Y.E. NESTEROV and M.J. TODD. Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22(1):1-42, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....cones. 1. Introduction An elegant, powerful and modern theory of interior point methods treats convex optimization problems in conic form, as the problem of optimizing a linear function over a convex cone intersected with an ane space (see Nesterov and Nemirovskii [12] Nesterov and Todd [13], 14] and the exposition by Renegar [17] Also see [23] where symmetric primal dual interior point methods are generalized to all convex optimization problems in conic form. In such formulations, we interpret the convex cone constraint as the di cult constraint and deal with the cone ....

....for SC(K;B) see [12] Whether the construction accounts for all optimal barriers is less well understood. When we restrict K to the set of symmetric cones, then we have the notion of self scaled barriers for K. In this context, the other most relevant results are those given by rst Nesterov Todd [13], 14] on the foundations of self scaled barriers) then by Hauser [8] Schmieta [20] Hauser G uler [9] and Hauser Lim [10] also see [22] about a geometric mean like characterization of self scaled barriers) It follows from these works that an optimal self scaled barrier is unique up to an ....

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Yu. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research 22, (1997) 1-46.

....the primal dual methods we have developed is proportional to m, which, for m n, is much better than the standard O( m n) complexity bound. 7.2. Long steps. We consider three related viewpoints: a) regularity of a s.c.b. 17] b) convexity of the gradient product hH (x) yi [18, 19]; c) normality of a s.c.b. 13] All of these properties are strengthenings of the fundamental property of the self concordant barriers which states that the Hessian of a s.c.b. behaves very well inside the Dikin ellipsoid (see SC.II) anywhere in the interior of the domain. Each of the three ....

.... is a symmetric cone) De ne x (h) sup ft : x th) 2 Kg 1 [1 t x ( h) x) H (x th) 1 t x (h) x) for every x 2 int K, h 2 E and t 2 [0; 1= x (h) This property was proven via establishing the convexity of the function hH (x) yi : int K R, for every y 2 K [18]. Later, this property was extended to all hyperbolic barriers [8] c) f is normal if for every x; z 2 Q, r Q;x (z x) 1 implies f(z th) 8h 2 E : It is known that all speci c examples discussed here are normal for moderate values of (see [13] Our approach is very ....

Nesterov, Yu., Todd, M. J. \Self-scaled barriers and interior-point methods for convex programming", Mathematics of Operations Research v. 22 (1997), 1-46.

....is easy to verify that (14) is also satisfied by the logarithmic determinant barrier function for the positive semidefinite matrix cone. Similarly, it can be shown that the additional inequality (14) is satisfied by all self scaled barrier functions for the self scaled cones (see Nesterov and Todd [6]) in a di#erent but equivalent framework, the self scaled barrier function is the log determinant barrier for the symmetric cones (see Faybusovich [1] or Sturm [8] One may therefore think that (14) is a property of the self scaled cones only. However, this guess is incorrect. Let us consider ....

Yu. Nesterov and M.J. Todd, Self--scaled barriers and interior--point methods for convex programming, Mathematics of Operations Research 22, 1--42, 1997.

.... # 1 Interior point methods for semidefinite programming were originally introduced by [11, 4] Early papers on primal dual methods include [17] and [6] A preliminary version of the present work appeared as [2] Convergence analysis of primal dual pathfollowing methods for SDP appeared first in [7, 13, 12]. We are primarily concerned with four methods, which we call the XZ, XZ ZX, Nesterov Todd (NT) and Q methods. The XZ method first appeared in [6, 7] The XZ ZX method was introduced in [2] and was recently analyzed in [8, 9] The NT method was given in [13, 12] and its implementation was ....

....for SDP appeared first in [7, 13, 12] We are primarily concerned with four methods, which we call the XZ, XZ ZX, Nesterov Todd (NT) and Q methods. The XZ method first appeared in [6, 7] The XZ ZX method was introduced in [2] and was recently analyzed in [8, 9] The NT method was given in [13, 12] and its implementation was recently discussed in [16] The Q method originally appeared in [1] Many other papers on semidefinite programming have recently been announced. The paper is organized as follows. In section 2 we introduce several algorithms in a common framework based on Newton s ....

Y. Nesterov and M. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res., 22 (1997), pp. 1--42.

....was given by Todd, Toh and Tutuncu [35] On the other hand, Alizadeh, Haeberly, and Overton [4] report that the path following algorithm using the AHO direction has empirically better convergence properties than the one using the HRVW KSH M direction. 1.4. The NT Direction. Nesterov and Todd [30, 31] proposed primal dual algorithms for more general convex programming than SDP, which includes SDP as a special case. Their search direction naturally produces a symmetric direction. The direction is the solution of (7) 8) and D#ZD= X, 14) where D S(n) is a unique solution of DZD= X . 15) ....

Y. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Technical Report 1091, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 14853-3801, 1995.

....over real numbers complex numbers quaternions, and the cone of 3 3 semide nite matrices over octonions. Toh and Trefethen showed that semide nite programming over the complex domain is solvable in polynomial time [28] G uler [14] showed that the self scaled cones for which Nesterov and Todd [24, 25] had provided polynomial time algorithms are exactly the symmetric cones. We propose to take advantage of these developments by using semide nite programming over the complex domain; we call this complex semide nite programming (CSDP) Previously we modelled binary decisions with the square ....

Y. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22:1-42, 1997.

....transformation given by 1 [pMp i p TMTpT] Hp(M) for any matrix M, and where the scaling matrix P determines the symmetrization strategy. Some popular choices for P are listed in Table 1. The resulting linear systems are now P Reference [X21 (X21 SX21 ) 21 X21 ] Nesterov and Todd (NT) [8]; X Monteiro [7] Kojima et al. 5] S Monteiro [7] Helmberg et al. 3] Kojima et al. 5] I Alizadeh, Haeberley and Overton (AHO) 1] Table 1: Choices for the scaling matrix P. solvable (for the AHO direction (P = I) solvability is only guaranteed if (X, S) lie in a 1 certain ....

Yu. Nesterov and M.J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22(1):1-42, 1997. 16

....grant DMS 0075722. 1 1 Introduction In recent years a theory of interior point methods for linear, semide nite, and second order cone programming was developed within the uni ed framework of self scaled conic programming. The origins of this theory can be traced to the work of Nesterov and Todd [15, 16]. G uler [4, 5] pointed out fundamental connections between the theory of self scaled optimization and the theory of Euclidean Jordan algebras. This work was mainly concerned with barrier functions and their relations to characteristic functions of homogeneous cones, and not with interior point ....

....function allows one to construct primaldual interior point methods in which the problems (P) and (D) are solved simultaneously. In each iteration of such an algorithm, primal and dual information is exchanged in a meaningful way, which leads to improved scaling of the search directions. In [15], Nesterov and Todd isolated two properties which are responsible for the above mentioned advantages of the barrier P n i=1 ln x i in the case of linear programming, and they generalised these properties axiomatically (see (1.4) and (1.5) below) They used the term self scaled barrier for ....

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Y. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research 22 (1997), 1-42.

....over real numbers complex numbers quaternions, and the cone of 3 3 semidefinite matrices over octonions. Toh and Trefethen showed that semidefinite programming over the complex domain is solvable in polynomial time [28] Guler [14] showed that the self scaled cones for which Nesterov and Todd [24, 25] had provided polynomial time algorithms are exactly the symmetric cones. We propose to take advantage of these developments by using semidefinite programming over the complex domain; we call this complex semidefinite programming (CSDP) Previously we modelled binary decisions with the square ....

Y. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22:1--42, 1997.

....conditions this active set is stable under small perturbations to the problem. See [5, 20] for active set al..gorithms. Of course, this particular problem could be rephrased with more of a semblance of smoothness, as a convex quadratic program, amenable to contemporary interior point techniques [16, 17, 1]. Nonetheless, as in linear programming, the active set is an important tool for understanding the problem. This phenomenon of nonsmoothness inducing a certain activity central to optimality conditions repeats many times throughout optimization. Consider the following examples. a) Classical ....

Y. Nesterov and M. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 33:1{ 43, 1998.

....appearing in the problem formulation are direct sums of cones of positive semidefinite symmetric real matrices and cones of the form epi(k k 2 ) See Example 4.5.1 in the next subsection. The standard barriers for these cones are self scaled, allowing for especially efficient algorithms, cmp. [68]. Note, however, that the use of a p norm with p 6= 2 (an important case in applications [72] does not allow for a self scaled cone, and that a self dual formulation for the corresponding problem is not readily at hand. Indeed, interior point methods proposed up to now for this class of problems ....

....epi( can be avoided by noting that b (x; t) ln(t 2 h x; Qx i) is a self concordant barrier for this cone with self concordancy parameter # B = 2 . Note that epi(k k 2 ) is just the standard second order cone, while b is the corresponding self scaled barrier for this cone, see [68]. 2 Example 4.5.2 (Polyhedral Gauges) Let be a polyhedral gauge whose unit ball is given by a set of k linear inequalities: B = fx 2 IR n j Ax gg , A 2 IR k n , g 2 IR k . Of course, the standard logarithmic barrier b B (x) P k i=1 ln(g i a i x) for the polytope B can be used to ....

Yu[rii] E. Nesterov and M[ichael] J. Todd. Self scaled barrier and interior-point methods for convex programming. Mathematics of Operations Research, 22:1--42, 1997. 110

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Yu. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22:1-42, 1997.

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Nesterov, Yu. E. and Todd, M. J. (1997) Self-scaled barriers and interiorpoint methods for convex programming, MathematicsofOperationsResearch, 22:142.

....is the adjoint transformation to A and K # : s # 0 for all x K is the cone dual to K. In the two cases above, K is self dual, so that K # = K (we have identified E and E # ) Given a possibly infeasible interior point (x, y, s) int K int K # , a primaldual IIP method (see, e.g. [19, 20, 21, 27, 2]) takes steps in the directions (#x, #y, #s) obtained from a linear system of the form A # #y #s = c # y = #g (6.34) for certain operators : E V and : E # V (V is another real vector space of the same dimension as E) and certain g, h V , depending on the current ....

....for SDP (see [8, 11, 17] has the identity, xvs 1 s 1 vx) 2, g x# n times s 1 , and h = x. It is easily seen that these choices satisfy the conditions of case (b) as well as the extra condition. Another instance of case (b) is the Nesterov Todd (NT) direction for SDP see [20, 21]. Here is the operator v wvw, where w : x [x sx ] 1 2 x is the unique positive definite matrix with wsw = x, and , g, and h are as above. Then, if ws w = x, it is easy to see that w = # #) w, so again the conditions are simple to check. The dual HRVW KSH M direction ....

Yu. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22:1--42, 1997.

....Wolkowicz et al. 67] We also mention that SDP is both an extension of LP and a special case of more general conic optimization problems. Nesterov and Nemirovski [44, 45] consider general convex cones, with the sole proviso that a self concordant barrier is known for the cone. Nesterov and Todd [46, 47] consider the subclass of self scaled cones, which admit symmetric primal dual algorithms (these cones turn out to coincide with symmetric (homogeneous self dual) cones) Another viewpoint is that of Euclidean Jordan Algebras, developed by Faybusovich [15, 16] and now investigated by a number of ....

....rediscovered from the perspective above by Monteiro [39] The second was also introduced by Kojima, Shindoh, and Hara [33] and rediscovered by Monteiro; since it arises by switching the roles of X and S, it is called the dual of the first direction. The last was introduced by Nesterov and Todd [46, 47], from yet another motivation, and shown to be derivable in this form by Todd, Toh, and Tutuncu [60] These and several other search directions are discussed in Kojima et al. 32] and Todd [59] In the first case, the Newton direction can be obtained from the solution of a linear system as in (10) ....

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Yu. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22:1--42, 1997.

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Y.E. NESTEROV and M.J. TODD. Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22(1):1-42, 1997.

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Y. E. Nesterov and M. J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22:1--42, 1997.

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Y.E. Nesterov and M.J. Todd. Self-Scaled Barriers and InteriorPoint Methods for Convex Programming. Math. of Oper. Res., 22:1-42, 1997.

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Yu. E. Nesterov and M. J. Todd. Self--scaled barriers and interior--point methods for convex programming. Mathematics of Operations Research, 22:1--42, 1997.

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Y. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research 22 (1997), no. 1, 1--42.

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Y. E. Nesterov and M. J. Todd, Self--scaled barriers and interior--point methods for convex programming, Mathematics of Operations Research 22 (1997), 1--42.

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Y. Nesterov and M. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res., 22:1-42, 1997.

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Yu. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research 22 (1997) 1-46.

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Y.E. NESTEROV and M.J. TODD. Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res., 22(1):1--42, 1997.

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Y. Nesterov and M.J. Todd. Self-scaled barriers and interior-point methods for convex programming. Mathematics of Operations Research, 22(1997) 1-42.

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