Optimization over structured subsets of positive semidefinite matrices via column generation

Ahmadi, Amir Ali; Dash, Sanjeeb; Hall, Georgina

doi:10.1016/j.disopt.2016.04.004

Cited by 31 publications

(59 citation statements)

References 33 publications

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“…Note that the SOCP in particular fills up almost the entire spectrahedron in a single iteration. [11] showing the successive improvement on the dd (left) and sdd (right) inner approximation of the feasible set of a random SDP via five iterations of the column generation method.…”

Section: Iterative Change Of Basismentioning

confidence: 99%

See 1 more Smart Citation

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Majumdar

Hall

Ahmadi

2020

Annu. Rev. Control Robot. Auton. Syst.

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Historically, scalability has been a major challenge to the successful application of semidefinite programming in fields such as machine learning, control, and robotics. In this paper, we survey recent approaches for addressing this challenge including (i) approaches for exploiting structure (e.g., sparsity and symmetry) in a problem, (ii) approaches that produce low-rank approximate solutions to semidefinite programs, (iii) more scalable algorithms that rely on augmented Lagrangian techniques and the alternating direction method of multipliers, and (iv) approaches that trade off scalability with conservatism (e.g., by approximating semidefinite programs with linear and second-order cone programs). For each class of approaches we provide a high-level exposition, an entry-point to the corresponding literature, and examples drawn from machine learning, control, or robotics. We also present a list of software packages that implement many of the techniques discussed in the paper. Our hope is that this paper will serve as a gateway to the rich and exciting literature on scalable semidefinite programming for both theorists and practitioners.

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Section: Iterative Change Of Basismentioning

confidence: 99%

“…In [11], the authors design another iterative method for inner approximating the set of psd matrices using linear and second order cone programming. Their approach combines DSOS/SDSOS techniques with ideas from the theory of column generation in large-scale linear and integer programming.…”

Section: Column Generationmentioning

confidence: 99%

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Majumdar

Hall

Ahmadi

2020

Annu. Rev. Control Robot. Auton. Syst.

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“…In [2], SDSOS relaxations were proposed using SDD matrices. A matrix B ∈ S n is SDD if and only if it can be expressed as…”

Section: Lp Relaxationsmentioning

confidence: 99%

“…It is well-known that any DD matrix is SDD, therefore, D n ⊂ SD n ⊂ S n + holds. ReplacingS ∈ S n+1 + in (14) byS ∈ SD n+1 corresponds to the first level of the hierarchy of SDSOS relaxation for QCQPs in [2], and it is the dual of (11):…”

Section: Lp Relaxationsmentioning

confidence: 99%

Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods

Kimizuka

Kim²,

Yamashita

2019

J Glob Optim

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The pooling problem is an important industrial problem in the class of network flow problems for allocating gas flow in pipeline transportation networks. For P-formulation of the pooling problem with time discretization, we propose second order cone programming (SOCP) and linear programming (LP) relaxations and prove that they obtain the same optimal value as the semidefinite programming relaxation. The equivalence among the optimal values of the three relaxations is also computationally shown. Moreover, a rescheduling method is proposed to efficiently refine the solution obtained by the SOCP or LP relaxation. The efficiency of the SOCP and the LP relaxation and the proposed rescheduling method is illustrated with numerical results on the test instances from the work of Nishi in 2010, some large instances, and Foulds 3, 4, 5 test problems.

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“…1) Column generation method [41]: For simplicity, we present here the linear programming-based version of the algorithm. An analogous method based on second-order cone programming can be found in [41]. To understand this method, the following characterization of diagonally dominant matrices is needed [44]: A symmetric matrix M is diagonally dominant if and only if it can be written as…”

Section: B Improving On Dsos and Sdsos Programmingmentioning

confidence: 99%

Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations

Ahmadi

Hall

Papachristodoulou

et al. 2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

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It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.

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Optimization over structured subsets of positive semidefinite matrices via column generation

Cited by 31 publications

References 33 publications

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Recent Scalability Improvements for Semidefinite Programming with Applications in Machine Learning, Control, and Robotics

Solving pooling problems with time discretization by LP and SOCP relaxations and rescheduling methods

Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations

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