Y. YE. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....analytic center, cutting plane method, semide nite feasibility problems, deep cuts 1 Introduction The analytic center cutting plane method (ACCPM) was rst proposed by G. Sonnevend [7] in 1984. Since then, many di erent variants had been proposed and their complexities had been analyzed, see e.g. [4, 2, 3, 11]. All the research work done before 2000 only considered feasibility problems in the Euclidean space IR . Recently, the ACCPM was extended to the semide nite feasibility problem (SFP in short) by Sun, Toh and Zhao [8, 9] and independently by Oskoorouchi and Gon [6] Such extensions are natural ....

Yinyu Ye. Interior Point Algorithm: Theory and Analysis. John Wiley & Sons, 1997. 20

....) 0 (5) iii) 0: The idea of the method is then to solve a sequence of problems of type (4) while decreasing the parameter towards zero. Notice that by using problem (4) we have restricted the solution to have positive components only. This follows an interior point approach (cf. 3] 6] [24]) in which the iterates are feasible and positive. We shall now introduce a further simplification by substituting the nonlinear barrier problems (4) by quadratically constrained quadratic problems, or trustregion subproblems, where the objective function will be a quadratic approximation to the ....

Y. Ye. Interior point algorithms: theory and analysis, Wiley-Interscience, New York, 1997.

....we omit the proofs. 7 Lemma 2.3 Let M be a P 0 matrix. Then the set is bounded if any one of the following conditions holds: i) The solution set of the LCP (1.1) is nonempty and bounded; ii) M is a R 0 matrix. Proof. See Theorem 3.1 [20] Remark 1. For monotone LCPs, homogeneous algorithms [1, 17] also do not require the strict feasibility condition to ensure that the iterates are bounded. However, it is not known whether homogeneous algorithms have results similar to those in Lemmas 2.2 and 2.3. 3 Search Direction and Step Length We choose a parameter # (0, 1) Let # 0 = 1. Then (# 0 , ....

Y. Ye, Interior Point Algorithm: Theory and Analysis, John Wiley & Sons, Inc., 1997.

....feasibility and optimality conditions, then an exact solution has been recovered. Otherwise, the interior point algorithm resumes. Research in finite termination can be categorized into two areas, optimal face identification (Taraos [46] Mehrotra [30] Mehrotra and Ye [35] and Ye [52] 54] [55]) and optimal basis identification (Andersen [1] Andersen and Ye [2] Bixby and Saltzman [5] Marsten, Saltzman, Shanno, Pierce and Ballintijn [24] Tapia and Zhang [45] Vavasis and Ye [47] Ye and Todd [50] and Ye [51] Optimal face identification techniques identify the face upon which the ....

....= 2,3 and A = l , Objective Figure 1. 2 Step 3 Projection of x[ onto the optimal primal face Contributions from the Literature Much work has been done in the area of finite termination for feasible interior point methods for linear programming, see Mehretra [30] 31] and Ye [52] 54] [55]. Finite termination procedures in infeasible interior point methods for linear programming have been studied by Petra [38] and Anstreicher, Ji, Petra, and Ye in [3] where a probabilistic analysis was given. Monteiro and Wright [36] as well as Ji and Potra [18] investigated finite termination ....

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Y. Ye. Interior-Point Algorithm: Theory and Analysis, John Wiley & Sons, New York, New York, 1997.

....does not appear in (1) it does not have to be given to begin the algorithm and it does not have to be computed at each iteration. For notational convenience, we denote M to be matrix on the left hand side of (1) A more detailed explanation and derivation of the algorithm can be found in [7] and [29]. Computing the matrix M in (1) and solving the equations are the two most computationally expensive parts of the algorithm. For arbitrary matrices A i and C, 1) can be computed by using O(n m n ) operations and solved by using O(m ) operations, although sparsity in the data may ....

....of X(S, at each iteration. Primal dual methods use a symmetric linearization of the equation XS = I and form a Schur complement system with a similar form and complexity as (1) With a feasible dual starting point and appropriate choices for and step length, convergence results in [5, 7, 29] show that either the new dual point (y, S) or the new primal point X is feasible and reduces the Tanabe Todd Ye primal dual potential function #(X,S) # ln(X . S) ln det X ln det S enough to achieve linear convergence. 3 PDSDP The DSDP4 [5, 6] software package served as the basis for ....

Y. Ye. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997. 16

....[5] A lot of the interest in SDP is from the interior point community who have completed so much successful work on linear programming. At the moment, interior point methods are the most successful algorithms for general SDP problems, see e.g. the above survey articles as well as the books [30] [46] and the recent theses [4] 1] 22] 35] 17] 32] 33] The above references provide some evidence of the current high level of research activity in these areas. The main contribution of this paper is a new approach to solving EDMCP. This approach is different than those in the literature ....

Y. YE. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience series in Discrete Mathematics and Optimization. John Wiley & Sons New York 1997.

....of the optimal symmetric edge and node weights, found by solving the SDP (17) Note that many weights are negative. 11 6 Computational methods 6. 1 Interior point method Standard interior point algorithms for solving SDPs work well for problems with up to a thousand or so edges (see, e.g. [29, 1, 40, 42, 41, 39, 11]) The particular structure of the SDPs encountered in FDLA problems can be exploited for some gain in efficiency, but problems with more than a few thousand edges are probably beyond the capabilities of current interior point SDP solvers. We consider a simple primal barrier method, with the ....

Y. Ye. Interior Point Algorithms: Theory and Analysis. Wiey-Interscience Series in Discrete Mathematics and Optimization. John Wiley a; Sons, New York, 1997. 2O

....variable is p c R. Consider the augmented variable z (p, v) c R L . Let 7 z [ Cz b z [ cz hi, i 1, m be a bounded polyhedron. Then its analytic center is defined as the unique maximizer of m Xlo 45, which can be computed very efficiently using Newton s method; see, e.g. [22, 8]. We start with a bounded polyhedron 7 ( z I C( z b( that contains an optimal solution z (p , V ) which may not be unique) The initial polyhedron can, for example, be a box in R L where and 7 are known lower and upper bounds for [he op[imal value V , and is a (componen[ wise) ....

....v(p ) T(p p( 3. Form the polyhedron 7 ( 7 ( 7 ) and update the upper bound V : min V, V(p ) 4. If the duality gap v , quit; else, let k : k 1. One iteration of the ACCPM is illustrated in figure 6. For computational details and convergence analysis of ACCPM, see [21, 22, 8] and references therein. We applied ACCPM to the example in section 5. Figure 5 also shows the dual objective function and the lower bound obtained by ACCPM versus number of iterations. Unlike the subgradient methods in section 6.3.1 where only the current subgradient is used at each step, in ....

Y. Ye. Interior Point Algorithms: Theory and Analysis. Discrete Mathematics and Optimization. Wiley, New York, 1997.

....vectors IF, L5) C ]4 which satisfy this last constraint can be transformed to the intersection of a rotated second order cone in ]4 2 and a hyperplane (e.g. 27] Hence, Problem 5. 1 is a convex symmetric cone programme [28, 29] which can be efficiently solved using interior point methods [30]. Furthermore, infeasibility can be reliably detected. If 5 represents the autocorrelation sequence of the filter, then an optimal filter can be obtained from the solution of Problem 5.1 by spectral factorization [9, 11] We now demonstrate the flexibility of this design method by solving a ....

Y. Ye, Interior Point Algorithms: Theory and Analysis, J. Wiley & Sons, New York, 1997.

....vectors IF, L) C ]a4 which satisfy this last constraint can be transformed to the intersection of a rotated second order cone in 1a4 2 and a hyperplane (e.g. 23] Hence, Problem 5. 1 is a convex symmetric cone programme [24, 25] which can be efficiently solved using interior point methods [26]. Furthermore, a certificate of infeasibility can be issued if there are no filters of the given length which satisfy the mask. If represents the autocorrelation sequence of the filter, then an optimal filter can be obtained from the solution of Problem 5.1 by spectral factorization [9, 11] We ....

Y. Ye, Interior Point Algorithms: Theory and Analysis, J. Wiley : Sons, New York, 1997.

....limited memory BFGS updates [39] can be used, but we will not consider this issue in this paper. Our motivation is based on practical considerations. During the last 15 years much progress has been realized on IP methods for solving linear or convex minimization problems (see the monographs [29, 10, 38, 44, 23, 42, 47, 49]) For nonlinear convex problems, these algorithms assume that the second derivatives of the functions used to define the problem are available (see [43, 35, 36, 12, 38, 26] In practice, how # Received by the editors September 15, 1998; accepted for publication (in revised form) January 26, ....

Y. Ye, Interior Point Algorithms---Theory and Analysis, Wiley-Intersci. Ser. Discrete Math. Optim., John Wiley, New York, 1997.

....our knowledge, such decomposition procedures have not been proposed before. In trust region methods for nonlinear programming, one often needs to solve problems of type (P) in Section 3, where D is a unit ball. The problem is known to be solvable in polynomial time; for detailed discussions, see [15]. Our result extends the polynomial solvability property to a nonconvex quadratic constraint (inequality or equality) and a non convex quadratic objective. Another case that we can handle is a non convex objective with a concave quadratic inequality constraint and an additional linear restriction. ....

Y. Ye. Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, New York, NY, USA, 1997. 28

....algorithm based on one of the formulations and present preliminary numerical results. 1 Introduction Since Karmarkar s 1984 ground breaking work [8] the area of interior point methods has matured considerably, as evidenced by a string of recently appeared books in this area (see, for example [19, 20, 24, 29, 30, 31]) which This research was supported in part by DOE Grant DE FG03 97ER25331. 1 contain comprehensive lists of references. Even so, there are still many research topics that need to be further studied, especially in extending and applying interior point methodology to solving practically ....

Y. Ye. Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, New York, 1997. 16

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Y. Ye, Interior-Point Algorithm: Theory and Analysis, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, 1997.

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Y. Ye, Interior Point Algorithms : Theory and Analysis (Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1997). 29

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Y. Ye. Interior Point Algorithms : Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997. 13

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Y. Ye. Interior Point Algorithms : Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997. 23

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Y. Ye. Interior Point Algorithms : Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997. 14

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Y. YE. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.

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Y. Ye, Interior Point Algorithms: Theory and Analysis. Wiley, 1997. 11

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Y. Ye, Interior Point Algorithms: Theory and Analysis, ser. Wiley Interscience series in discrete Mathematics and Optimization. John Wiley & Sons, 1997.

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Y. Ye, Interior Point Algorithms: Theory and Analysis, ser. Wiley Interscience series in discrete Mathematics and Optimization. John Wiley & Sons, 1997.

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Y. Ye, Interior Point Algorithms: Theory and Analysis. Wiley, 1997.

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Y. YE. Interior Point Algorithms: Theory and Analysis. Wiley-Interscience series in Discrete Mathematics and Optimization. John Wiley & Sons, New York, 1997.

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Y. Ye, Interior Point Algorithm Theory and Analysis, (John Wiley and Sons, New York, 1998).

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