Examples of ill-behaved central paths in convex optimization

Gilbert, Jean Charles; Gonzaga, Clóvis C.; Karas, Elizabeth W.

doi:10.1007/s10107-003-0460-0

Cited by 12 publications

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References 33 publications

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“…The barrier version of the interior point method was found for these problems to offer considerable advantage over the primal-dual method: for the rapidly decaying kernels used, the central paths (Gilbert et al 2005) taken by these methods are very different and sometimes very different in length.…”

Section: Discussionmentioning

confidence: 99%

Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions

Teal

Eccles²

2015

Inverse Problems

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The two most successful methods of estimating the distribution of nuclear magnetic resonance relaxation times from two dimensional data are data compression followed by application of the Butler-Reeds-Dawson algorithm, and a primal-dual interior point method using preconditioned conjugate gradient. Both of these methods have previously been presented using a truncated singular value decomposition of matrices representing the exponential kernel. In this paper it is shown that other matrix factorizations are applicable to each of these algorithms, and that these illustrate the different fundamental principles behind the operation of the algorithms. These are the rank-revealing QR (RRQR) factorization and the LDL factorization with diagonal pivoting, also known as the Bunch-Kaufman-Parlett factorization. It is shown that both algorithms can be improved by adaptation of the truncation as the optimization process progresses, improving the accuracy as the optimal value is approached. A variation on the interior method viz, the use of barrier function instead of the primal-dual approach, is found to offer considerable improvement in terms of speed and reliability. A third type of algorithm, related to the algorithm known as Fast iterative shrinkage-thresholding algorithm, is applied to the problem. This method can be efficiently formulated without the use of a matrix decomposition.

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Section: Discussionmentioning

confidence: 99%

Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions

Teal

Eccles²

2015

Inverse Problems

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“…The fact that problem (P f ) is equivalent to the linear optimization problem (P ′ f ) makes it possible to solve (P f ) in polynomial time, for example by using an interior point method [37,39,1,23].…”

Section: Solution Computationmentioning

confidence: 99%

On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery

Gilbert

2016

J Optim Theory Appl

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This paper first proposes another proof of the necessary and sufficient conditions of solution uniqueness in 1-norm minimization given recently by H. Zhang, W. Yin, and L. Cheng. The analysis avoids the need of the surjectivity assumption made by these authors and should be mainly appealing by its short length (it can therefore be proposed to students exercising in convex optimization). In the second part of the paper, the previous existence and uniqueness characterization is extended to the recovery problem where the ℓ 1 norm is substituted by a polyhedral gauge. In addition to present interest for a number of practical problems, this extension clarifies the geometrical aspect of the previous uniqueness characterization. Numerical techniques are proposed to compute a solution to the polyhedral gauge recovery problem in polynomial time and to check its possible uniqueness by a simple linear algebra test.

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Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems

Ernst

2018

Vietnam J. Math.

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Examples of ill-behaved central paths in convex optimization

Abstract: Abstract. This paper presents some examples of ill-behaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data.

Cited by 12 publications

References 33 publications

Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions

Adaptive truncation of matrix decompositions and efficient estimation of NMR relaxation distributions

On the Solution Uniqueness Characterization in the L1 Norm and Polyhedral Gauge Recovery

Apart Sets and Functions: an Application to the Stability of Penalized Optimization Problems

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