Yu. E. Nesterov and A. S. Nemirovsky, Self-concordant functions and polynomial time methods in convex programming, Technical report, Centr. Econ. & Math. Inst. USSR Adad. Sci., Moscow, USSR, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... (where # is part of the input, its size being proportional to log(1 #) For this purpose one can use the ellipsoid algorithm (cf. G L S] other polynomial time algorithms for convex programming [Vai] or the interior point methods [Ali, N N1, N N2] Since the work of Nesterov and Nemirovskii [N N1], and Alizadeh [Ali] there has been much development in the design and analysis of interior point methods for semidefinite programming, yielding practically useful algorithms whose performance is comparable, and even superior, to the performance of the simplex method. 4.2 Max cut and semidefinite ....

Y. Nesterov, A. Nemirovskii, Self-Concordant Functions and Polynomial Time Methods in Convex Programming, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, 1989.

....problems that arise in control theory. They also describe a saddle point method for eigenvalue mimimization due to Pyatnitski and Skorodinsky [PS83] Interior point methods for eigenvalue minimization have recently been developed by several researchers. The first were Nesterov and Nemirovsky [NN88, NN90b, NN90a, NN91a, NN93]; others include Alizadeh [Ali92b, Ali91, Ali92a] Jarre [Jar91a] and Vandenberghe and Boyd [VB93] 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 ....

....inequalities In this section we discuss ways of representing convex constraints on the variable x in the form of an affine matrix inequality C(x) 0. The idea that affine matrix inequalities can be used to represent a wide variety of convex constraints can be found in Nesterov and Nemirovsky [NN90b, NN90a, NN93] (who formalize the idea of a positive definite representable function) and Alizadeh [Ali92b, Ali91] 2.1 Multiple constraints We first note that multiple constraints on x, expressed as the affine matrix inequalities C i (x) 0, i = 1; l, are equivalent to the single affine matrix ....

Yu. Nesterov and A. Nemirovsky. Self-concordant functions and polynomial time methods in convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, April 1990. 38

....least for small ) due to the global convergence of all trajectories of the vector field V c to the optimal solution. Observe that conditions 4. 1 are satisfied provided the functions ln f i ; i = 1; m are self concordant functions (this is always the case when f i are linear or quadratic [10] ) The properties of this type provide a special status for logarithmic barrier functions at least for now. Our analysis can be generalized to certain infinite dimensional situations with interesting applications to control problems like linear quadratic regulators with quadratic constraints ....

Yu.E. Nesterov and A.S. Nemirovsky. Self-concordant functions and polynomial-time methods in convex programming. Technical report, Centr. Econ. Inst.,Moscow, USSR,1989.

....Table 3: Solving Model 2 Problems with the Termination Test 6 2.3. Progress in Convex Nonlinear Optimization Several interior point algorithms currently exist for convex nonlinear optimization (e.g. 10] 14] 18] 20] 24] We augmented those with a potential reduction algorithm (e.g. 16][22][Y5] Y10] Y14] which has exhibited promising practical efficiency. Han, Pardalos and myself implemented and tested the algorithm using vectorization on an IBM 3090 600S computer with Vector Facilities using the VS Fortran compiler. All numerical results were obtained in double precision. ESSL ....

Ju. E. Nesterov and A. S. Nemirovsky. Self-concordant functions and polynomial-time methods in convex programming, USSR Academy of Sciences, Central Economic and Mathematics Institute (Moscow, USSR, 1989).

....a thorough treatment. The ellipsoid method, however, has not proven practical in most applications, including SP. A more recent development is the possibility of using interior point methods to obtain polynomial time algorithms for semidefinite programs. See the work by Nesterov and Nemirovskii [20], and by Alizadeh [1] 4.1 Semidefinite Programming how it was used The application of Semidefinite Programming to obtaining approximation algorithms for NP hard problems was pioneered by Goemans and Williamson [12] This technique involves relaxing an integer quadratic program (which is ....

Y. Nesterov, A. Nemirovskii, "Self-Concordant Functions and Polynomial Time Methods in Convex Programming", Central Economical and Mathematical Institute, USSR Academy of Science, Moscow, 1990. 18

....of the form find some x such that A(x) 0 where x 2 R m is a vector of decision variables. In turn this can be formulated as the following convex nondifferentiable program: min x2R m max (A(x) for which efficient (polynomial time) convex optimization algorithms are now available (see e.g. [NN90, BG92] and references therein) For the synthesis of stabilizing control laws for uncertain LTI systems, convex LMI techniques are only applicable when the full state is available for feedback. When this is not the case, available formulations of the output feedback problem are not convex in the ....

Yu. Nesterov and A. Nemirovsky. Self-concordant functions and polynomial time methods in convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, April 1990.

....with the problems we first present some lemmas that are quite helpful in recognizing self concordant functions. 159 3.5.2 Some composition rules for self concordant functions The following lemma provides some composition rules for self concordant functions. Lemma 3. 50 (Nesterov and Nemirovskii [33, 34]) ffl (addition and scaling) Let i be i self concordant on F 0 i , i = 1; 2, and ae 1 ; ae 2 2 IR then ae 1 1 ae 2 2 is self concordant on F 0 1 F 0 2 , where = maxf 1 p ae 1 ; 2 p ae 2 g. ffl (affine invariance) Let be self concordant on F 0 and let B(y) By ....

Nesterov, Y.E. and Nemirovsky, A.S. (1989), Self--Concordant Functions and Polynomial Time Methods in Convex Programming, Report, Central Economical and Mathematical Institute, USSR Academy of Science, Moscow, USSR.

....self concordance. 1 Introduction The efficiency of a barrier method for solving convex programs strongly depends on the properties of the barrier function used. A key property that is sufficient to prove fast convergence for barrier methods is the property of self concordance introduced in [17]. This condition not only allows a proof of polynomial convergence, but numerical experiments in [1, 11, 14] and others further indicate that numerical algorithms based on self concordant barrier functions are of practical interest and effectively exploit the structure of the underlying problem. A ....

....of problems treated in [5, 12, 23] and in Section 7 we show that the smoothness conditions used in [7, 9, 13, 16, 25] imply self concordance of the barrier function. 2 Some general composition rules Let us first give the precise definition of self concordance as given by Nesterov and Nemirovsky [17]: Definition of self concordance: Let F 0 be an open convex subset of IR n . A function : F 0 IR is called self concordant on F 0 , 0, if is three times continuously differentiable in F 0 and if for all x 2 F 0 and h 2 IR n the following inequality holds: r 3 (x) h; h; ....

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Nesterov, Y.E. and Nemirovsky, A.S. (1989), Self-Concordant Functions and Polynomial Time Methods in Convex Programming, Technical Report, Central Economical and Mathematical Institute, Academy of Science, Moscow. To Appear in Lecture Notes in Mathematics.

....has a simple expression. We further observe that the absolute value of this derivative can be used as a new measure of proximity to the central path. Based on these remarks, the 1 convergence analysis turns out to be very much the same as [26] and [2] and even more simple. See also [8] and [9] [23]. To make those statements more precise, we introduce the linear programming problem (P ) min fc T x : Ax = b; x 0g; and its dual problem (D) max fb T y : A T y s = c; s 0g: Here A denotes an m Theta n matrix of rank m, and b; c; x; y and s are vectors of appropriate sizes. We ....

....one, and independent of the current iterates) This is repeated until the barrier parameter has a sufficiently small value. In this way we have explicit control on the value of the barrier parameter, just as in the logarithmic barrier approach to the primal and the dual case. See, e.g. 26] 8] [23] [9] Before we deal with the algorithms, we will introduce our proximity measure. This is the subject of the next section, and we shall see that also in this respect our approach differs from the usual one. 3 Proximity measure The above formulas for the search direction vectors d x ; d s and d ....

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Nesterov, Y.E. and Nemirovsky, A.S. (1989), Self--Concordant Functions and Polynomial Time Methods in Convex Programming, Report, Central Economical and Mathematical Institute, USSR Academy of Science, Moscow, USSR.

....semidefinite, finds a hyperplane separating X from the set of positive semidefinite matrices. This subroutine, combined with the ellipsoid method, provided a polynomial time algorithm for semidefinite programming. However, the complexity of this algorithm is quite high. Nesterov and Nemirovskii [112] showed how to use the interior point methods to solve semidefinite programs. Alizadeh [5] showed how an interior point algorithm for linear programming could be directly generalized to handle semidefinite programming. Since the work of Alizadeh, there has been a great deal of research into such ....

....program. Although a semidefinite program is a nonlinear program, surprisingly it preserves many nice properties of a linear program. For instance most methods for solving integer programs (e.g. simplex [48] ellipsoid [92] and interior point [89, 156] can be generalized to semidefinite programs [119, 68, 146, 5, 112, 113]. As a result, a semidefinite program can be approximately solved to within an additive error ffl in polynomial time unless the optimal solution itself is exponentially large [5] For example if the semidefinite program (2.1) has a feasible solution, then it takes time O( p n 3 log ffl ....

Y. Nesterov and A. Nemirovskii. Self-Concordant Functions and Polynomial Time Methods in Covex Programming. Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, 1989.

....has not proven practical in most applications, including SDP. A more recent development is the possibility of using interior point methods to obtain polynomial time algorithms for semidefinite programs. The earliest work in this direction to our knowledge is that of 1 Nesterov and Nemirovskii [NN90]. In this important work the authors develop a general approach for using interior point methods for solving convex programming problems which is based on the concept of p selfconcordant barrier functions. See the more recent [NN92] for a complete treatment of this subject. Nesterov and ....

....the authors extend the revolutionary result of Karmarkar [Kar84] to a rather general class of convex programs. In this article we study interior point methods for semidefinite programs from an alternative point of view. Our work [Ali91] started somewhat later than, and independent of, that of [NN90]. Nesterov and Nemirovskii obtain their complexity theorems by specializing their general results to SDP. We, on the other hand, take a specific interior point algorithm for linear programming (i.e Ye s projective potential reduction method [Ye90] and extend it to SDP. Furthermore, we argue that ....

[Article contains additional citation context not shown here]

Y. Nesterov and A. Nemirovskii. Self-Concordant Functions and Polynomial Time Methods in Convex Programming. Moscow, 1990.

.... survey by Gonzaga [Gon92] Interior point methods were subsequently extended to handle convex quadratic programming, and to certain linear complementarity problems (see, e.g. Kojima, Megiddo, Noma and Yoshise [KMNY91] An important breakthrough was achieved by Nesterov and Nemirovsky in 1988 [NN88, NN90b, NN90a, NN91a, NN91a]. They showed that interior point methods for linear programming can, in principle, be generalized to all convex optimization problems. The key element is the knowledge of a barrier function with a certain property: self concordance. To be useful in practice, the barrier (or really, its first and ....

Yu. Nesterov and A. Nemirovsky. Self-concordant functions and polynomial time methods in convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, April 1990.

.... (mon teiro isye.gatech.edu, paulo isye.gatech.edu) 1 1 Introduction Several authors have discussed generalizations of interior point algorithms for linear programming (LP) to the context of semidefinite programming (SDP) The landmark work in this direction is due to Nesterov and Nemirovskii [26, 27] where a general approach for using interior point methods for solving convex programs is proposed based on the notion of self concordant functions. See their book [29] for a comprehensive treatment of this subject. They show that the problem of minimizing a linear function over a convex set can ....

Y. E. Nesterov and A. S. Nemirovskii. Self-concordant functions and polynomial time methods in convex programming. preprint, Central Economic & Mathematical Institute, USSR Acad. Sci. Moscow, USSR, 1989.

.... Several authors have discussed generalizations of interior point algorithms for linear programming (LP) to the context of semidefinite programming (SDP) and the more general semidefinite linear complementarity problem (SDLCP) The landmark work in this direction is due to Nesterov and Nemirovskii [19, 20] where a general approach for using interior point methods for solving convex programs is proposed based on the notion of self concordant functions. See their book [22] for a comprehensive treatment of this subject. They show that the problem of minimizing a linear function over a convex set can ....

Y. E. Nesterov and A. S. Nemirovskii. Self-concordant functions and polynomial time methods in convex programming. preprint, Central Economic & Mathematical Institute, USSR Acad. Sci. Moscow, USSR, 1989.

....and f is convex. Remark. The expressions for the derivatives with respect to the diagonal elements of X were also given by Fletcher in [8] Since det(D) det(L DL T ) det(X) it follows that # n j =1 log(d j (X) is the usual self concordant barrier function for SDP (see, e.g. [11]) 4. AN INTERIOR POINT METHOD FOR CONVEX OPTIMIZATION. In this section, we describe an interior point algorithm that can be easily modified to handle SDPs in which the positive semidefiniteness constraint is expressed using (5) We describe the algorithm for problems in which all constraints are ....

Y.E. Nesterov and A.S. Nemirovsky. Self--concordant functions and polynomial--time methods in convex programming. Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR, 1989.

....) Is P feasible Consider a condition number of A, defined as oe p = min j2B fmax x j : xB 2 P p g oe d = min j2N fmax s j : y; s) 2 P d g oe(A) min(oe p ; oe d ) 1) 3 (we assign oe p (oe d ) to 1 if B (N) is null. It has been shown that the interiorpoint algorithm, such as [2] 4] 10][11][13] generates a sequence of partitions (B k ; N k ) A such that, after O( p n(j log oe(A)j log n) iterations, we have convergence to B k = B and N k = N (see [18] It seems that this result could be used to answer question (P ) to determine whether A = N or not. However, this ....

Ju. E. Nesterov and A. S. Nemirovsky. Self-concordant functions and polynomial-time methods in convex programming, USSR Academy of Sciences, Central Economic and Mathematics Institute (Moscow, USSR, 1989).

No context found.

Yu. E. Nesterov and A. S. Nemirovsky, Self-concordant functions and polynomial time methods in convex programming, Technical report, Centr. Econ. & Math. Inst. USSR Adad. Sci., Moscow, USSR, 1989.

No context found.

Y. Nesterov and A. Nemirovskii. Self-Concordant Functions and Polynomial Time Methods in Convex Programming. Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, 1989.

No context found.

Y. Nesterov and A. Nemirovski. Self-concordant functions and polynomialtime methods in convex programming. Technical report, Center Econ. Math. Inst., USSR Acad. Sci., Moscow, USSR, 1990.

No context found.

Y. Nesterov and A. Nemirovskii. Self-Concordant Functions and Polynomial Time Methods in Convex Programming. Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, 1989.

No context found.

Ju. E. Nesterov and A. S. Nemirovsky, "Self-concordant functions and polynomial-time methods in convex programming," Report, Central Economical and Mathematical Institute, USSR Acad. Sci. (Moscow, USSR, 1989).

No context found.

Yu. Nesterov and A. Nemirovsky. Self-concordant functions and polynomial time methods in convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, April 1990.

No context found.

Y.E. NESTEROV and A.S. NEMIROVSKI. Self-concordant functions and polynomial time methods in convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Acad. Sci., Moscow, USSR, April 1990.

No context found.

Y.E. NESTEROV and A.S. NEMIROVSKI. Self--concordant functions and polynomial--time methods in convex programming. Book--Preprint, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR, 1989. Published in Nesterov and Nemirovsky [661].

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Y. Nesterov, A. Nemirovskii, Self-Concordant Functions and Polynomial Time Methods in Convex Programming, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, 1989.

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