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Handbook on Semidefinite, Conic and Polynomial Optimization pp 715–754Cite as

On the Implementation and Usage of SDPT3 – A Matlab Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

Abstract

This software is designed to solve primal and dual semidefinite-quadratic-linear conic programming problems (known as SQLP problems) whose constraint conic is a product of semidefinite conics, second-order conics, nonnegative orthants and Euclidean spaces, and whose objective function is the sum of linear functions and log-barrier terms associated with the constraint conics. This includes the special case of determinant maximization problems with linear matrix inequalities. It employs an infeasible primal-dual predictor-corrector path-following method, with either the HKM or the NT search direction. The basic code is written in Matlab, but key subroutines in C are incorporated via Mex files. Routines are provided to read in problems in either SDPA or SeDuMi format. Sparsity and block diagonal structure are exploited. We also exploit low-rank structures in the constraint matrices associated with the semidefinite blocks if such structures are explicitly given. To help the users in using our software, we also include some examples to illustrate the coding of problem data for our solver. Various techniques to improve the efficiency and robustness of the main solver are incorporated. For example, step-lengths associated with semidefinite conics are calculated via the Lanczos method. The current version also implements algorithms for solving a 3-parameter homogeneous self-dual model of the primal and dual SQLP problems. Routines are also provided to determine whether the primal and dual feasible regions of a given SQLP have empty interiors. Numerical experiments show that this general-purpose code can solve more than 80% of a total of about 430 test problems to an accuracy of at least 10− 6 in relative duality gap and infeasibilities.

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Notes

  1. 1.

    This strategy works well on most of the problems we tested. However, it should be noted that the occasional failure of the software on problems with poorly chosen initial iterates is likely due to the lack of a neighborhood enforcement in the algorithm.

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Author information

Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Blk S17, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Kim-Chuan Toh

  2. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA

    Michael J. Todd

  3. Quantitative Investment Strategies, Goldman Sachs Asset Management, New York, NY, 10282, USA

    Reha H. Tütüncü

Authors
  1. Kim-Chuan Toh
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  2. Michael J. Todd
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  3. Reha H. Tütüncü
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Correspondence to Kim-Chuan Toh .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Toh, KC., Todd, M.J., Tütüncü, R.H. (2012). On the Implementation and Usage of SDPT3 – A Matlab Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_25

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