I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Math. Dokl., 8(1967), 674-675.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....omit them here. When 0 the corresponding method is called affine scaling algorithm. Most of the existing SDP literature considers path following algorithm. In this paper, we restrict our attention to affine scaling algorithm. The affine scaling algorithm was originally proposed for LP by Dikin [8], and independently rediscovered by Barnes [5] Vanderbei, Meketon and Freedman [39] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial timecomplexity has not been proved yet for this algorithm, global convergence using so called long steps ....

I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. (Translated in: Soviet Mathematics Doklady 8(1967)674-675.).

....That is why it and its following variants are called interior point methods. Karmarkar s paper led many researchers into this area. Soon, Vanderbei, Meketon, and Freedman [36] and Barnes [1] proposed a natural simpli cation of Karmarkar s algorithm. It turned out that as early as 1967, Dikin [4] had a very similar proposal. Nowadays, it is called the ane scaling method. Then, in 1990, Monteiro, Adler, and Resende [22] described a polynomialtime primal dual ane scaling algorithm. Renegar [27] was the rst to prove an O( nL) iteration bound for a path following method. The path following ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747-748, 1967.

....the potential function decreases quite linearly. The typical reduction in 5 per iteration increases with v. Increasing v places more weight on cost reduction (versus centering) and will at first speed up convergence. For large values of , the algorithm behaves like Dikin s affine scaling method [11]; convergence slows down again, because the iterates conhe too close to the boundary. The paths approach a limiting value of V5 that depends on g. In other words, the centrality of the iterates eventually remains nearly constant: for = 1, the iterates eventually remain in or very near the region ....

I. Dikin, "Iterative solution of problems of linear and quadratic programlning", Soviet Mathematics Doklady 8 (1967) 674 675.

....solve mathematical programs problems. Even though Huard proved that these algorithms converge to an optimal solution, the success of Dantzig s simplex method apparently thwarted any attempt of implementation for linear programming at the time. During the same time period in the Soviet Union, Dikin [10, 11] developed another interior point algorithm which was implemented and used to solve economic problems. Although Dikin s method has a simple interpretation as steepest descent on a scaled space, the polynomality of this algorithm is still not proven and it is not believed to be polynomial. In 1968 ....

....bounds on these derivatives. The rst of these results show that the derivatives of x (r) along b are uniformly bounded. The second of 78 these results shows that D (r) is uniformly bounded over fw 2 IR : k k = 1g. Both of these results use an observation due to Stewart [86] see also [10, 14, 87]) which is stated in lemma 3.10. De ne D to be the set of positive diagonal matrices in IR m m . For any D 2 D set AD = D A and PD = AA D D . The following lemma shows that PD is uniformly bounded over D . Lemma 3.10 (Stewart [86] There is a number k 0, such that kPD k k. ....

I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747-748, 1967.

....and preconditioned conjugate gradients, in the context of this approach. In addition, we demonstrate the benefits of our affine scaling, reflection and subspace techniques with computational results. First, for (hl) our affine scaling technique outperforms the classical Dikin scaling [7], at least in the context of our algorithm. Second, we examine our method with and without reflection. We show the reflection technique can substantially reduce the number of minimization itera tions. Third, our computational experiments support the notion that the subspace trust region method is ....

....the scaled gradient D 2gk converges to zero (i.e. first order optimality) The scaling matrix used in our approach is related to, but different from, the scaling typically used in affine scaling methods for linear programming. The affine scaling matrix D ffine de=f diag(min(z lk, u z) [7], commonly used in affine scaling methods for linear programming, is formed from the distance of variables to their closest bounds. Our scaling matrix D equals to D ffine only when min(z l, uk z) Ivl. Note that even in this case we employ the square root of the quantities used to define D ....

I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademiia Nauk SSSR, 174:747 748, 1967.

....Of course, general interior point methods (and the method of centers in particular) have a long history. Early work includes the SUMT book by Fiacco and McCormick [FM68] the method of centers described by Huard et al. LH65, Hua67] and Dikin s interior point method for linear programming [Dik67]. Interest in interior point methods, mostly for lin1 ear and quadratic programs, surged in 1979 when Khachyian used the ellipsoid method developed by Shor, Nemirovsky, and Yudin to prove that linear programs can be solved in polynomial time [Kha79, GL81] Interest surged again in 1984 when ....

....Newton steps required per iteration. In practice, this can lead to substantially faster convergence of to (measured in total Newton steps) However, the dual bounds are often worse than for = 1. For large, the method of centers will approach an analog of Dikin s affine scaling algorithm [Dik67]. ffl Switching to a quadratically convergent local method. We note the possibility of combining the method of centers with a quadratically convergent local method. The method of centers identifies the active eigenvalues and eigenvectors (via the dual matrices U and V ) as it proceeds (or more ....

I. Dikin. Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl., 8(3):674--675, 1967.

....that under standard assumptions the barrier Newton method converged quadratically. The purpose of this paper is to apply our results to linear programming(LP) problem. After some simplifications and after choosing a particular exponential space transformation function we obtain Dikin s algorithm [4] from the barrier projection method sometimes called the variation of Karmarkar s algorithm . However, there are some differences between our approach: 1. We use mainly quadratic space transformation and owing to it we get faster local convergence. 2. We developed a stable version of the ....

....on X 0 . If we use the exponential space transformation (11) and set = 0, then from (20) 22) we obtain dx dt = D 2 (x) A T u(x) Gamma c) AD 2 (x)A T u(x) AD 2 (x)c: 28) The discrete and continuous versions of this method were investigated in various papers (see, for example, [1, 3, 4, 13, 14, 19, 20]) In [1] the discrete version was called a variation on Karmarkar s algorithm . We should remark that method (28) does not possess the local convergence property. Here the convergence takes place only if x 0 belongs to the relative interior of X. Theorem 3.1 cannot be used for the exponential ....

Dikin, I., "Iterative solution of problems of linear and quadratic programming", Sov. Math. Dokl., Vol.8 (1967), pp. 674--675

....of the trajectories. Therefore, we need not introduce any penalty coefficients. In the second section we consider a family of barrier projection methods. In the linear programming case after simplifications and choosing an exponential space transformation function we obtain Dikin s algorithm [6], sometimes called the variation on Karmarkar s algorithm . The analysis of this algorithm was given in numerous papers (see, for example, 2, 4, 21, 23, 24, 34, 35] However, there are four main differences with our approaches: 1. We considered LP and NLP problems. 2. From 1983 we developed ....

....Method ( was proposed in 1978 (see [11, 8] Nonlocal convergence analysis of this method was made in [14] on the basis of the second Lyapunov method. System ( was also considered in [21] The discrete and continuous versions of method ( were investigated in numerous papers (see, for example, [2, 4, 6, 21, 23, 34, 35]) In [2] the discrete version was called a variation on Karmarkar s algorithm . We should remark that method ( does not possess the local convergence property. Here the convergence takes place only if x 0 belongs to the relative interior of X. This result was proved by G.Smirnov on the basis of ....

I. Dikin, Iterative solution of problems of linear and quadratic programming, Sov. Math. Dokl., 8, 674-675 (1967).

....barrier or penalty functions and this feature provides a high rate of convergence. Different numerical methods are obtained by different choices of the space transformations. For example, if we choose an exponential space transformation in the linear programming case, we obtain the Dikin algorithm [3] from the family of primal barrier projection methods. This algorithm, however, does not posses local convergence properties and it converges only for Research supported by the grant N 94 01 01379 from Russian Scientific fund. 1 starting points inside the feasible set. Furthermore, the ....

I.I. Dikin, Iterative solution of problems of linear and quadratic programming, Sov. Math. Dokl., 8, 674-675 (1967).

....that under standard assumptions the barrier Newton method converged quadratically. The purpose of this paper is to apply our results to linear programming (LP) problem. After some simplifications and after choosing a particular exponential space transformation function we obtain Dikin s algorithm [4] from the barrier projection method sometimes called the variation of Karmarkar s algorithm . However, there are some di#erences between our approach: 1. We use mainly quadratic space transformation and owing to it we get faster local convergence. 2. We developed a stable version of the ....

....on X 0 . If we use the exponential space transformation (11) and set # = 0, then from (20) 22) we obtain dx dt = D 2 (x) A T u(x) c) AD 2 (x)A T u(x) AD 2 (x)c. 28) The discrete and continuous versions of this method were investigated in various papers (see, for example, [1, 3, 4, 13, 14, 19, 20]) In [1] the discrete version was called a variation on Karmarkar s algorithm . We should remark that method (28) does not possess the local convergence property. Here the convergence takes place only if x 0 belongs to the relative interior of X . Theorem 3.1 cannot be used for the exponential ....

Dikin, I., "Iterative solution of problems of linear and quadratic programming", Sov. Math. Dokl., Vol.8 (1967), pp. 674--675

....can now be developed which differ mainly in the way the combinatorial structure of he problem is exploited to gain additional computational efficiency. 17 4 Interior point methods for linear and network programming 4. 1 Linear programming Interior point methods were first described by Dikin [29] in the mid 1960s, and current interest in them started with Karmarkar s algorithm in the mid 1980s [72] As the name suggests, these algorithms generate a sequence of iterates which moves through the relative interior of the feasible region, in marked contrast to the simplex method [22] where ....

....primal dual optimal solution, etc. In this subsection, we present in detail these components, illustrating their implementation in the dual affine scaling network flow algorithm dlnet of Resende and Veiga [130] 4.3. 1 The dual affine scaling algorithm The dual affine scaling (das) algorithm [12, 29, 135, 141] was one of the first interior point methods to be shown to be competive computationally with the simplex method [2, 3] As before, let A be an m Theta n matrix, c, u, and x be n dimensional vectors and b an m dimensional vector. The das algorithm solves the linear program min fc x j Ax = b; ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademiia Nauk SSSR, 174:747--748, 1967. English Translation: Soviet Mathematics Doklady, 1967, Volume 8, pp. 674--675. 84

....that (3.1) is feasible. What remains to be analyzed is the size of the solution. A key ingredient in our analysis is the following lemma. Lemma 3.1 Suppose that A has full row rank. Then, A) supfkDA T (ADA T ) Gamma1 k j D diagonal and D 0g 1: Lemma 3. 1 was first shown by Dikin [4] and was used in his convergence analysis for affine scaling methods. Among others, Stewart [19] and Todd [23] rediscovered this result later. The meaning of Lemma 3.1 can be interpreted as follows. It is well known that Null(A) fx j Ax = 0g and Range(A T ) fx j 9y 2 m x = A T yg are ....

I.I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady 8 (1967) 674-675.

....1984 1989 more than 1300 papers were published on the subject. 2 Originally the aim of the research was to get a better understanding of the so called Projective Method of Karmarkar. Soon it became apparent that this method was related to classical methods like the Ane Scaling Method of Dikin [8, 9, 10], the Logarithmic Barrier Method of Frisch [11, 12] and the Center Method of Huard [17] and that the last two methods, when tuned properly, could also be proved to be polynomial. Moreover, it turned out that the IPM approach to LO has a natural generalization to the related eld of convex ....

I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747-748, 1967. Translated in : Soviet Mathematics Doklady, 8:674-675, 1967.

....of diagonally weighted linear leastsquares problem can be expressed as a certain convex combination, is the basis for our results. As far as we know, it was originally given by Dikin [8] who used it in the convergence analysis of the interior point method for linear programming he proposed [7]. The proof of the theorem is based on the Cauchy Binet formula and Cramer s rule. Theorem 2.1 (Dikin [8] Let A be an m n matrix of full row rank, let g be a vector of dimension n, and let D be a positive de nite diagonal n n matrix. Then, ADA T ) 1 ADg = X J2J (A) det(D J ) det(A ....

I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady, 8 (1967), pp. 674-675.

....that there exists ffl 0 such that for any x 2 int(Q) x 2 int(Q) k k x ffl: 4.1) If 4. 1 holds and x 2 int(P ) then (x) x V c (x) k V c (x) k x 2 int(P ) jj ffl: Given x 2 int(P ) consider iterations x; x) 2 (x) This procedure is known as Dikin s algorithm [2] for the case where all constraints are linear. It is known to be an efficient tool for solving linear programming problems. To the best of our knowledge, no convergence results have been reported for the case of nonlinear constraints. It is clear, however, that convergence to the optimal solution ....

I. Dikin. Iterative solution of problems of linear and quadratic programming, Soviet. Math. Dokl. 8 (1967), 674-675.

....a certain metric and we describe this precisely. The motivation here, of taking as a neighborhood of the current interior point not an ellipsoid but rather an ellipsoidal cone, was apparently first discussed by Megiddo [10] in the context of a modification of the affine scaling algorithm of Dikin [2]. See also p. 171 in [7] note that f in (30) should be r ) and Padberg [11] We also prove directly that a fixed decrease in the potential function can be obtained by taking a step of an appropriate length in this direction. This proof is again very similar to those of Gonzaga [5, 7] We ....

I. I. Dikin, "Iterative solution of problems of linear and quadratic programming," Soviet Mathematics Doklady 8 (1967) 674-675.

....primal dual affine scaling algorithms for LCP. The aim of this paper is to show that this class of algorithms can also be used to solve sufficient LCP s. Scaling is one of the most important techniques in modern polynomial time optimization methods. The first affine scaling algorithm, of Dikin [3], remained unnoticed for a long time. After Karmarkar [10] initiated the dynamically developing field of interior point methods (IPM s) for linear programming (LP) affine scaling became one of the basic concept in IPM s for LP. Primal or dual affine scaling methods were studied by e.g. Barnes ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. Translated in : Soviet Mathematics Doklady, 8:674--675, 1967.

....that smaller neighborhoods generally restrict all iterates moved by a short step and they might be too conservative for solving real LP problems. A second important class of algorithms are the affine scaling ones. For the primal LP problem, the idea of the algorithm was first introduced by Dikin [3], and later rediscovered by Barnes [1] and Vanderbei et al. 19] Monteiro et al. 14] extended the idea to the primal dual setting and obtained an O(nL 2 ) complexity. Recently, Jansen et al. 6] provided a more natural application of the affine scaling idea to the primal dual setting; this ....

I.I. Dikin, "Iterative Solution of Problems of Linear and Quadratic Programming," Doklady Akademiia Nauk SSSR 174, (1967) 747--748.

....till this point are feasibility (5) and Condition 3.2. We will now derive a family of affine scaling directions as in Jansen et al. 13] The directions are obtained by minimizing the complementarity (suitably scaled) over an ellipsoid, which is the idea of Dikin s primal affine scaling algorithm [3]. Consider the problem minimize x T s subject to (x; s) 2 F : The NCP is equivalent to the above problem in the sense that (x; s) is a solution of the NCP if and only if it is a minimum solution of the above problem with objective value zero. According to the search mapping defined by (2) and ....

I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. (Translated in : Soviet Mathematics Doklady, 8:674--675, 1967).

.... Fan [Fan93] Hiriart Urruty and Ye [HUY95] Shapiro and Fan [SF94] and Pataki [Pat94] Interior point methods for LPs were introduced by Karmarkar in 1984 [Kar84] although many of the underlying principles are older (see, e.g. Fiacco and McCormick [FM68] Lieu and Huard [LH66] and Dikin [Dik67]) Karmarkar s algorithm, and the interior point methods developed afterwards, combine a very low, polynomial, worst case complexity with excellent behavior in practice. Karmarkar s paper has had an enormous impact, and several 4 variants of his method have been developed (see, e.g. the survey ....

I. Dikin. Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl., 8(3):674--675, 1967. 51

....state feasibility of X, Z and y. The last condition is called complementary slackness. Interior point methods Brief history The ideas underlying interior point methods for convex optimization can be traced back to the sixties; see e.g. Fiacco and McCormick [21] Lieu and Huard [22] and Dikin [23]) Interest in them was revived in 1984, when Karmarkar introduced a polynomial time interior point method for LP [24] In 1988 Nesterov and Nemirovskii [25] showed that those interior point methods for linear programming can, in principle, be generalized to all convex optimization problems. The ....

I. Dikin, "Iterative solution of problems of linear and quadratic programming", Soviet Math. Dokl., vol. 8, pp. 674--675, 1967.

....hand, ane scaling interior point methods for nonlinear optimization were developed by Coleman and Li (see, e.g. 3] 6] and [12] for minimization problems with simple bounds. The Coleman Li ane scaling incorporates dual information and relates to the DikinKarmarkar ane scaling (see, e.g. [13], 18] 22] and [23] One attractive feature of ane scaling interior point methods is that they can be appropriately tailored to speci c classes of problems. They have been applied to discretized optimal control problems by Dennis et al. [10] and to in nite dimensional control problems by ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Soviet Math. Dok., 8:674-675, 1967.

....H = Q A T S 1 ZS 1 A: 17) 6 Observe that the second term of H is simply a diagonal matrix. The quadratic approximation of around w k is given by: Q(w) 1 2 (w w k ) T H (w w k ) h T (w w k ) w k ) As approximation of the feasible polytope, the Dikin ellipsoid [3] is used. The Dikin ellipsoid around the interior solution w k is given by E( fw 2 IR n j (w w k ) T A T S 2 A(w w k ) 2 g; where for 1 the ellipsoid is inscribed in the feasible region. Denoting w = w w k we obtain the following optimization problem: P E ) min 1 2 ....

I.I. Dikin (1967), "Iterative solution of problems of linear and quadratic programming", Doklady Akademiia Nauk SSSR 174, 747-748. Translated into English in Soviet Mathematics Doklady 8, 674-675.

....is possible for such a method to yield a minimizer with larger objective function value than that of a feasible initial point. Depending on the problem context, this can be a serious limitation for practical applications. The a#ne scaling method for linear programming was first introduced by Dikin [6]; it is simpler than its later competitors such as Karmarkar s projective scaling [12] and primal and dual a#ne scaling methods [13] The monotonic decrease property of this a#ne scaling method is appealing for nonconvex programming problems. In this paper, we consider the a#ne scaling method ....

I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademiia Nauk SSSR, 174 (1967), pp. 747--748.

....(63) This method also converges exponentialy fast, but in contrast to (59) it does not have asymptotical stability property with respect to equality constrain Ax = b. This method was proposed in 1977 (see [8] It was called a barrier projection method . Method (59) reminds Dikin s algorithm [4], sometimes called the variation on Karmarkar s algorithm [3] It has better local convergence property than Dikin s method. The differences between these methods are analysed in [17] Recently this method was reinvented by [19] 24] 22] 5. If we set in (31) ff = 1; 0 and x = e, then we ....

I.I.Dikin. Iterative solution of problems of linear and quadratic programming, Sov. Math. Dokl. 8, 674-675, (1967).

....them here. When # 0 the corresponding method is called affine scaling algorithm. Most of the existing SDP literature considers path following algorithm. In this paper, we restrict our attention to affine scaling algorithm. The affine scaling algorithm was originally proposed for LP by Dikin [8], and independently rediscovered by Barnes [5] Vanderbei, Meketon and Freedman [39] and others, after Karmarkar [16] proposed the first polynomial time interior point method. Though polynomial time complexity has not been proved yet for this algorithm, global convergence using so called long ....

I.I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. (Translated in: Soviet Mathematics Doklady 8(1967)674-675.).

....and (2.6) at each iteration. Several entries of D k converge to zero as the method approaches a solution while other entries tend to infinity. The same type of systems appear in other interior point methods such as the dual and primal versions [55] For instance, on the primal affine version [13] we have D k = X k ) 2 and the right hand side is given by (c t o t ) t on (2.5) and for the dual affine method [1] D k = Z k ) 2 and the right hand side is given by (0 t b t ) t . 11 Notice that for the affine methods the centering parameter is zero. These methods do ....

DIKIN, I. I. Iterative Solution of Problems of Linear and Quadratic Programming. Soviet Mathematics Doklady, Vol. 8, pp.674-675, 1967.

....row rank. Then the set fQ(D) j D is a positive diagonal matrixg is bounded. Therefore, the angle 6 (x; DA T y) with Ax = 0, x 6= 0, y 6= 0 and D a positive diagonal matrix is bounded below from zero by a positive constant, which is independent of x, y and D. The above result is well known (cf. [6, 7, 24, 26]) and is widely used for the analysis of interior point methods in linear programming. The minimum angle between KerA and ImgDA T is also used as a condition number for linear programs whose constraint matrix is A; see Vavasis and Ye [27] 14 If Q is a limit point of the (bounded) set fQ(D) j ....

Dikin, I.I., Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady 8 (1967) 674-675. 23

....in a comparable way as they solve linear programming (LP) problems, see [2, 8, 14, 15, 16, 18, 22, 21, 24, 26, 33] For linear programming (LP) the primal affine scaling algorithm is one of the more popular interior point methods, since it is both simple and efficient. Although proposed by Dikin [4, 5] as early as in 1967, the affine scaling algorithm only received the proper attention when Barnes [3] and Vanderbei et al. 32] rediscovered it as a natural simplification of Karmarkar s algorithm [13] Namely, they derived the affine scaling algorithm by simply replacing the projective ....

....may obtain the cone affine search direction of Padberg [23] and Goldfarb and Xiao [6] by optimizing the original linear objective over a conic section, using merely an affine transformation. This conic section contains the ellipsoid that is used in the classical affine scaling algorithm of Dikin [4, 5]. Monteiro et al. 17] proposed a variant of the affine scaling algorithm for LP that is symmetric in the duality, henceforth called a primal dual affine scaling algorithm. Other primal dual affine scaling algorithms were proposed by Jansen et al. 10, 11, 12] and Sturm and Zhang [28] Although ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. Translated in : Soviet Mathematics Doklady, 8:674--675, 1967.

....flow problem on a bipartite connected graph [29] Furthermore each basic solution of the TP problem (2) is related with a spanning tree of this graph. Based on these facts, Resende and Veiga [31, 30] have developed an efficient implementation of the Dual Affine (DA) interior point algorithm [8] for the solution of large scale TP problems where the vectors c and d have integer components. This procedure incorporates a Preconditioned Conjugate Gradient (PCG) algorithm [9] to find the ascent direction that is required in each iteration of the DA algorithm. Resende and Veiga [31, 30] have ....

....describe briefly the three interior point algorithms (the dual affine scaling, the primal dual and the primal dual predictor corrector) that are considered in this paper. We include some additional references for the interested reader. 2.1. Dual affine scaling algorithm. The dual affine algorithm [8, 5, 32, 33] was the first interior point algorithm that has been shown to be competitive [1, 3] with the simplex method for the solution of large scale linear programs. The algorithm finds an optimal solution by solving the linear program in the standard dual form (3) Having a dual interior starting point ....

I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Mathematics Doklady, 8 (1967), pp. 674--675.

....by F = f (x; s) j Gamma Mx s = q; x 0; s 0 g; F 0 = f (x; s) j Gamma Mx s = q; x 0; s 0 g: We shall assume throughout that F 0 6= Scaling is one of the most important techniques in modern polynomial time optimization methods. The first affine scaling algorithm, of Dikin [2], remained unnoticed for a long time. After Karmarkar [6] initiated the dynamically developing field of interior point methods (IPMs) affine scaling became one of the basic concept in IPMs. Primal or dual affine scaling methods were studied by e.g. Barnes [1] Vanderbei et al. 13] Tsuchiya ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. Translated in : Soviet Mathematics Doklady, 8:674--675, 1967.

.... Beale [5] Bland [6] Charnes [7] Dantzig [8] Gal [10, 11] Greenberg [12] Hoffman [15] Magnanti and Orlin [16] Megiddo [17] Monteiro and Mehrotra [18] Ward and Wendell [29] Williams [30] Wolfe [31] with the convergence of the affine scaling interior point algorithm (Barnes [4] Dikin [9], Hall and Vanderbei [14] Monteiro and Tsuchiya [19] Monteiro, Tsuchiya and Wang [20] Tsuchiya [24, 25, 26] Vanderbei et al. 28] Vanderbei and Lagarias [27] and etc. The paper by Guler et al. 13] surveys the theoretical and practical issues related to degeneracy in the context of interior ....

I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR, 174 (1967), pp. 747--748. Translated in : Soviet Mathematics Doklady, 8:674--675, 1967.

....problem. Key words: Linearly constrained problem, affine scaling algorithm, trust region method, interior point method. AMS 1991 subject classification: 49M37, 49M45, 65K05, 90C25, 90C26, 90C30. 1 Introduction The affine scaling (AS) algorithm for linear programming was first introduced by Dikin [6] in 1967 but remained unknown to the western community until the late 80 s. The method was later rediscovered independently by Barnes [3] and Vanderbei et al. 44] Since then, there have appeared a number of papers which study its global and local convergence [7, 8, 12, 21, 37, 38, 39, 41, 42, ....

I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR, 174 (1967), pp. 747--748. Translated in: Soviet Mathematics Doklady, 8 (1967), pp. 674--675.

....or Newton s method. Later, researchers simpli ed this algorithm, removing the need for projective transformations, and obtained a class of methods called a ne scaling algorithms. It was later discovered that these methods had been previously proposed by Dikin in Russia, 17 years before Karmarkar [Dik67]. A ne scaling algorithms do not explicitly follow the central path and do not even refer to it. The basic idea underlying these methods is the following: consider for example the primal problem (P) P) min x2R n c T x s.t. Ax = b x 0 : This problem is hard to solve because of the ....

I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR 174 (1967), 747-748.

....or Newton method. Later, researchers simpli ed this algorithm, removing the need for projective transformations, and obtained a class of methods called a ne scaling algorithms. It was later discovered that these methods had been previously proposed by Dikin in Russia, 17 years before Karmarkar [Dik67]. A ne scaling algorithms do not explicitly follow the central path and do not even refer to it. The basic idea underlying these methods is the following: consider for example the primal problem (P) min x2R n c T x s.t. Ax = b x 0 : P) This problem is hard to solve because of the ....

I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR 174 (1967), 747-748.

....When solving large problems this is unacceptable. Therefore we will introduce a potential function that yields a more sparse Hessian in Section 2.2. Minimizing (2) subject to x 2 P is NP complete, but when the polytope P is substituted by an inscribed ellipsoid, the so called Dikin ellipsoid [5], the resulting approximate problem can be solved in polynomial time (Ye, 20] Lemma 1 (Karmarkar, 12] Let x k 2 P 0 . Then the ellipsoid E(r) fx 2 IR m j (x Gamma x k ) T A T S Gamma2 A(x Gamma x k ) r 2 g is an inscribed ellipsoid in P , i.e. E(r) ae P, for r 1. ....

I.I. Dikin (1967), "Iterative solution of problems of linear and quadratic programming", Doklady Akademiia Nauk SSSR 174, 747--748. Translated into English in Soviet Mathematics Doklady 8, 674--675.

....primal affine scaling algorithm is not believed to run in polynomial time in the worst case. It was proposed by many people independently (e.g. Barnes [6] Vanderbei, Meketon and Freedman [48] Later, however, it was discovered that the same primal affine scaling algorithm was proposed by Dikin [14] in the 1960 s. Gill, Murray, Saunders, Tomlin and Wright [17] established the connection between Karmarkar s algorithm and the classical logarithmic barrier function method of nonlinear programming. Dual versions and primal dual versions of the affine scaling method have also been studied. After ....

I. I. Dikin, "Iterative solution of problems of linear and quadratic programming," Soviet Math. Dokl. 8 (1967) 674--675.

....(17) Note that the density of H is determined by the density of the matrix Q B T B. The quadratic approximation of around x k is given by: Q(x) 1 2 (x Gamma x k ) T H (x Gamma x k ) h T (x Gamma x k ) x k ) As approximation of the polytope P, the Dikin ellipsoid [4] is used. The Dikin ellipsoid around x k 2 P 0 is given by E(r) fx 2 IR m j (x Gamma x k ) T A T S Gamma2 A(x Gamma x k ) r 2 g; where for r 1 the ellipsoid is inscribed in P. Denoting Deltax = x Gamma x k we obtain the following optimization problem: FAP E ) min 1 ....

I.I. Dikin (1967), "Iterative solution of problems of linear and quadratic programming", Doklady Akademiia Nauk SSSR 174, 747--748. Translated into English in Soviet Mathematics Doklady 8, 674--675.

....method (Monteiro, Adler and Resende, 1990) Key words: interior point method, affine scaling method, primal dual method. iii 1 Introduction In this paper we present a new primal dual affine scaling method for linear programming (LP) The notion of affine scaling has been introduced by Dikin [2], in 1967, as a tool for solving the (primal) problem in standard format (P ) minfc T x : Ax = b; x 0g: The underlying idea is to replace the the nonnegativity constraints x 0 by the ellipsoidal constraint kX Gamma1 (x Gamma x)k 1; where x denotes some given interior feasible point, ....

....slack vector s = c Gamma A T y. Let Deltax; Deltay; Deltas denote the search directions in the respective spaces. Neglecting for the moment the nonnegativity conditions, feasibility will be maintained by requiring that A Deltax = 0; A T Deltay Deltas = 0: Following the idea of Dikin [2], we replace the nonnegativity conditions by requiring that the next iterates (x Deltax; y Deltay; s Deltas) belong to a suitable ellipsoid. We define this ellipsoid by requiring that kX Gamma1 Deltax S Gamma1 Deltask 1; and call this the (primal dual) Dikin ellipsoid. It may ....

Dikin, I.I. (1967), Iterative Solution of Problems of Linear and Quadratic Programming, Doklady Akademiia Nauk SSSR 174, 747--748.

....reformulate this problem as minfc T Deltax : A Deltax = 0; x Deltax 0g; because if Deltax solves this problem, then x Deltax solves the original problem. The difficulty in solving the last problem is caused by the inequality constraint x Deltax 0. Assuming that x is positive, Dikin [3] proposed to replace this inequality constraint by the ellipsoidal constraint fl fl X Gamma1 Deltax fl fl 1, where X = diag (x) The resulting problem, given by minfc T Deltax : A Deltax = 0; fl fl X Gamma1 Deltax fl fl 1g; is now easily solvable, and yields the so called affine ....

....We refer to [7] for a survey of these and related results, including a thorough discussion of degeneracy. It may be mentioned that the occurance of degeneracy complicates the analysis of the affine scaling method. We recall some results from [7] which are relevant for the present paper. Dikin [3] assumed primal nondegeneracy and proved global convergence for the unit step size (fi = 1) Tsuchiya [15] got the first result without any assumption on degeneracy. He showed global convergence when the step size is 1 8. From the practical point of view, Tsuchiya s result is unsatisfactory. Most ....

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Doklady Akademii Nauk SSSR, 174:747--748, 1967. Translated in : Soviet Mathematics Doklady, 8:674-- 675, 1967.

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I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Math. Dokl., 8(1967), 674-675.

No context found.

I. Dikin, Iterative solution of problems of linear and quadratic programming, Soviet Math. Dokl., 8(1967), 674-675.

No context found.

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Sov. Math. Doklady, 8(66):674--675, 1967.

No context found.

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Sov. Math. Doklady, 8(66):674--675, 1967.

No context found.

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Sov. Math. Doklady 8, pp. 674-675, 1967.

No context found.

I. I. Dikin, "Iterative solution of problems of linear and quadratic programming," Soviet Mathematics Doklady8 (1967) 674--675.

No context found.

I. I. Dikin. Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl.,8:, 1967. 10

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I. Dikin. Iterative solution of problems of linear and quadratic programming. Soviet Math. Dokl., 8(3):674--675, 1967. 51

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I. I. Dikin, Iterative solution of problems of linear and quadratic programming, Doklady Akademii Nauk SSSR 174 (1967), 747-748.

No context found.

I.I. Dikin (1967). "Iterative solution of problems of linear and quadratic programming." Soviet Mathematics Doklady, 8, 674-675.

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