, and to update M during the iterations of the algorithm even if we start with a smaller M. Applications of the theorem are given to a polynomial-time potential reduction algorithm for positive semi-definite linear complementarity problems, and to an artificial self-dual linear program which has a close relation with the primal-dual interior point algorithm using Lustig's limiting feasible direction vector.
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