M. Kojima, N. Megiddo, and S. Mizuno. Theoretical convergence of large-step-primal-dual interior point algorithms for linear programming. Mathematical Programming, 59:1-22, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....feasible. The remedy is to take a damped Newton step z 0 = z z, with damping factor ; this factor can be chosen such that z 0 is feasible and at the same time the proximity decreases suciently to get a provable polynomial method. In practice this approach yields very ecient methods ([2, 14, 15, 16, 18, 22, 23, 32, 34]) 6.3 Predictor corrector method This is the most popular method. We describe a simple variant. It is based on a very greedy strategy that uses the Newton step targeting at the zero vector. So, in the de nition of the Newton step one takes = 0. The resulting direction is called the ane scaling ....

M. Kojima, N. Megiddo, and S. Mizuno. Theoretical convergence of large-step-primal-dual interior point algorithms for linear programming. Mathematical Programming, 59:1-22, 1993.

....p 1 2) 2 kp v k 2 (1 fl ) 1 2)kp v k 2 : This completes the proof of the lemma. 8 The above lemmas give us some tools for analyzing primal dual algorithms applying to NCP. In the linear case, Lemma 3. 1 is important to provide the polynomiality of many primal dual algorithms ( [25, 24, 34, 35, 26, 32, 33, 20, 19, 18], etc. On the other hand, Lemma 3.4 suggests us that these analyses may be extended to nonlinear cases. Before proceeding we mention that for the monotone NCP the bounds in Lemma 3.4 can be improved by using Deltax T Deltas( 1 2 ( Deltax) T (f(x Deltax) Gamma f(x) 0: 4 ....

M. Kojima, N. Megiddo, and S. Mizuno. Theoretical convergence of large--step--primal--dual interior point algorithms for linear programming. Mathematical Programming, 59:1--21, 1993.

....restriction limits large improvement in practical computation. To overcome this deficiency, some long step algorithms are proposed for linear problems (linear programs and linear complementarity problems) which allow a line search in a large neighborhood of the current solution, see for instance [4, 5, 8, 9, 23, 24]. Long step algorithms are especially important in nonlinear problems because it costs more to determine an improving direction for these problems. Another attractive feature of interior point methods is the possibility to achieve quadratic rate of local convergence. However, in order to obtain ....

M. Kojima, N. Megiddo and S. Mizuno,"Theoretical convergence of large-step primaldual interior point algorithms for linear programming", Mathematical Programming, 59 (1992) 1-21.

....Saigal (1971, 1976) Todd (1976b) Complementarity problems can also be considered from an interior point algorithm viewpoint, see Section 4. 9, hence by following a smooth path, see, e.g. Kojima, Mizuno and Noma (1990b) Kojima, Mizuno and Yoshise (1991d) Kojima, Megiddo and Noma (1991b) Kojima, Megiddo and Mizuno (1990a) Mizuno (1992) We present the Lemke algorithm as an example of a piecewise linear algorithm since it played a crucial role in the development of subsequent piecewise linear algorithms. Let us consider the following linear complementarity problem: Given an affine map g : R n R n , find ....

M. Kojima, N. Megiddo and S. Mizuno (1991a), Theoretical convergence of largestep primal-dual interior point algorithms for linear programming, preprint, Tokyo Inst. of Techn.

....apply to the dual affine scaling methods. The situation is much clearer for short step path following methods [18, 51, 52] etc. short step primal dual path following methods [32, 49] some long step path following algorithms [19, 20, 26] long step primal dual path following methods [30], and primal dual potential reduction algorithms [34, 63, 74] As shown in [24] all these algorithms either explicitly generate a primal dual solution sequence (x k ; s k ) satisfying condition (7) or can generate such a sequence. It turns out that we can obtain more information about the ....

Kojima, M., Megiddo, N., and Mizuno, S. (1990). Theoretical convergence of large--step primal--dual interior point algorithms for linear programs. RJ 7872, IBM Almaden Research Center, San Jose, California. To appear in Mathematical Programming.

....point algorithms use large neighborhoods which allow iterates to move more freely so as to possibly take large steps, so they are called large step algorithmsy in the literature) Some works on large step interior point methods are listed in the references. Among them are [2] 3] 4] 5] [8], 10] 12] 13] 14] 16] 20] and [23] It is well known that the upper bound of the complexity for largestep algorithms is, in general, greater than that for short step ones, cf. 1] 15] 18] 19] and [21] for short step algorithms) Using high order approximations, the iteration ....

.... ) ffi also allows some components of u and u Gamma1 to be as large as O( p n) So the neighborhood implicitly imposed in Algorithm 3. 1 (for = 1=2) may be larger than the generally used neighborhood with fi = O(1) N Gamma 1 (fi) f(x; s) j nxs hx; si fi Gamma1 eg; cf. 4] [8] and [12] We will show that the iteration complexity of Algorithm 3.1 is bounded by O(n ln( 0 =ffl) for = 1=2) which is the same complexity bound as achieved for the algorithms using the neighborhood N Gamma 1 (fi) The second aim of the paper is to extend Algorithm 3.1 to a high order ....

M. Kojima, N. Megiddo and S. Mizuno, "Theoretical convergence of large-step primaldual interior point algorithms for linear programming," Mathematical Programming 59 (1993) 1--21.

....with a factor ff 0 = 0:99995 to prevent hitting the boundary. The use of such a stepsize rule saves about 40 of the iterations number compared with the case when the theory of the long step methods is strictly followed. The use of different stepsizes in different spaces is due to Kojima et al. [45]. However, to preserve the polynomial complexity, some additional safeguards were needed. In another paper Kojima, Megiddo and Mizuno [44] prove the global convergence of the infeasible primal dual method. In their stepsize selection, only nonnegativity of variables is concerned like in (26) but ....

Kojima, M., Megiddo, N. and Mizuno, S. (1993) Theoretical Convergence of Large--Step--Primal--Dual Interior Point Algorithms for Linear Programming, Mathematical Programming, 59, 1--21.

....IP methodology to (1.2) include global polynomial time convergence and local superlinear convergence; both are very desirable features in numerical optimization. Of particular interest to us among possible IP methods for MVIP is the long step pathfollowing algorithms developed by Kojima et al. [7, 8] and Mizuno et al. 10] which were originally designed for linear programming problems. The long step methods usually have an O[n log( 0 =ffl) iteration complexity, namely that starting from an initial feasible solution of duality gap 0 , they find an ffl optimal solution in at most O[n log( ....

M. Kojima, N. Megiddo and S. Mizuno,"Theoretical convergence of large-step primaldual interior point algorithms for linear programming", Mathematical Programming, 59 (1992) 1-21.

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