ZHANG, Y., TAPIA, R. A. and DENNIS, J. E. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal on Optimization 2, 1992, pp. 303-324.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 ,Z 0 ) with (X 0 ,y 0 ,Z 0 ) X,y,Z) #, the iterates converge Q quadratically to (X,y,Z) The proof of Corollary 3.2 is immediate from the standard convergence theory for Newton s method. It is clear that Corollary 3.2 holds also for less restrictive assumptions on #, #, and #. See [20] for relevant results for LP. There is no requirement that (X 0 ,y 0 ,Z 0 ) lie in a horn shaped neighborhood of the central path, or even in the feasible region. Note that the assumptions of Corollary 3.2 do not guarantee positive definite iterates. These are not required to make (2.10) well ....

Y. Zhang, R. A. Tapia, and J. E. Dennis, On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms, SIAM J. Optim., 2 (1992), pp. 304--324.

....from the generalized Kuhn Tucker stationary condition. We present some results about convergence rate analysis for continuous and discrete versions and show that superlinear and quadratic convergence can be attained. In a particular case our algorithm bears a resemblance to the algorithm of [38], which was developed for LP. But there are also significant differences: 1. Our algorithms can start the computation from the infeasible region, although they preserve feasibility. 2. Our algorithms enable us to take different stepsizes in the primal space and in the dual space, which is proved ....

....and minimal components of the vector j k , respectively. We will adopt the convention that, if j k 1, then k = 1, and, if j k 0, then ff k = 1. If we set ff = 1 and substitute b = Ax in (74) then formulas (74) coincide with the primal dual interior point algorithms proposed in [38], if in the latter we ignore the barrier (perturbation) term. In this algorithm the starting point z 0 must be strictly interior. Algorithm (74) does not require feasibility of starting and current points, but according to (75) it preserves feasibility. Another important advantage of algorithm ....

Y. Zhang, R. Tapia, and J. Dennis, On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms, Technical Report TR90-6, Rice University, Houston, Texas (1990). 26

....incorporate the objective function into a barrier function explicitly, motivated in part by the successful implementation of such algorithms for linear programming. Long step linear programming algorithms 3 are not only polynomial in complexity, but can also be designed to exhibit superlinear [31] or quadratic [30] convergence asymptotically. Good computational results have been obtained with interior point cutting plane methods. They have been used to solve stochastic programming problems [7] multicommodity network flow problems [8] and integer programming problems [21] as well as ....

Y. Zhang, R. A. Tapia, and J. E. Dennis. On the superlinear and quadratic convergence of primal--dual interior point linear programming algorithms. SIAM Journal on Optimization, 2(2):304--324, 1992. 36

....methods, and also small update methods since can take only small values in such methods. The measure ffi K was introduced by Kojima et al. 10] and used in many other papers (cf. McShane, Monma and Shanno [13] Mehrotra [15] Mizuno [17, 18] Monteiro and Adler [20] Todd [24] Zhang and Tapia [29]) and also in the books of Wright [26] and Ye [28] The third measure, ffi, was introduced by Jansen et al. 7] and thoroughly used in [22] and Zhao [30] It was also used in the analysis of interior point methods for semidefinite optimization by Jiang [8] de Klerk [4] and by de Klerk et al. ....

Y. Zhang and R.A. Tapia. Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. Journal of Optimization Theory and Applications, 73(2):229-- 242, 1992.

....methods, and also small update methods since can take only small values in such methods. The measure ffi K was introduced by Kojima et al. 10] and used in many other papers (cf. McShane, Monma and Shanno [13] Mehrotra [15] Mizuno [17,18] Monteiro and Adler [20] Todd [24] Zhang and Tapia [29]) and also in the books of Wright [26] and Ye [28] The third measure, ffi, was introduced by Jansen et al. 7] and thoroughly used in [22] and Zhao [30] It was also used in the analysis of interior point methods for semidefinite optimization by Jiang [8] de Klerk [4] and by de Klerk et al. ....

Y. Zhang and R.A. Tapia. Superlinear and quadratic convergence of primal-dual interiorpoint methods for linear programming revisited. Journal of Optimization Theory and Applications, 73(2):229--242, 1992.

....methods, and also small update methods since can take only small values in such methods. The measure ffi K was introduced by Kojima et al. 8] and used in many other papers (cf. McShane, Monma and Shanno [10] Mehrotra [12] Mizuno [14, 15] Monteiro and Adler [16] Todd [20] Zhang and Tapia [25], Zhao [26] and also in the books of Wright [22] and Ye [24] The third measure, ffi, was introduced by Jansen et al. 5] and thoroughly used in [18] Full Newton step methods have the best theoretical performance, with an iteration bound O Gamma p n log n Delta , but practically they ....

Y. Zhang and R.A. Tapia. Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. Journal of Optimization Theory and Applications, 73(2):229-- 242, 1992.

....at the first glance since it may not even hold for the complementarity problems arising from linear programming. However, as we shall show in Proposition 4. 1 below, it is actually not stronger than the nondegeneracy assumption used in the analysis of local convergence of linear programming, e.g. [22]. ffl Assumption 3 is valid if F i is twice differentiable with bounded Hessian on F . This can be proved by using the second order Taylor expansion of F i . ffl It is easy to see that j 0 because of the existence of strictly complementary solution. ffl It follows from Lemma 4.2 below and ....

Y. Zhang and R.A. Tapia, "Superlinear and quadratic convergence for primal-dual interior point algorithms for linear programs revisited ",Journal on Optimization Theory and Applications 73(1992) 229-242.

.... or SDP one usually performs some form of a line search on the Newton step in order to force the variables to remain in the positive cone (LP) or positive semidefinite cone (SDP) The convergence rate of the resulting algorithm is very sensitive to the particular scheme that used in the line search [10,14]. We have found that the use of the full Newton step followed by a shift, if necessary, to be very efficient: quadratic convergence was achieved in every test problem. 5. Numerical Results The algorithm was implemented in Matlab. We observe fast local convergence generically, at a quadratic ....

Y. Zhang, R.A. Tapia, and J.E. Dennis, "On the superlinear and quadratic convergence of primaldual interior point linear programming algorithms," SIAM Journal on Optimization, 2:304--324, 1992.

....0. Provided the initial guess X satisfies the semidefinite constraint, the barrier term prevents subsequent values from leaving the positive semidefinite cone. Primal dual interior point methods are of particular interest, since these have been shown to be very efficient for solving LP (see e.g. [3,4,11,12]) One iteration of the primaldual method can be derived by applying Newton s method to three equations: primal feasibility, dual feasibility, and complementarity centering. The primal and dual feasibility equations are the equality constraints in (2) 4) The complementarity centering equation ....

Y. Zhang, R.A. Tapia, and J.E. Dennis. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal on Optimization, 2:304--324, 1992.

....k ) to the linear system D 2 k H k E k D 2 k rg k rg k 0 x = D 2 k r x k k e g k ; 13) given the approximation H k to the Hessian matrix r 2 xx (x k ; k ) and k 0. k is a perturbation parameter for centralization purposes, see [14] 26] and [30]. 2.2 Set k = k min i=1; n n 1; min n (x k ) i ( x k ) i : x k ) i 0 oo , where k 2 [ 1] and 2 (0; 1) 2.3 Set the new iterates: x k 1 = x k k x k ; k 1 = k k : For the analysis, it is convenient to use the following notations: w k = x k k ; ....

Y. Zhang, R. A. Tapia, and J. E. Dennis. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM J. Optim., 2:304-324, 1992. 13

....parameter, X and Z are diagonal matrices where the diagonal elements are the components of x and z respectively, and e is a vector of ones with n components. The equation XZe = e is a relaxation of the complementarity condition x z = 0 that includes a perturbation term e (see [12] and [32] for more details) This algorithm is also of interior point type, meaning that x and z are required always to be strictly feasible with respect to the bound constraints, i.e. x and z have to satisfy x 0 and z 0. We will assume that x 0 and z 0 throughout this section. The linearization ....

Y. Zhang, R. A. Tapia, and J. E. Dennis, On the superlinear and quadratic convergence of primal--dual interior point linear programming algorithms, SIAM J. Optim., 2 (1992), pp. 304--324.

....review of the primal dual methods together with other path following methods had been given in [11] 12] 13] The numerical experiments carried out by Lustig et al..l [14] 17] showed their good practical performance even if the perturbation coefficients are taken very small. It was proved in [18] [20] that these algorithms possesses the superlinear and Q quadratic rate of convergence. In this report we will consider the new version of the primal dual method in which the pure without perturbation system of optimality conditions is solved by Newton s method. We suppose that starting points ....

Y.Zhang, R.Tapia, J.Dennis. On the Superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. Technical Report TR90-6. Rice university, Houston, Texas. 1990. 16

.... [38] and developed in [30] These methods have the same computational effort per iteration as the other classes of interior point methods but they have better theoretical properties for worst case complexity analysis as for instance in [41] and also for asymptotic convergence rate analysis (see [58] among others) These results were partially motivated by the superior performance in practice of the primal dual method over the primal and the dual approaches as in [34, 37] The majority of the primal dual interior point methods found in the literature can be seen as variants of Newton s method ....

....k = 1. This class of primal dual interior point methods with an initial feasible point and for wise choices of oe and enjoy very good theoretical properties including, polynomial complexity (see [27] for references to the more important results) and as showed 8 more recently, fast convergence [58]. In practice it is common to start with an infeasible point since in general it is very expensive to find a feasible one. A feasible point is a point whose nonnegative variables are strictly positive and the residuals r p and r d are zero. The theory for these methods with an infeasible starting ....

[Article contains additional citation context not shown here]

ZHANG, Y. and TAPIA, R. A. Superlinear and Quadratic Convergence of Primal-Dual Interior Point Methods for Linear Programming Revisited. Journal of Optimization Theory and Applications, Vol. 73 pp.229-242, 1992.

....(feasible and infeasible) interior point algorithms for linear programs (LP) convex quadratic programs (QP) monotone linear complementarity problems (LCP) and monotone nonlinear complementarity problems (NCP) that are superlinearly or quadratically convergent. For LP and QP, these works include [1, 4, 5, 19, 23, 25, 27, 28, 29]. For LCP, we mention the papers [10, 11, 12, 13, 21, 24] and for NCP, we cite [3, 14, 15, 22] In this paper we are interested in the superlinear convergence analysis of infeasible interior point algorithms for solving the linearly constrained convex program minimize x f(x) subject to Ax = b; x ....

Y. Zhang, R. A. Tapia, and J. E. Dennis, On the superlinear and quadratic convergence of primal--dual interior point linear programming algorithms, SIAM Journal on Optimization, 2 (1992), pp. 304--324.

....(feasible and infeasible) interior point algorithms for linear programs (LP) convex quadratic programs (QP) monotone linear complementarity problems (LCP) and monotone nonlinear complementarity problems (NCP) that are superlinearly or quadratically convergent. For LP and QP, these works include [1, 4, 5, 19, 23, 25, 27, 28, 29]. For LCP, we mention the papers [10, 11, 12, 13, 21, 24] and for NCP, we cite [3, 14, 15, 22] In this paper we are interested in the superlinear convergence analysis of infeasible interior point algorithms for solving the linearly constrained convex program minimize x f(x) subject to Ax = b; x ....

Y. Zhang and R. A. Tapia, Superlinear and quadratic convergence of primal--dual interior-- point methods for linear programming revisited, Journal of Optimization Theory and Applications, 73 (1992), pp. 229--242.

....with jj(X 0 ; y 0 ; Z 0 ) Gamma (X; y; Z)jj ffl, the iterates converge Q quadratically to (X; y; Z) The proof of Corollary 1 is immediate from the standard convergence theory for Newton s method. It is clear that Corollary 1 holds also for less restrictive assumptions on oe; ff and fi. See [ZTD92] for relevant results for LP. There is no requirement that (X 0 ; y 0 ; Z 0 ) lie in a horn shaped neighborhood of the central path, or even in the feasible region. Note that the assumptions of Corollary 1 do not guarantee positive definite iterates. These are not required to make (20) ....

Y. Zhang, R.A. Tapia, and J.E. Dennis. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal on Optimization, 2:304--324, 1992.

....is drastically different. Under nondegeneracy assumptions, convergence is global, with a local Q quadratic rate. The importance of a fast rate of convergence, even for problems that can be solved in finitely many iterations (such as LP and CQP problems) has been stressed by several authors (e.g. [16,17] and references therein) and recently a primal dual logarithmic barrier method has been shown to be both polynomial time and Q quadratically convergent [18] The algorithm proposed here however has the advantage of greater simplicity (no centering step, i.e. no barrier parameter to be iteratively ....

Y. Zhang, R.A. Tapia & J.E. Dennis, "On the Superlinear and Quadratic Convergence of Primal-Dual Interior Point Linear Programming Algorithms," SIAM J. on Optimization 2 (1992), 304--324.

....assumptions the method converges globally and superlinearly to the solution set of (3) even in some situations in which the solution does not satisfy a strong uniqueness and nondegeneracy condition. Superlinear convergence for interior point methods was discussed first by Zhang, Tapia, and Dennis [22]; see also Ye, Guler, Tapia, and Zhang [21] A recent paper by Sun and Zhao [11] presents a feasible interior point method for monotone variational inequalities where the set C is polyhedral, that achieves global and local quadratic convergence. Infeasible interior point methods for the latter ....

Y. Zhang, R. A. Tapia, and J. E. Dennis, On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms, SIAM Journal on Optimization, 2 (1992), pp. 304--324.

....(7) and w ) It is easily seen that (20) implies that condition (8) will be satisfied for k large enough which contradicts the assumption that I( is infinite. Let us call ff ; and 2 (0; 1) 14 It is known (see the proof of Theorem 3. 1 in Zhang, Tapia, and Dennis [23]) that Assumptions (a1) and (a2) imply that ff is bounded away from zero. If ff = ff in infinitely many iterations, then by using(ii) of Proposition 4.2 we deduce immediately that (20 holds. Now, suppose that ff ff for k sufficiently large. It is known (e.g. see Theorem 6.3.3 in Dennis ....

ZHANG, Y., TAPIA, R. A. and DENNIS, J. E. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal on Optimization 2, 1992, pp. 303-324.

No context found.

ZHANG, Y. and TAPIA, R. A. Superlinear and quadratic convergence of primal-dual interior point methods for linear programming revisited. Journal of Optimization Theory and Applications 73 (1992), pp. 229-242.

....term rr is an algorithmic parameter, but the step length c is not. However, the step length c is dependent upon the algorithmic parameter r . Hence, the theory for Q superlinear convergence of the duality gap sequence (xk)rz is stated in terms of rr and r. Theorem 5. 2 (Zhang, Tapia, Dennis [60]) Let (x,zk) be generated by Algorithm 1 and (x,S,z ) x , S , z ) Assume (i) strict complementarity, ii) the sequence (x )r z k (n min( X Z e) is bounded, iii) r 1 and rr 0. Then the duality gap sequence (xk)rz converges to zero Q superlinearly. That is, the Q factor ....

Y. Zhang, R. Tapia, and J. Dennis. On the Superlinear and Quadratic Con- vergence of Primal-Dual Interior-Point Linear Programming Algorithms. SIAM Journal of Optimization, 2:304-324, 1992.

.... (x , S , z ) Assume (i) strict complementarity, ii) the sequence (x )r z k (n min( X Z e) is bounded, iii) r 1 and rr 0. Then the duality gap sequence (xk)rz converges to zero Q superlinearly. That is, the Q factor xk lTzk l Q1 = lim sup 0. k. c xk Tz k Zhang and Tapia [59] improved Theorem 5.2 by replacing the convergence of the iteration sequence (x , y, z) assumption with the assumption that the duality gap sequence (x k)rz converges to zero. 5.2, Zhang, Tapia, Potra [61] Under the assumptions of Theorem lim oz k = 1. 69 Mehrotra [30] used observed ....

Y. Zhang and R. Tapia. Superlinear and Quadratic Convergence of Primal- Dual Interior-Point for Linear Programming Revisited. Journal of Optimization Theory and Applications, 73:229-242, 1992.

....Moreover, for a specific choice of the norm, it can be implemented as a sequential linear programming method, and hence be solved by a simplex type method or by the more recent primal dual feasible point methods for linear programming. See Kojima, Mizuno, and Yoshise [10] Zhang and Tapia [19], Zhang, Tapia, and Dennis [20] and Lusting, Marsten, and Shanno [12] Also, recently there has been considerable activity in the area of feasible point methods for general nonlinear programming problems. We cite for example Yamashita [18] Wright [16] Wright [17] Lasdon, Yu, and Plumruer ....

Y. ghang, R.A. Tapia, Superlinear and quadratic convergence of primal-dual interior-point algorithms for linear programming revisited, J. Optim. Theo. Appl., vo. 73, 1992, pp.229-242.

No context found.

Y. Zhang, R. A. Tapia, and J. E. Dennis. On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM Journal of Optimization, pages 304--324, 1992.

....led researchers to extending the primal dual interior point algorithmic framework to more general problems. One of the key ingredients that contribute to the success of primal dual methods is their superlinear convergence rate that was first studied for these methods by Zhang, Tapia, and Dennis [10]. To prove Q superlinear convergence, strict complementarity is assumed in most theoretical results that followed the work of Zhang, Tapia and Dennis. For linear programming problems, the existence of a strict complementary solution was proved by Goldman and Tucker [3] However, for monotone ....

Y. Zhang, R. A. Tapia, and J. E. Dennis. On the superlinear and quadratic convergence of primal-dual interior-point linear programming algorithms. SIAM J. on Optimization, 2:304-- 324, 1992.

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