Showing papers in "Mathematics of Operations Research in 1996"
TL;DR: It is shown that the problem of 1 + en-rounding of A can be solved in Om3.5 lnme-1 operations to a relative accuracy of e in the volume, and that bounds hold for the real number model of computation.
Abstract: Let A be a set of m points in Rn. We show that the problem of 1 + en-rounding of A, i.e., the problem of computing an ellipsoid E ⊆ Rn such that [1 + en]-1E ⊆ conv. hullA ⊆ E, can be solved in Omn2e-1 + ln n + ln ln m arithmetic operations and comparisons. This result implies that the problem of approximating the minimum volume ellipsoid circumscribed about A can be solved in Om3.5 lnme-1 operations to a relative accuracy of e in the volume. The latter bound also applies to the 1 + en-rounding problem. Our bounds hold for the real number model of computation.
277 citations
TL;DR: This paper provides a mathematical justification for sample-path optimization by showing that under certain assumptions, the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function.
Abstract: Sample-path optimization is a method for optimizing limit functions occurring in stochastic modeling problems, such as steady-state functions in discrete-event dynamic systems It is closely related to retrospective optimization techniques and to M-estimation The method has been computationally tested elsewhere on problems arising in production and in project planning, with apparent success In this paper we provide a mathematical justification for sample-path optimization by showing that under certain assumptions---which hold for the problems just mentioned---the method will almost surely find a point that is, in a specified sense, sufficiently close to the set of optimizers of the limit function
264 citations
TL;DR: It is shown that not all work-conserving policies are stable for such networks; however, all workThe stability and instability of Kelly-type networks are shown to be stable in a ring network.
Abstract: Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai Dai, J. G. 1995. On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid models. Ann. Appl. Probab.5 49--77. which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served FBFS and Last-Buffer-First-Served LBFS disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.
183 citations
TL;DR: The universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone, which demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone.
Abstract: We show that the universal barrier function of a convex cone introduced by Nesterov and Nemirovskii is the logarithm of the characteristic function of the cone. This interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone. This provides a simple method for calculating the universal barrier for homogeneous cones. We identify some known barriers as the universal barrier scaled by an appropriate constant. We also calculate some new universal barrier functions. Our results connect the field of interior point methods to several branches of mathematics such as Lie groups, Jordan algebras, Siegel domains, differential geometry, complex analysis of several variables, etc.
181 citations
TL;DR: A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X if and only if the function f is differentiable at the vector λX, and the formula for the derivative is given.
Abstract: A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X, λ1X ≥ λ2X ≥... ≥ λnX, and hence may be written fλ1X, λ2X,..., λnX for some symmetric function f. Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is differentiable at X if and only if the function f is differentiable at the vector λX, and we give a concise formula for the derivative. We then apply this formula to deduce an analogous expression for the Clarke generalized gradient of the spectral function. A similar result holds for real symmetric matrices.
173 citations
TL;DR: The coordination complexity of approximate price-directive decomposition PDD for the general block-angular convex resource sharing problem in K blocks and M nonnegative block-separable coupling constraints is studied and the fastest currently-known deterministic approximation algorithm for minimum-cost multicommodity flows is obtained.
Abstract: The general block-angular convex resource sharing problem in K blocks and M nonnegative block-separable coupling constraints is considered. Applications of this model are in combinatorial optimization, network flows, scheduling, communication networks, engineering design, and finance. This paper studies the coordination complexity of approximate price-directive decomposition PDD for this problem, i.e., the number of iterations required to solve the problem to a fixed relative accuracy as a function of K and M.
First a simple PDD method based on the classical logarithmic potential is shown to be optimal up to a logarithmic factor in M in the class of all PDD methods that work with the original unrestricted blocks. It is then shown that logarithmic and exponential potentials generate a polylogarithmically-optimal algorithm for a wider class of PDD methods which can restrict the blocks by the coupling constraints.
As an application, the fastest currently-known deterministic approximation algorithm for minimum-cost multicommodity flows is obtained.
119 citations
TL;DR: These properties of the Euclidean projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC1 function in terms of its B-derivative.
Abstract: This paper is concerned with properties of the Euclidean projection map onto a convex set defined by finitely many smooth, convex inequalities and affine equalities. Under a constant rank constraint qualification, we show that the projection map is piecewise smooth PC1 hence Bouligand-differentiable, or directionally differentiable; and a relatively simple formula is given for the B-derivative. These properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps, using a new characterization of invertibility of a PC1 function in terms of its B-derivative. An extension of the implicit function theorem which does not require local uniqueness is also presented. Degree theory plays a major role in the analysis of both the locally unique case and its extension.
115 citations
TL;DR: A dependent sample path coupling and a Foster-Lyapunov drift inequality are used to derive convergence rate bounds and it is shown that the bounds obtained are frequently the best possible.
Abstract: Let {Φn} be a Markov chain on the state space [0, ∞ that is stochastically ordered in its initial state; that is, a stochastically larger initial state produces a stochastically larger chain at all other times. Examples of such chains include random walks, the number of customers in various queueing systems, and a plethora of storage processes. A large body of recent literature concentrates on establishing geometric ergodicity of {Φn} in total variation; that is, proving the existence of a limiting probability measure π and a number r > 1 such that $$ \lim_{n\to \infty} r^n \sup_{A\in {\cal B}[0, \infty} \vert P_x [\Phi_n\in A]-\piA\vert = 0 $$ for every deterministic initial state Φ0 ≡ x. We seek to identity the largest r that satisfies this relationship. A dependent sample path coupling and a Foster-Lyapunov drift inequality are used to derive convergence rate bounds; we then show that the bounds obtained are frequently the best possible. Application of the methods to queues and random walks are included.
113 citations
TL;DR: A polynomial time algorithm which computes a Hamiltonian circuit with the given vertices in the space whose length approximates, with relative error less than δ, the largest length of a Hamiltonians circuit with these vertices is presented.
Abstract: For any norm in a Euclidean space and for any number δ > 0 we present a polynomial time algorithm which computes a Hamiltonian circuit with the given vertices in the space whose length approximates, with relative error less than δ, the largest length of a Hamiltonian circuit with these vertices. We also present an algorithm which for a given graph with n vertices computes the number of Hamiltonian circuits in this graph with 2n+Olog n time complexity and a polynomial in n space complexity. As a by-product of our approach we prove that the permanent of a matrix can be computed in polynomial time provided the rank of the matrix is fixed.
104 citations
TL;DR: This paper proves the existence and structure of optimal policies, and establishes Pareto optimality of policies from the two classes described above for multi-criteria problems, and describes an algorithm to compute optimal policies with properties i--iii for constrained problems.
Abstract: This paper deals with constrained optimization of Markov Decision Processes with a countable state space, compact action sets, continuous transition probabilities, and upper semicontinuous reward functions. The objective is to maximize the expected total discounted reward for one reward function, under several inequality constraints on similar criteria with other reward functions.
Suppose a feasible policy exists for a problem with M constraints. We prove two results on the existence and structure of optimal policies. First, we show that there exists a randomized stationary optimal policy which requires at most M actions more than a nonrandomized stationary one. This result is known for several particular cases. Second, we prove that there exists an optimal policy which is i stationary nonrandomized from some step onward, ii randomized Markov before this step, but the total number of actions which are added by randomization is at most M, iii the total number of actions that are added by nonstationarity is at most M.
We also establish Pareto optimality of policies from the two classes described above for multi-criteria problems. We describe an algorithm to compute optimal policies with properties i--iii for constrained problems.
The policies that satisfy properties i--iii have the pleasing aesthetic property that the amount of randomization they require over any trajectory is restricted by the number of constraints. In contrast, a randomized stationary policy may require an infinite number of randomizations over time.
97 citations
TL;DR: This work forms the nonlinear complementarity problem as a bound-constrained nonlinear least squares problem, and algorithms based on this formulation are applicable to general non linear complementarity problems, and each iteration only requires the solution of systems of linear equations.
Abstract: Global methods for nonlinear complementarity problems formulate the problem as a system of nonsmooth nonlinear equations, or use continuation to trace a path defined by a smooth system of nonlinear equations. We formulate the nonlinear complementarity problem as a bound-constrained nonlinear least squares problem. Algorithms based on this formulation are applicable to general nonlinear complementarity problems, can be started from any nonnegative starting point, and each iteration only requires the solution of systems of linear equations. Convergence to a solution of the nonlinear complementarity problem is guaranteed under reasonable regularity assumptions. The converge rate is Q-linear, Q-superlinear, or Q-quadratic, depending on the tolerances used to solve the subproblems.
TL;DR: The horizontal linear complementarity problem is studied, and general convergence properties for algorithms based on Newton iterations are derived, which provide a simple and general framework for most existing primal-dual interior point methods.
Abstract: The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences.
This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primal-dual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases whenever the convergence is not too fast in a certain sense, the sequences converge to the analytic center of the optimal face.
TL;DR: In this article, the authors studied the asymptotic behavior of the derivative of the trajectory of solutions and obtained information on the way the trajectory approached the solution set of the HLCP.
Abstract: Given a certain monotone horizontal linear complementarity problem HLCP, we can naturally construct a family of systems of nonlinear equations parametrized by a parameter t ∈ 0, 1] with the property that, as t tends to 0, the corresponding system “converges” to the HLCP. Under reasonable conditions, it has been shown that each system of the family has a unique solution and that, as t tends to 0, these solutions converge to a specific solution of the HLCP. The main purpose of this paper is to study the asymptotic behavior of the derivative of the trajectory of solutions and therefore obtain information on the way the trajectory approaches the solution set of the HLCP. We show that the trajectory of solutions converges to the solution set along a unique and well-characterized direction. Moreover, if the HLCP has a solution satisfying strict complementarity then the direction forms a definite angle with any face of the feasible region which contains the limit point; otherwise, the direction is tangent to some face of the feasible region.
TL;DR: A topology on information structures is proposed which is the minimal one that satisfies continuity requirements in games with incomplete information and is defined in terms of the common belief players have about the proximity of each player's information.
Abstract: Existing notions of proximity of information fail to satisfy desired continuity properties of equilibria in games with incomplete information. We demonstrate this failure and propose a topology on information structures which is the minimal one that satisfies those continuity requirements. This topology is defined in terms of the common belief players have about the proximity of each player's information.
TL;DR: In this article, the authors consider repeated zero-sum games with one-sided information and standard signaling and show that the normal density in the n-1/2-term of the asymptotic expansion of vn can be explained by the Central Limit Theorem.
Abstract: This paper is concerned with the repeated zero-sum games with one-sided information and standard signaling. We introduce here dual games that allow us to analyze the “Markovian” behavior of the uninformed player, and to explicitly compute his optimal strategies. We then apply our results on the dual games to explain the appearance of the normal density in the n-1/2-term of the asymptotic expansion of vn as a consequence of the Central Limit Theorem.
TL;DR: This paper investigates asymptotic convergence to stationary points of gradient descent algorithms where the functions involved are not available in closed form but are approximated by sequences of random functions.
Abstract: This paper investigates asymptotic convergence to stationary points of gradient descent algorithms where the functions involved are not available in closed form but are approximated by sequences of random functions. The algorithms take large stepsizes and use progressively finer precision at successive iterations. Two kinds of optimization algorithms are considered: one, where the limiting function to be minimized is differentiable but the random approximating functions can be nondifferentiable, and the other, where both the limiting and approximating functions are nondifferentiable and convex. Such functions often arise in discrete event dynamic system-models in various application areas.
The analysis brings together ideas and techniques from the disciplines of nonlinear programming and nondifferentiable optimization on one hand, and stochastic processes and especially uniform laws of large numbers, on the other hand. A general algorithmic frame-work is first developed and analyzed, and then applied to the specific algorithms considered. The analysis extends the results derived to date for similar algorithms, and has a potential for further extensions in proving convergence of other algorithms as well.
TL;DR: It is proved that the special case of minimizing maximum lateness subject to release dates that lie in the interval [dj-pj-A, dj-A], for all j = 1,..., n and for some constant A, is solvable in On log n time if no machine idle time is allowed and in On2 log nTime if machine idleTime is allowed.
Abstract: We consider the following single-machine bicriteria scheduling problem. A set of n independent jobs has to be scheduled on a single machine that can handle only one job at a time and that is available from time zero onwards. Each job Jj requires processing during a given positive uninterrupted time pj, and has a given target start time sj and due date dj with A ≤ dj-sj ≤ A + pj for some constant A. For each job Jjj = 1,..., n, its promptness Pj is defined as the difference between the target start time and the actual start time, and its lateness Lj as the difference between the completion time and the due date. We consider the problem of finding a schedule that minimizes a function F of maximum promptness Pmax = max1≤j≤nPj and maximum lateness Lmax = max1≤j≤nLj; we assume that F is nondecreasing in both arguments. We present an On2 algorithm for the variant in which idle time is not allowed and an On2 log n algorithm for the special case in which the objective function is linear. We prove that the problem is NP-hard if neither of these restrictions is imposed. As a side result, we prove that the special case of minimizing maximum lateness subject to release dates that lie in the interval [dj-pj-A, dj-A], for all j = 1,..., n and for some constant A, is solvable in On log n time if no machine idle time is allowed and in On2 log n time if machine idle time is allowed.
TL;DR: Limiting the definition to prediction of near future events and to a full sequence of times yields the new notion of almost weak merging AWM, presented here and shows many cases with no absolute continuity that exhibit AWM.
Abstract: Two measures, μ and μI, are updated as more information arrives. If with μ-probability 1, the predictions of future events according to both measures become close, as time passes, we say that μI merges to μ. Blackwell and Dubins Blackwell, D., L. Dubins. 1962. Merging of opinions with increasing information. Ann. Math. Statist.38 882--886. showed that if μ is absolutely continuous with respect to μI then μI merges to μ. Restricting the definition to prediction of near future events and to a full sequence of times yields the new notion of almost weak merging AWM, presented here. We introduce a necessary and sufficient condition and show many cases with no absolute continuity that exhibit AWM. We show, for instance, that the fact that μI is diffused around μ implies AWM.
TL;DR: The globally convergent framework proposed by Kojima, Noma, and Yoshise is used to construct an infeasible-interior-point algorithm for monotone nonlinear complementarity problems that achieves superlinear convergence when the solution is nondegenerate.
Abstract: We use the globally convergent framework proposed by Kojima, Noma, and Yoshise to construct an infeasible-interior-point algorithm for monotone nonlinear complementarity problems. Superlinear convergence is attained when the solution is nondegenerate and also when the problem is linear with a strictly complementary solution. Numerical experiments confirm the efficacy of the proposed approach.
TL;DR: It is proved the existence of a one-to-one correspondence between coalitional games bounded with respect to the composition norm and countably additive measures defined on an appropriate space.
Abstract: A coalitional game is a real-valued set function ν defined on an algebra F of subsets of a space X such that νO = 0. We prove the existence of a one-to-one correspondence between coalitional games bounded with respect to the composition norm and countably additive measures defined on an appropriate space.
TL;DR: An algorithm for balanced partitioning is constructed, with the same rate of convergence to zero as the algorithm LDM largest differencing method, which divides a list of n random items into two blocks.
Abstract: The algorithm LDM largest differencing method divides a list of n random items into two blocks. The parameter of interest is the expected difference between the two block sums. It is shown that if the items are i.i.d. and uniform then the rate of convergence of this parameter to zero is n-Θlog n. An algorithm for balanced partitioning is constructed, with the same rate of convergence to zero.
TL;DR: A new primal-dual affine scaling method for linear programming that yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof.
Abstract: In this paper we present a new primal-dual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing the objective function which is the duality gap in the primal-dual case, over some suitable ellipsoid. This gives rise to completely new primal-dual affine scaling directions, having no obvious relation with the search directions proposed in the literature so far. The new directions guarantee a significant decrease in the duality gap in each iteration, and at the same time they drive the iterates to the central path. In the analysis of our algorithm we use a barrier function which is the natural primal-dual generalization of Karmarkar's potential function. The iteration bound is OnL, which is a factor OL better than the iteration bound of an earlier primal-dual affine scaling method Monteiro, Adler and Resende [Monteiro, R. D. C., I. Adler, M. G. C. Resende. 1990. A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res.15 191--214.].
TL;DR: A variety of new entropy measures which attempt to control derivative values of the densities are presented, including relaxed moment matching with Fisher information and best interpolation on a strip.
Abstract: Maximum entropy spectral density estimation is a technique for reconstructing an unknown density function from some known measurements by maximizing a given measure of entropy of the estimate. Here we present a variety of new entropy measures which attempt to control derivative values of the densities. Our models apply among others to the inference problem based on the averaged Fisher information measure. The duality theory we develop resembles models used in convex optimal control problems. We present a variety of examples, including relaxed moment matching with Fisher information and best interpolation on a strip.
TL;DR: This paper presents a modified version of Algorithm MCF proposed by Goldberg, Plotkin and Tardos for the generalized circulation problem that has a worst-case complexity that is better than the complexities of the MCF and Fat-Path combinatorial algorithms.
Abstract: This paper presents a modified version of Algorithm MCF proposed by Goldberg, Plotkin and Tardos for the generalized circulation problem. This new combinatorial algorithm has a worst-case complexity that is better than the complexities of the MCF and Fat-Path combinatorial algorithms of Goldberg, Plotkin, and Tardos Goldberg, A. V., S. A. Plotkin, E. Tardos. 1991. Combinatorial algorithms for the generalized circulation problem. Math. Oper. Res.16 351--379..
TL;DR: Two classes of matrices are identified which are analogues of the class of copositive-plus and L-matrices in the study of the linear complementarity problem and it is proved that the algorithm processes ACx = a when A is the linear transformation associated with such matrices.
Abstract: We explain and justify a path-following algorithm for solving the equations ACx = a, where A is a linear transformation from Rn to Rn, C is a polyhedral convex subset of Rn, and AC is the associated normal map. When AC is coherently oriented, we are able to prove that the path following method terminates at the unique solution of ACx = a, which is a generalization of the well known fact that Lemke's method terminates at the unique solution of LCPq, M when M is a P = matrix. Otherwise, we identity two classes of matrices which are analogues of the class of copositive-plus and L-matrices in the study of the linear complementarity problem. We then prove that our algorithm processes ACx = a when A is the linear transformation associated with such matrices. That is, when applied to such a problem, the algorithm will find a solution unless the problem is infeasible in a well specified sense.
TL;DR: The main result is that the existence of a solution ψ* to a partial differential equation with appropriate boundary conditions and regularity properties implies the uniform convergence of ψn to the Fenchel conjugate of ω*.
Abstract: Let vnp denote the value of the n-times repeated zero-sum game with incomplete information on one side and full monitoring and let up be the value of the average game Gp. The error term enp = vnp-cavup is then converging to zero at least as rapidly as 1/√n.
In this paper, we analyze the convergence of ψnp = √nenp in the games with square payoff matrices such that the optimal strategy of the informed player in the average game Gp is unique, is completely mixed and does not depend on p.
Our main result is that the existence of a solution ψ* to a partial differential equation with appropriate boundary conditions and regularity properties implies the uniform convergence of ψn to the Fenchel conjugate of ψ*.
In particular cases, the P.D.E. problem is linear and its solution ψ* is then related to the multidimensional normal distribution.
TL;DR: This paper describes a fast algorithm to find the nucleolus of any game with a nonempty imputation set based on the algorithm scheme of Maschler et al. for the generalucleolus.
Abstract: This paper describes a fast algorithm to find the nucleolus of any game with a nonempty imputation set. It is based on the algorithm scheme of Maschler et al. Maschler, M., J. Potters, S. Tijs. 1992. The general nucleolus and the reduced game property. Internal. J. Game Theory21 83--106. for the general nucleolus.
TL;DR: The modified nucleolus coincides with the just mentioned solutions in these special cases and can be seen as a canonical representation in the general weighted majority case.
Abstract: A new solution concept for cooperative transferable utility games is introduced, which is strongly related to the nucleolus and therefore called modified nucleolus. It has many properties in common with the prenucleolus and can be considered as the canonical restriction of the prenucleolus of a certain replicated game. For weighted majority games this solution concept induces a representation of the game. In the special case of weighted majority constant-sum games and homogeneous games respectively the nucleolus and the minimal integer representation respectively are adequate candidates for a canonical representation see Peleg [Peleg, B. 1968. On weights of constant-sum majority games. SIAM J. Appl. Math.16 527--532.] and Ostman [Ostmann, A. 1987a. On the numerical representation of homogeneous games. Int. J. Game Theory16 69--81.]. Fortunately the modified nucleolus coincides with the just mentioned solutions in these special cases and can, therefore, be seen as a canonical representation in the general weighted majority case.
TL;DR: This paper modifies a recent sufficient condition for the existence of optimal solutions proposed by Buttazzo and Tomarelli to obtain a condition that is necessary and sufficient.
Abstract: This paper modifies a recent sufficient condition for the existence of optimal solutions proposed by Buttazzo and Tomarelli, to obtain a condition that is necessary and sufficient. Applications to mathematical programming are given and stability results established.
TL;DR: In this article, the authors considered the continuous polling system, where customers arrive according to a homogeneous Poisson process or a more general stationary point process and wait on a circle in order to be served by a single server.
Abstract: We consider the following continuous polling system: Customers arrive according to a homogeneous Poisson process or a more general stationary point process and wait on a circle in order to be served by a single server. The server is “greedy,” in the sense that he always moves with constant speed towards the nearest customer. The customers are served according to an arbitrary service time distribution, in the order in which they are encountered by the server. First-order and second-order Taylor-expansions are found for the expected configuration of customers, for the mean queue length, and for expectation and distribution function of the workload. It is shown that under light traffic conditions the greedy server works more efficiently than the cyclically polling server.