On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming

Ji, Jun; Potra, Florian A.; Sheng, Rongqin

doi:10.1137/s1052623497316828

Cited by 18 publications

(15 citation statements)

References 32 publications

(31 reference statements)

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“…For example, superlinear and quadratic convergence results for interior-point methods depend on the strict complementarity assumption, e.g. [43,30,3,37,32]. This is also the case for convergence of the central path to the analytic center of the optimal face, [27].…”

Section: Auxiliary Problem and Regularizationmentioning

confidence: 99%

Strong duality and minimal representations for cone optimization

Tunçel

Wolkowicz

2012

Comput Optim Appl

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The elegant results for strong duality and strict complementarity for linear programming, LP , can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primal-dual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones.We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs ; duality and characterizations of optimality that hold without any CQ ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQ and failure of strict complementarity.One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ . We include a discussion on obtaining these representations efficiently. A parallel theme is the loss of strict complementarity and the corresponding theoretical and numerical difficulties that arise; a discussion on avoiding these difficulties is included. We include results and examples on the surprising theoretical connection between duality, strict complementarity, and nonclosure of sums of closed cones.Our emphasis is on results that deal with Semidefinite Programming, SDP .

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Section: Auxiliary Problem and Regularizationmentioning

confidence: 99%

Strong duality and minimal representations for cone optimization

Tunçel

Wolkowicz

2012

Comput Optim Appl

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“…Strict complementarity is a desirable property of an SDP instance; in fact the strict complementarity of an optimal solution is a necessary condition for superlinear convergence of interior-point methods that take Newton-like steps, see [16], and much recent research has explored what conditions in addition to strict complementarity are needed to guarantee superlinear convergence for different interior-point algorithms [9,10,11,12,15]. However, even for linear programming (which must have a strictly complementary solution), there are instances for which the optimal solutions are nearly non-strictly complementary, and can be made arbitrarily badly so.…”

Section: Non-strict Complementaritymentioning

confidence: 99%

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

Freund¹,

Ordóñez

Toh

2006

Math. Program.

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We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.

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“…Four particular choices for P and M are: P = M = I, giving the Alizadeh-Haeberly-Overton (AHO) direction [3] [13,19]; and P = W −1/2 , M = W −1 , where W is the scaling matrix of (12), giving the Nesterov-Todd (NT) direction [27,28,31].…”

Section: The Monteiro-zhang and Monteiro-tsuchiya Familiesmentioning

confidence: 99%

“…Convergence results for long-step algorithms can be found in Monteiro [19] (who proves a bound of O(n 3/2 ln(1/ )) iterations for two particular search directions), in Monteiro and Zhang [22] (who extend the results of [19] and establish a bound of O(n ln(1/ )) iterations for another search direction) and Monteiro and Tsuchiya [21] (who give a bound of O(n 3/2 ln(1/ )) iterations for all directions in a subclass of the Monteiro-Tsuchiya family). Ji, Potra, and Sheng [12] describe the literature on local convergence and prove convergence of Q-order 1.5 or 2 for predictor-corrector algorithms using certain search directions from the Monteiro-Zhang family. Nor do we discuss the amount of computational work involved in computing our search directions in much detail; see Monteiro and Zanjacomo [24] and Toh [32] for flop counts for some of these directions.…”

Section: Introductionmentioning

confidence: 99%

A study of search directions in primal-dual interior-point methods for semidefinite programming

Todd

1999

Optimization Methods and Software

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We discuss several different search directions which can be used in primal-dual interior-point methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primal-dual symmetry, and whether they always generate well-defined directions. Among the directions satisfying all but at most two of these desirable properties are the Alizadeh-Haeberly-Overton, HelmbergRendl-Vanderbei-Wolkowicz/Kojima-Shindoh-Hara/Monteiro, Nesterov-Todd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also.

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On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming

Abstract: An interior point method for monotone linear complementarity problems acting in a wide neighborhood of the central path is presented. The method has O( √ nL)-iteration complexity and is superlinearly convergent even when the problem does not possess a strictly complementary solution.

Cited by 18 publications

References 32 publications

Strong duality and minimal representations for cone optimization

Strong duality and minimal representations for cone optimization

Behavioral measures and their correlation with IPM iteration counts on semi-definite programming problems

A study of search directions in primal-dual interior-point methods for semidefinite programming

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