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

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • Kim-Chuan Toh

    (National University of Singapore)

  • Michael J. Todd

    (Cornell University)

  • Reha H. Tütüncü

    (Quantitative Investment Strategies, Goldman Sachs Asset Management)

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.

Suggested Citation

  • Kim-Chuan Toh & Michael J. Todd & Reha H. Tütüncü, 2012. "On the Implementation and Usage of SDPT3 – A Matlab Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 715-754, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_25
    DOI: 10.1007/978-1-4614-0769-0_25
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    Citations

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    Cited by:

    1. Kurt M. Anstreicher, 2018. "Maximum-entropy sampling and the Boolean quadric polytope," Journal of Global Optimization, Springer, vol. 72(4), pages 603-618, December.
    2. Sungwoo Park & Dianne P. O’Leary, 2015. "A Polynomial Time Constraint-Reduced Algorithm for Semidefinite Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 558-571, August.
    3. Dawen Yan & Xiaohui Zhang & Mingzheng Wang, 2021. "A robust bank asset allocation model integrating credit-rating migration risk and capital adequacy ratio regulations," Annals of Operations Research, Springer, vol. 299(1), pages 659-710, April.
    4. Kristijan Cafuta, 2019. "Sums of Hermitian squares decomposition of non-commutative polynomials in non-symmetric variables using NCSOStools," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 397-413, June.
    5. Kurt M. Anstreicher, 2020. "Efficient Solution of Maximum-Entropy Sampling Problems," Operations Research, INFORMS, vol. 68(6), pages 1826-1835, November.
    6. He, Qie & Zhang, Xiaochen & Nip, Kameng, 2017. "Speed optimization over a path with heterogeneous arc costs," Transportation Research Part B: Methodological, Elsevier, vol. 104(C), pages 198-214.

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