LUSTIG, I. J., MARSTEN, R., and SHANNO, D. F., Interior-Point Methods for Linear Programming: Computational State of the Art, ORSA Journal of Computing, Vol. 6, pp. 1--14, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equality constraints (see [4] but for simplicity of exposition we will only consider here inequality constrained problems of the form subject to g(x) 0; 1.1) where f : R R and g : R R are smooth functions. Following the strategy of interior point methods (see for example [13, 28, 19]) we associate with (1.1) the following barrier problem in the variables x and s x;s subject to g(x) s = 0; 1.2) where 0 and where the vector of slack variables s = s ; s is implicitly assumed to be positive. The main goal of this paper is to propose and analyze an ....

I.J. Lustig, R.E. Marsten, and D.F. Shanno (1994). "Interior point methods for linear programming: Computational state of the art". ORSA Journal on Computing, 6, pp. 1--14.

....by these claims [46] Some delved into the theory and implementation of this algorithm to investigate the validity of these claims. It is now understood that interior point methods are viable algorithms and appear computationally superior to simplex based approaches when the problems are large [49, 50, 51, 52]. In 1986, Sonnevend [80] introduced the mathematical programming community to the concept of the analytic center. In [81, 82, 83, 84, 85] Sonnevend connects this concept to Karmarkar s algorithm, develops his own algorithm, demonstrates applications, and together with Stoer, shows several ....

I. Lustig, R. Marsten, and D. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6:1-14, 1994.

....which is characterized by the barrier parameter IZ 0. Ideal primal dual subproblem solutions at a particular point I along the barrier trajectory also satisfy the perturbed optimality conditions (12) 14) Primaltual methods have enjoyed great success in linear programming problems [29] [37] and have only recently been proposed for nonlinear optimization [6] 14] These methods maintain strict primal feasibility and dual feasibility to within a specified tolerance while reducing the duality gap (i.e. the complementary slackness) as b is reduced and the optimal primal dual solution ....

I. Lustig, R. E. Marsten, and D. F. Shanno, "Interior point methods for linear programming: Computational state of the art," ORSA J. Comp., vol. 6, pp. 1 14, 1994.

....gray scale image. Whether a particular value is exactly zero or just very close to zero is immaterial. Slight inaccuracies below the gray scale threshold are inconsequential; obtaining an image rapidly is a neccessity. Primal dual methods have enjoyed considerable success in linear programming [18, 32, 40], and have recently been proposed for nonlinear programming [5, 13, 41] LARGE SCALE PRIMAL DUAL IMAGE RECONSTRUCTION 7 Although they are closely related to the logarithmic barrier method, primal dual methods may pose some advantages. In the logarithmic barrier method, the Lagrange multiplier ....

I. Lustig., R.E. Marsten and D.F. Shanno, Interior point methods for linear programming: computational state of the art, ORSA J. Comp, 6 (1994), pp. 1-14.

....can thus be determined for a list of candidate starting points and used to choose the most suitable one. We also discuss how our theoretical developments made for feasible path following method can be applied in the infeasible method known to be the most e#cient interior point method in practice [13, 2]. One of the di#culties in the implementation of interior point methods is the choice of the starting point, cf. 2] and the references therein. Most implementations of interior point methods use some variation of Mehrotra s starting point [15] Unlike the simplex method that can take advantage of ....

I. J. Lustig, R. E. Marsten, and D. F. Shanno, Interior point methods for linear programming: computational state of the art, ORSA Journal on Computing, 6 (1994), pp. 1--14.

....for linear programming include [132, 140, 146, 148] The dual affine scaling method is similar to Dikin s original method and is discussed in section 4.3.1. In this section, we consider a slightly more complicated interior point method, namely the primal dual predictor corrector method pdpcm [92, 90]. This is perhaps the most popular and widely implemented interior point method. The basic idea with an interior point method is to enable the method to take long steps, by choosing directions that do not immediately run into the boundary. With the pdpcm this is achieved by considering a ....

....the optimality conditions (8 10) then the duality gap will be n. We assume we have a strictly positive feasible solution x and a dual feasible solution (y; z) satisfying A T y z = c, z 0. If these assumptions are not satisfied, the algorithm can be modified appropriately; see, for example, [90] or Zhang [150] An iteration of pdpcm consists of three parts: ffl A Newton step to move towards the solution of the linear program is calculated (but not taken) This is known as the predictor step. 19 ffl This predictor step is used to update . ffl A corrector step is taken, which combines ....

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I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6(1):1--14, 1994. See also the following commentaries and rejoinder.

....to the solution of linear programs: simplex method [10] and interior point method [27] There exist e#cient implementations of both methods for general problems. An advantage of interior point methods (IPMs) is that they require a number of iterations that is almost independent of the problem size [2, 22]. Hence for very large problems they are usually faster. The implementations of both direct approaches might exploit the structure in many di#erent ways. Some general purpose solvers use heuristics to detect certain structures and then take advantage of them. For example, simplex solvers look for ....

....several network optimization problems that have been used to illustrate the e#ciency of our new interior point code. In Section 7 we discuss numerical results and finally, in Section 8, we give our conclusions. 2 Linear Algebra in Interior Point Methods The theory [27] and the implementation [2, 22] of interior point methods for general linear programs are very well understood. It seems that for very large LPs with unknown structure interior point methods are often a better choice than the simplex method. The e#ciency of IPM strongly depends on the linear algebra. The dominant computational ....

I. J. Lustig, R. Marsten, and D. Shanno, Interior point methods for linear programming: computational state of the art, ORSA Journal on Computing, 6 (1994), pp. 1--14.

....be improved so that the chances of numerical troubles are substantially reduced [7, 18] In IPMs, obtaining sparse factorization of ## # is also considered under preprocessing . The importance of this step cannot be underestimated. This is one of the most heavily researched area in IPMs (see [50] or [5] for an overview) The success of this step fundamentally determines the performance of the IPM code. 5.2.2.2 Starting procedures. They are used to nd a good initial solution with little computational e ort. How is good de ned It can mean several di erent things. It is usually said ....

I.J. Lustig, R.E. Marsten, and D.F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6(1):1-14, 1994. 32 ###### ###### Optimization Software

....Over the last ten years there has been a rapid expansion in the size of linear programs that have been successfully solved using digital computers. A good overview of the recent rapid progress in this field and the current state of the art is a#orded by the article of Lustig, Marsten, and Shanno [23] and the accompanying discussions by Bixby [1] Saunders [36] Todd [38] and Vanderbei [39] Much of the rapid expansion in the size of linear programs solved is due to the interior point revolution initiated by Karmarkar s proof that a pseudo polynomial time algorithm could be based on an ....

.... size of linear programs solved is due to the interior point revolution initiated by Karmarkar s proof that a pseudo polynomial time algorithm could be based on an interior point method [20] Since then a very wide array of interior point algorithms have been proposed and considerable practical [21, 23, 27] and theoretical [30] understanding is now available. In this section we describe our algorithm and our experience with it. 6.1. Duality theory. We consider the linear program in the standard form min c T x subject to Ax = b, x # 0. 6.1) This is often called the primal linear program. The ....

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I. J. Lustig, R. E. Marsten and D. F. Shanno, Interior point methods for linear programming: Computational state of the art, ORSA J. Comput., 6 (1994), pp. 1--14.

....speed improvement due to hardware technologies. 3 . Shown to be several times slower than interior point methods, then of the art commercial mathematical programming software MINOS 4.0 was put out of date. Better version was spawn to match up with the performance of the interior point methods (Lustig, Marsten, Shanno, 1994). Robert E. Bixby, a professor at the Department of Mathematical Sciences of Rice University, the director of the CPLEX Optimization, Inc, and the author of simplex method routines for the now very popular CPLEX, admitted that the state of art implementation of the interior point methods were at ....

....is asymptotically quadratically convergent. It can also be shown that with full Newton step, the algorithm also requires at most O( p nL) iterations, and it is still the best up to date bound achieved. Predictor corrector algorithm is probably the most popular interior point methods in practice (Lustig et al. 1994). Even though, theoretically, multiple corrector steps per iteration are possible, it has been shown, that a single corrector step is probably the best choice. Practice showed that for some problems, with more than 4 steps, the algorithm begins to diverge. 25 Input: A proximity parameter , 0 ....

Lustig, I. J., Marsten, R., & Shanno, D. F. (1994). Interior point methods for linear programming: computational state of the art. ORSA Journal on Computing, 6, 1 -- 14.

....algorithms enjoyed great publicity for two reasons. First, these algorithms solve LP problems in polynomial time, as proved by Karmarkar and many others. Secondly, interior point algorithms have demonstrated excellent practical performance when solving large scale LP problems, see Lustig et al. [37]. It was soon realized (see Gill et al. 25] that Karmarkar s method was closely related to the logarithmic barrier algorithm for general nonlinear programming studied by Fiacco and McCormick [23] and others in the sixties. Hence, it is natural to investigate the efficiency of the interior point ....

I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA J. on Comput., 6(1):1--15, 1994. 38

....have become a Department of Management, Odense University, DK 5230 Odense M, Denmark. E mail: eda busieco.ou.dk. URL: http: www.busieco.ou.dk eda 1 strong competitor. We will not review the development of interior point methods here, but instead we refer the reader to the survey papers [4, 15]. The main difference between interior point and simplex type methods is that the interior point methods move through the interior of the feasible region to the optimal solution, whereas the simplex algorithm generates a sequence of adjacent extreme point solutions. Consequently, the simplex ....

I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA J. on Comput., 6(1):1--15, 1994.

....exploit parallel computers efficiently, because parallel computers now offer a lot of computing power cheaply. The two most popular methods for solution of general LPs are the classical simplex method and the more recent interior point methods. However, recent computational results presented in [2, 8, 24] indicate that the interior point methods for many medium to large scale LPs are significantly more efficient than the simplex method. Therefore, in this paper we restrict the attention to parallelization of an interior point method. Several researchers have already considered the parallelization ....

I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA J. on Comput., 6(1):1--15, 1994.

....problem reduces to SDP and the options for interior point methods become more numerous. In [1] it is argued that essentially any interior point method designed for LP can be extended to solve SDP. In LP it is now generally agreed that primal dual interior point methods are especially efficient [7]. A specific primal dual method for SDP with a proof of global convergence was given by [8] A related method was given by [9] A different approach to primal dual interior point methods for SDP is given in [10] 4. Quadratically Convergent Local Methods In [2] the authors derived a ....

....that of [2] The new method can be viewed as a local primal dual interior point method. This is because primal dual interior point methods for LP and SDP can be viewed as applying Newton s method to a set of nonlinear equations which define primal feasibility, dual feasibility, and complementarity [7,9,10]. Let vec denote the map from the space S n Thetan of symmetric matrices onto n(n 1) 2 satisfying M ffl N = vec(M ) T vec(N ) for any M;N 2 S n Thetan . Let l = n(n 1) m 1 and let z = x; h; u) 2 l where x 2 m , 2 , and h; u 2 n(n 1) 2 . The optimality ....

I. J. Lustig, R. E. Marsten and D. F. Shanno, "Interior Point Methods for Linear Programming: Computational State of the Art," ORSA Journal of Computing, 6:1--14, 1994.

....These methods enjoy several properties that make them especially attractive. ffl Practical efficiency. It is now generally accepted that interior point methods for LPs are competitive with the simplex method and even faster for problems with more than 10; 000 variables or constraints (see, e.g. [LMS94]) Similarly, our experience with system and control applications suggests that interior point methods for semidefinite programs are competitive with other methods for small problems, and substantially faster for medium and large scale problems. As a very rough rule of thumb, interior point ....

....Affix = GammaA T S Gamma2 d: 85) Since S is diagonal, the product in (85) is usually still sparse. This depends on the sparsity pattern of A, however. Dense rows in A, for example, have a catastrophic effect on sparsity. Equation (85) can be solved using a sparse Cholesky decomposition [LMS94]. The second strategy is to solve the sparse system (67) directly. Several researchers have argued that this method has better numerical properties (See Fourer and Mehrotra [FM91] Gill, Murray, Ponceleon, and Saunders [GMPS92] and Vanderbei and Carpenter [VC93] Moreover, directly solving (67) ....

I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6(1), 1994.

....language. A certain amount of incumbent advantage is probably acceptable or desirable. But publication and funding decisions are rather sensitive to initial computational results, and the technology of commercial codes can discourage the development of new approaches. Lustig, Marsten and Shanno [13] suggest, for example, that if interior point methods had come along a couple of years later than they did after the recent upswing in simplex technology now embodied in such codes as CPLEX they might have been judged too unpromising to pursue. A second cluster of evils concern the choice of ....

Lustig, I. J., R. E. Marsten and D. F. Shanno, Interior point methods for linear programming: Computational state of the art, ORSA Journal on Computing 6 (1994) 1-14.

....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 ....

I.J. Lustig, R.E. Marsten, and D.F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6:1--14, 1994.

....maximum variance problem by using a deterministic logarithmic barrier approach. Recently a variety of deterministic approaches for the global concave optimization problem have been suggested [10] Recent breakthroughs of interior point methods (IPM) for solving large scale optimization problems [15, 16] suggest that IPMs can be used to develop new fast learning laws in ANN. Ideas along these lines have already been published [20, 21] Other work has used ane scaling, trust region, and primal dual techniques for ANN training [19] This paper will present a logarithmic barrier method with ....

I.J.Lustig, R.E Marsten and D.F.Shanno (1994), "Interior point methods for linear programming: Computational state of the art", ORSA J. Comput. 6, 1-14.

....power series approximations. This approach has the added advantage of providing a rational scheme for reducing the value of . It is the predictor corrector based primal dual interior method that is considered the current winner in interior point methods. The OB1 code of Lustig, Marsten and Shanno [52] is based on this scheme. CPLEX 4.0 [17] a general purpose linear (and integer) programming solver, also contains implementations of interior point methods. Saltzman [70] describes a parallelization of the OB1 method to run on shared memory vector multiprocessor architectures. Recent ....

I.J.Lustig, R.E.Marsten, and D.F.Shanno, Interior Point Methods for Linear Programming: Computational State of the Art, ORSA J. on Computing, Vol. 6, No. 1, (1994) 1-14.

....(and then enforced throughout) Instead, it is only achieved as one approaches an optimal solution. The modifier interior point refers to the fact that the slack variables are required to be strictly positive at the beginning (and then throughout) As has been documented many places (see e.g. [10]) infeasible interior point methods represent the state of the art in linear programming. There are two salient points concerning LOQO that we feel may not be well understood, and both are important in the context of this paper. The first is that the decision variables x are always free ....

....them will completely invalidate the algorithm. Here, however, the w i s, p i s, and g i s are slack variables and interior point methods can shift slacks without seriously impacting the progress of the algorithm. Indeed, this is common practice in interior point codes for linear programming [10]. In the current LOQO, a slack variable, say w i , is shifted whenever it is smaller than 10 4 and #w i w i 10 4 . All such slack variables are shifted by the same amount s which is given by the harmonic average of the values of the variables that are not being shifted: s = k # i 1 w ....

I.J. Lustig, R.E. Marsten, and D.F. Shanno. Interior point methods for linear programming: computational state of the art. ORSA J. on Computing, 6:1--14, 1994. 4, 12

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LUSTIG, I. J., MARSTEN, R., and SHANNO, D. F., Interior-Point Methods for Linear Programming: Computational State of the Art, ORSA Journal of Computing, Vol. 6, pp. 1--14, 1994.

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I.J. Lustig, R.E. Marsten and D.F. Shanno. Interior point methods for linear programming: computational state of the art. ORSA Journal on Computing, vol. 6, pp. 1--14, 1994.

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I. J. Lustig, R. E. Marsten, and D. F. Shanno. Interior point methods for linear programming: Computational state of the art. ORSA Journal on Computing, 6(1), 1994.

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I.J. Lustig, R.E. Marsten, and D.F. Shanno, 1994. Interior point methods for linear programming: Computational state of the art, ORSA Journal on Computing 6, 1--14.

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I. J. Lustig, R. E. Marsten, and D. F. Shanno, 1994. Interior Point Methods for Linear Programming: Computational State of the Art. ORSA Journal on Computing 6, 1--14.

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