Nemirovski, A.S. & Yudin, D.B. (1983) Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a joint Lyapunov function and state feedback that optimize some objective (see for example [BGFB93, FBB92, FBBEG92, EGBFB92, BFBG92] The problem is quasiconvex and so can be solved reliably by several methods, for example, the ellipsoid algorithm developed by Shor, Nemirovsky, and Yudin [Sho85, NY83, BGT81, BB91] or Kelley s cutting plane algorithm [Kel60, BB91] In this paper we describe an interior point algorithm that solves the problem, and appears to be very efficient compared to these methods. We give a simple proof of convergence for our algorithm, but we do not give a detailed complexity ....

A. Nemirovsky and D. Yudin. Problem Complexity and Method Efficiency in Optimization. John Wiley & Sons, 1983.

....value of a design objective, a local minimization method would yield a possibly conservative upper bound for the global minimum. In this case, the cost associated with not finding the global minimumwould usually be acceptable. For a quantitative description of the term hard , see for example, [1]. There exist several popular methods for solving global optimization problems (P1) or (P2) see [3] for example) Simulated Annealing (see [4] and the references therein) describes a family of iterative methods where every iteration consists of taking a step in parameter space with a ....

A. Nemirovsky and D. Yudin, Problem Complexity and Method Efficiency in Optimization, John Wiley & Sons, (1983).

....programming. The IEM minimizes a convex function over an n dimensional cube to relative accuracy in O(n ln(n= iterations, each requiring evaluation of the function and a subgradient. The order of this complexity, also achieved by the volumetric cutting plane algorithm [1, 15] is optimal [13]. Another application of fl maximal ellipsoids is to provide a rounding of P. It is known that for the maximum volume inscribed ellipsoid (MVIE) E , E ae P ae nE ; where for an ellipsoid E and positive scalar , E denotes the dilation of E about its center by the factor . For a ....

A.S. Nemirovskii and D.B. Yudin, Problem complexity and method efficiency in optimization, John Wiley, Chichester, 1983.

....convex programming. The IEM minimizes a convex function over an n dimensional cube to relative accuracy in O(n ln(n= iterations, each requiring evaluation of the function and a subgradient. The order of this complexity, also achieved by the volumetric cutting plane algorithm [1, 13] is optimal [11]. Another application of fl maximal ellipsoids is to provide a rounding of P. It is know that for the maximum volume inscribed ellipsoid (MVIE) E , E ae P ae nE ; where for an ellipsoid E and positive scalar , E denotes the dilation of E about its center by the factor . In the worst ....

A.S. Nemirovskii and D.B. Yudin, Problem complexity and method efficiency in optimization, John Wiley, Chichester, 1983.

....which is defined implicitly by the oracle. The classical centering methods that have been suggested for the above convex feasibility problem include the center of gravity method of Levin [12] the largest sphere method of Elzinga and Moore [5] the ellipsoid method of Yudin and Nemirovskii [16, 10, 18], the maximum volume ellipsoid method of Tarasov, Khacian and Erlich [20, 11] and the method of volumetric centers of Vaidya [21] among others. Sonnevend [19] Goffin, Haurie and Vial [7] and Ye [22] proposed a column generation or cutting plane algorithm that computes y as the analytic ....

A. Nemirovsky and D. Yudin, "Problem complexity and method efficiency in Optimization", Wiley-Interscience, NY 1983.

....A popular approach to solve the latter problem is to use column generation method or the methods of centers. The classical centering methods include the center of gravity method of Levin [20] the largest sphere method of Elzinga and Moore [5] the ellipsoid method of Yudin and Nemirovskii [28, 18, 31], the maximum volume ellipsoid method of Tarasov, Kahachiyan and Erlich [33, 19] and the method of volumetric centers of Vaidya [35] among others. Recently, the use of analytic center in the column generation algorithms was proposed [6, 34] and subsequently a number of studies have presented the ....

A. Nemirovsky and D. Yudin, "Problem Complexity and Method Efficiency in Optimization ", Wiley-Interscience, NY (1983).

....of this is x = x 1=4 2 ; showing that the sequence generated by the algorithm converges to x = 1=4. For convergence to be guaranteed one must make the assumption that the feasible set is described by a Lipschitzian convex constraint. Nevertheless, the complexity of this method is very poor [67]. If the problem to be solved is that of optimization min ff(x) j x 2 Cg ; then the next iterate x k 1 is defined as the solution of the LP: minfz j z f(x i ) hg i ; x Gamma x i i; 8i k ; x 2 Cg where g i 2 f(x i ) The oracle will then answer: f(x k 1 ) and g k 1 2 f(x k 1 ) and this ....

....plane will be added to the current approximation to f . The piecewise linear function f k (x) max ik f(x i ) hg i ; x Gamma x i i is a lower approximation to f , and clearly f k f k 1 f . The method converges, but the estimate in O(1=ffl n ) is very poor [9] and can be as bad as this [67]. 3.2.2. Chebyshev centers or largest sphere. In the pure feasibility case, the next point where the separation oracle is called, x k 1 , solves the linear program minfz j z h g i kg i k ; x Gamma x i i; 8i kg: 9 This is a linear program, and x k 1 is the center of the largest sphere ....

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A.S. Nemirovskii and D.B. Yudin, "Problem Complexity and Method Efficiency in Optimization ", John Wiley, Chichester (1983).

....paper we study time complexity for the GP method for optimal routing in wide area networks. There is an extensive literature devoted to numerical methods for optimization problems; yet, far less attention has been paid to complexities of optimization problems or the algorithms for solving them [1]. In their book, Nemirovsky and Yudin [1] have addressed both the problem complexity and algorithm complexity for general convex optimization; in this paper, we focus on a specific gradient projection algorithm for the path formulated optimal routing problem. The optimal routing problem and the GP ....

....GP method for optimal routing in wide area networks. There is an extensive literature devoted to numerical methods for optimization problems; yet, far less attention has been paid to complexities of optimization problems or the algorithms for solving them [1] In their book, Nemirovsky and Yudin [1] have addressed both the problem complexity and algorithm complexity for general convex optimization; in this paper, we focus on a specific gradient projection algorithm for the path formulated optimal routing problem. The optimal routing problem and the GP algorithm are important topics. The ....

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A.S. Nemirovsky and D.B. Yudin, Problem Complexity and Method Efficiency in Optimization, translated by E.R. Dawson, John Wiley & Sons, 1983.

....involve multivariate functions belonging to F r have been proven computationally intractable in the number of variables in the worst case setting. These include nonlinear equations [10] partial differential equations [19] function approximation [7] integral equations [19] and optimization [5]. Material on the computational complexity of optimization will be presented in the second half of this article. Very high dimensional integrals occur in many disciplines. For example, problems with dimension ranging from the hundreds to the thousands occur in mathematical finance. Path ....

....I. Sloan H. Wo zniakowski information partial partial information contaminated contaminated information L. Plaskota priced priced information optimal algorithms radius of information file: traub1 date: March 18, 1999 4 In their seminal book, A.S. Nemirovsky and D.B. Yudin [5] study a constrained optimization problem. They wish to minimize a nonlinear function subject to nonlinear constraints. Let f = f 0 ; f 1 ; f m ] where f 0 denotes the objective function and f 1 ; f m denote constraints. Let F be the product of m 1 copies of F r . Then ....

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Nemirovsky, A.S., and Yudin, D.B.: Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience, New York, 1983.

....of the variable (x; y) Therefore, unless there are additional properties on f such as the uniqueness of the optional solution, the sequence f(x k ; y k )g may not converge to an optimal solution. This is the problem of well posedness that is discussed at length by Nemirovsky and Yudin [15]. 6 3. Applications and Numerical Examples Block angular linear programming (BALP) and block angular quadratic programming (BAQP) are two special cases of (1.1) In this section, we show how GPPA can be applied to solve BALP and BAQP and propose decomposition algorithms. We also give numerical ....

A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, Interscience Series in Discrete Mathematics. John Wiley and Sons, New York, 1983.

....0 2 f(x k ) This optimal point strategy assumes that the relaxation is a good approximation of the original problem, but this is only true when a big number of cuts has been generated. The method is globally convergent [48] but in practice, it sometimes shows a slow pattern of convergence [36]. Centre of Gravity Method. This the method was first proposed by Levin [33] as the first cutting plane method that generates query points at the centre of the localization set. Choosing the centre of gravity to generate query points seems to be the natural choice. In fact cutting through the ....

A. S. Nemirovskii and D. B. Yudin, "Problem complexity and method efficiency in optimization", John Wiley, Chichester (1983).

.... f : kk 1 1g) with stepsizes fl t = O(t Gamma1 ) is completely inappropriate here (the Hessian matrix Phi = lim t 1 OE t OE T t is extremely ill conditioned) However, Irisa On Minimax Prediction for Nonparametric Autoregressive Models 5 one can use the robust version of this algorithm (cf. [7]) with the gain fl t = O(t Gamma1=2 ) with the estimate N obtained by the averaging: N = 1 N N Gamma1 X t=0 t : The prediction error of the latter algorithm attains E[OE T N ( Gamma N ) 2 = O(N Gamma1=2 ) 11) for and does not depend on the conditional number of ....

....number of Phi. However, the constant factor in the right hand side of (11) is proportional to the l 2 level of noise Eke t OE t k 2 2 which is O(M ) The method we use here is the non Eucludian stochastic approximation procedure associated with the L 1 norm. It has been first introduced in [7] and is referred to as mirror descent algorithm. Let q = 2 ln M and fl t ; t = 1; N be a positive sequence (we give the precise definition of this sequence below) We set W (z) kzk 2 q =2. In order to obtain the estimate t of the parameter given t observations (10) we use the ....

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A. Nemirovski and D. Yudin, Problem complexity and method efficiency in optimization -- J. Wiley & Sons, 1983. Irisa

....of the functions in the class F d . Problems which suffer the curse of dimension in the worst case setting include integration, approximation, global optimization, integral and partial differential equations for classes of functions whose rth derivatives are uniformly bounded in L1 , see [1, 6, 8, 9, 13, 18, 27]. In the average case and randomized settings, the curse of dimension is present for approximation over the class of functions with r continuous derivatives which is equipped with the folded isotropic Wiener measure, see [15, 22] for the average case, and [7, 10, 21] for the randomized setting. ....

A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience, New York, 1983.

....problem as a batch processing problem. This provides an alternate approach to quantifying the communication and computational requirements of resource allocation processes, that is different from the measures that have been used in the decomposition, message space and team theory literatures. Nemirovsky and Yudin (1983) and Ibaraki and Katoh (1988) provide a number of results on the complexity of resource allocation and other constrained optimization. However, Friedman and Oren (1995) were the first to study an algorithm that resembles decentralized communication procedures. They study the complexity of a price ....

Nemirovsky, A. S. and Yudin, D. B. (1983). Problem Complexity and Method Efficiency in Optimization. New York: Wiley.

....Without loss of generality, we assume that a is normalized so that kak = 1. The classical centering methods that have been suggested for the above include the center of gravity method of Levin [12] the largest sphere method of Elzinga and Moore [5] the ellipsoid method of Yudin and Nemirovskii [17, 10], the maximum volume ellipsoid method of Tarasov, Khacian and Erlich [20, 11] and the method of volumetric centers of Vaidya [22] among others. Sonnevend [19] Goffin, Haurie and Vial [7] and Ye [24] proposed a column generation or cutting plane algorithm that computes y as an approximate ....

A. Nemirovsky and D. Yudin, "Problem complexity and method efficiency in Optimization", Wiley-Interscience, NY 1983.

....problem: max n u(x)jp 0 x y o ; x 2 R d (3:1) An ffl approximation to this global optimization problem is a vector x satisfying p 0 y and ju(x) Gamma u(x )j ffl, where x is an vector which attains a global maximum in equation (3. 1) The following result is from Nemirovsky and Yudin (1983): Theorem 3.1: Let F denote the set of r times continuously differentiable functions on R d . Then a lower bound on the worst case deterministic complexity of the consumer s problem (3.1) is given by: comp(ffl; d; r) o i (1=ffl) d=r j (3:2) If F is the set of all concave functions ....

Nemirovsky, A.S. and D.B. Yudin (1983) Problem Complexity and Method Efficiency in Optimization Wiley, New York.

....f on the lattice X Theta Y . We refer the reader to the survey of Orlitsky and El Gamal (1988) for a summary of this approach and many possible extensions. More closely related to the approach of the present paper is the work done in the area of information based complexity (Traub et al. 1988) (Nemirovsky and Yudin, 1983), and in particular the work of Sukharev (1992) This line of work is concerned with the efficiency of algorithms for problems defined on the infinite dimensional spaces, such as the function integration problem, approximation, and optimization. In this context, the processing element can obtain ....

Nemirovsky, A. S., Yudin, D. B. (1983), "Problem Complexity and Method Efficiency in Optimization," Wiley, New York.

....Since the query point is an extreme point of the localization set in Kelley s approach, the next iteration may select again the same query point, and the method stalls. An illuminating example that confirms the possible slow behavior of Kelley s cutting plane method is fully described in [28]. The motivation for choosing central query point is very much to obviate the above bad feature of Kelley s strategy. Since centers never touch the boundary of the localization set, the query points are never repeated. Besides, the query point will eventually fall within the feasible set, if the ....

A.S. Nemirovskii and D.B. Yudin (1983), "Problem Complexity and Method Efficiency in Optimization", John Wiley, Chichester.

....and Wolfe [6] This class of methods has been reported to perform well on some problems, but poorly on some others; the performance may vary for different formulations of the same problem. Specific implementations of the cutting plane method have been shown to have extremely poor complexity bounds [22]. Many methods have been proposed that have as goal achieving better practical performance as well as better complexity bounds. The analytic center cutting plane method [13, 11, 8] was designed with such a goal in mind. Reasonable theoretical bounds were obtained [2, 12, 19] Extensive numerical ....

A.S. Nemirovsky and D.B. Yudin, "Problem Complexity and Method Efficiency in Optimization ", John Wiley, Chichester (1983).

....of the functions in the class F d . Problems which suffer the curse of dimension in the worst case setting include integration, approximation, global optimization, integral and partial differential equations for classes of functions whose rth derivatives are uniformly bounded in L1 , see [1, 4, 8, 9, 17, 24, 32]. In the average case and randomized settings, the curse of dimension is present for approximation over the class of functions with r continuous derivatives which is equipped with the folded isotropic Wiener measure, see [20, 28] for the average case and [7, 10, 27] for the randomized setting. ....

Nemirovsky, A. S. and Yudin, D. B., Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience, New York, 1983.

....[9] This class of methods has been reported to perform well on some problems, but poorly on some others [14] and the performance may vary for different formulations of the same problem. Specific implementations of this cutting plane method have been shown to have extremely poor complexity bounds [35]. The second approach is based on interior point methodology, and uses the analytic center of a set of localization. See Goffin et al. 19, 13] This novel cutting plane method has been shown to achieve competitive practical performance (see [2, 3, 21, 28] as well as better complexity bounds ....

A.S. Nemirovsky and D.B. Yudin, "Problem Complexity and Method Efficiency in Optimization", John Wiley, Chichester (1983).

....Ax b 0 x c is (K; Q) strongly convex if the Hessian matrix r 2 f(x) 2 A is positive definite. Another example is the function f(x 1 ; x 2 ) 2 x 2 1 4 x 2 2 2 x 1 Gamma 2 x 2 2 x 1 arctan x 1 Gamma log(x 2 1 1) 4 cos x 2 : 2) In case of twice differentiable functions, Nemirovsky and Yudin (1983), p. 255, have offered a simple condition to verify the (K; Q) strong convexity of some function f( Delta) Let 1 be the smallest and 2 be the largest eigenvalue of the Hessian matrix. If there exist positive constants K and L such that 0 K 1 2 L 1 for all x 2 S then the function f(x) ....

Nemirovsky, A. S. and D. B. Yudin (1983). Problem complexity and method efficiency in optimization. Chichester: Wiley.

.... of large M seemingly the best, from the viewpoint of overall computational complexity (i.e. total # of arithmetic operations) procedure for solving the stochastic counterpart within accuracy ffl is the k Delta k 1 version of the Mirror Descent method for largescale convex minimization, see [13]. The method finds ffl solution to (25) in O(N ) iterations; the computational effort at a single iteration is dominated by necessity to compute the value and the gradient of the objective 0 at the current iterate. In order to implement the method efficiently, we should first compute N ....

.... it is the cardinality of some multi dimensional grid; as a result, the constant factor O(M ) in the right hand side of (29) makes the robust versions of the standard SA useless for our purposes. What seems to meet our needs, is the non Euclidean robust SA associated with the L 1 norm ([13]) As we shall see in a while this version of SA yields the efficiency estimate (29) with the constant factor in the right hand side O( Delta) proportional to ln M , which fits our goals incomparably better than the versions of SA discussed above. 4.2 The algorithm The robust SA algorithm, ....

A. Nemirovski and D. Yudin, Problem complexity and method efficiency in optimization -- J. Wiley & Sons, 1983.

No context found.

Nemirovski, A. and Yudin, D. (1983). Problem Complexity and Method Efficiency in Optimization, J. Wiley & Sons.

.... of large M seemingly the best, from the viewpoint of overall computational complexity (i.e. total # of arithmetic operations) procedure for solving the stochastic counterpart within accuracy ffl is the k Delta k 1 version of the Mirror Descent method for largescale convex minimization, see [13]. The method finds ffl solution to (25) in O(N ) iterations; the computational effort at a single iteration is dominated by necessity to compute the value and the gradient of the objective N 0 at the current iterate. In order to implement the method efficiently, we should first compute N ....

....large it is the cardinality of some multi dimensional grid; as a result, the constant factor O(M ) in the right hand side of (29) makes the robust versions of the standard SA useless for our purposes. What seems to meet our needs, is the non Euclidean robust SA associated with the L 1 norm ([13]) As we shall see in a while this version of SA yields the efficiency estimate (29) with the constant factor in the right hand side O( Delta) proportional to p ln M , which fits our goals incomparably better than the versions of SA discussed above. 4.2 The algorithm The robust SA algorithm, ....

A. Nemirovski and D. Yudin, Problem complexity and method efficiency in optimization -- J. Wiley & Sons, 1983.

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Nemirovski, A.S. & Yudin, D.B. (1983) Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience.

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A. Nemirovsky and D. Yudin, Problem Complexity and Method Efficiency in Optimization. Wiley, 1983.

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A. S. Nemirovsky and D. B. Yudin. Problem Complexity and Method Efficiency in Optimization. Wiley-Interscience, New York, 1983.

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A.S. Nemirovski, D.B. Yudin, Problem complexity and method efficiency in optimization, Wiley-Interscience, (1983). 17

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