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A solution approach for cardinality minimization problem based on fractional programming

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Abstract

This paper proposes a new algorithm for solving the linear Cardinality Minimization Problem (CMP). The algorithm relies on approximating the nonconvex and non-smooth function,\(card(x)\), with a linear fractional one. Therefore, the cardinality minimization problem is converted to the sum-of-ratio problem. The new model is solved with an optimization algorithm proposed for finding the optimal solution to the ratio problem. In the numerical experiments, we focus on two types of CMP problems with inequality and equality constraints. We provide a series of examples to evaluate the performance of the proposed algorithm, showing its efficiency.

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References

  • Abdi MJ (2013) Cardinality Optimization Problems. phd thesis,School of Mathematics The University of Birmingham

  • Beck A, Vaisbourd Y (2016) The sparse principal component analysis problem: optimality conditions and algorithms. J Optim Theory Appl 1:119–143

    Article  MathSciNet  MATH  Google Scholar 

  • Bertsimas D, King A, Mazumder R (2016) Best subset selection via a modern optimization lens. Ann Stat 44(2):813–852

    Article  MathSciNet  MATH  Google Scholar 

  • Bian W, Chen X (2020) A smoothing proximal gradient algorithm for nonsmooth convex regression with cardinality penalty. SIAM J Numer Anal 58(1):858–883

    Article  MathSciNet  Google Scholar 

  • Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev 51(1):34–81

    Article  MathSciNet  MATH  Google Scholar 

  • C. F. Wang and. X. Y. Chu, (2017) A new branch and bound method for solving sum of linear ratios problem. Int J Appl Math 47(3):276–281

    MathSciNet  Google Scholar 

  • Candès EJ, Wakin MB, Boyd SP (2008) Enhancing sparsity by reweighted l1 minimization. J Fourier Anal Appl 14:877–905

    Article  MathSciNet  MATH  Google Scholar 

  • Chen SS, Donoho DL, Saunders MA (1998) Atomic Decomposition by Basis. SIAM J Sci Comput 20(1):33–61

    Article  MathSciNet  MATH  Google Scholar 

  • Chen Z, Huang C, Lin S (2019) A new sparse representation framework for compressed sensing MRI. Knowl-Based Syst 188:1–10

    Article  Google Scholar 

  • Chen X, Zhou W (2014) Convergence of the reweighted ℓ 1 minimization algorithm for ℓ 2–ℓ p minimization. Comput Optim Appl 59:47–61

    Article  MathSciNet  Google Scholar 

  • Cui A, Peng J, Li H, Wen M (2019) Nonconvex fraction function recovery sparse signal by convex optimization algorithm. J Latex Class Files 31(5):1626–1637

    Google Scholar 

  • Dai W, Milenkovic O (2009) Subspace pursuit for compressive sensing signal reconstruction. IEEE Trans Inform Theory 55:2230–2249

    Article  MathSciNet  MATH  Google Scholar 

  • Dinh T, Xin J (2020) Convergence of a Relaxed Variable Splitting Method forLearning Sparse Neural Networks via l1, l0, andTransformed-l1 Penalties. arXiv:1812.05719

  • Donoho DL (1995) De-noising by soft-thresholdinng. IEEE Trans Inform Theory 41:613–627

    Article  MathSciNet  MATH  Google Scholar 

  • Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52(4):1289–1306

    Article  MathSciNet  MATH  Google Scholar 

  • Freund RW, Jarre F (2001) Solving the sum-of-ratios problem by an interior-point method. J Global Optim 19:83–102

    Article  MathSciNet  MATH  Google Scholar 

  • Ge D, Jiang X, Ye Y (2011) A note on the complexity of Lp minimization. Math Program 129:285–299

    Article  MathSciNet  MATH  Google Scholar 

  • Gotoh J-Y, Takeda A, Tono K (2017) DC formulations and algorithms for sparse optimization problems. Math Program 169(1):141–176

    Article  MathSciNet  MATH  Google Scholar 

  • Gruzdeva TV, Strekalovsky AS (2017) On solving the sum-of-ratios problem. Appl Math Comput 1–10

  • Gulpinar N, An LTH (2010) Robust investment strategies with discrete asset choice constraints using DC programming. J Math Program Oper Res 59:45–62

    MathSciNet  MATH  Google Scholar 

  • Jia P, Zhang M, Shen Y (2019) Hypergraph learning and reweighted ℓ1-norm minimization for hyperspectral unmixing. IEEE J Sel Top Appl Earth Observ Rem Sens 12(6):1898–1904

    Article  Google Scholar 

  • Jiao HW, Liu SY (2015) A practicable branch and bound algorithm for sum of linear ratios problem. Eur J Oper Res 243(3):723–730

    Article  MathSciNet  MATH  Google Scholar 

  • Jiao H, Liu S (2017) An efficient algorithm for quadratic sum-of-ratios fractional programs problem. Numer Funct Anal Optim 38(11):1426–1445

    Article  MathSciNet  MATH  Google Scholar 

  • Jokar S, Pfetsch M (2008) Exact and approximate sparse solutions of underdetermined linear equations. SIAM J Sci Comput 31:23–44

    Article  MathSciNet  MATH  Google Scholar 

  • Kim Y, Jong Y, Yu J (2021) A parametric solution method for a generalized fractional programming problem. Indian J Pure Appl Math

  • Lai M-J, Wang J (2011) An unconstrained lq minimization with 0<q<=1 for sparse solution of underdetermined linear system. SIAM J Optim 21(1):82–101

    Article  MathSciNet  MATH  Google Scholar 

  • Mangasarian O (1999) Minimum-support solutions of polyhedral concave programs. J Math Program Oper Res 45(1–4):149–162

    MathSciNet  MATH  Google Scholar 

  • Markovsky I, Huffel SV (2007) Overview of total least-squares methods. Signal Process 87(10):2283–2302

    Article  MATH  Google Scholar 

  • Mehranian A, Saligheh Rad H, Ay MR, Rahmim A (2013) Smoothly clipped absolute deviation (SCAD) regularization for compressed sensing MRI using an augmented Lagrangian scheme. In: IEEE nuclear science symposiwn and medical imaging conference record (NSS/MIC), pp 3646–3653

  • Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24(2):227–234

    Article  MathSciNet  MATH  Google Scholar 

  • Needell D, Tropp JA (2009) Iterative signal recovery from incomplete and inaccurate samples. Appl Comput Harmon Anal 26:301–321

    Article  MathSciNet  MATH  Google Scholar 

  • Selesnick I (2017) Sparse regularization via convex analysis. IEEE Trans Signal Process 65(17):4481–4494

    Article  MathSciNet  MATH  Google Scholar 

  • Shen P, Huang B (2019) Range division and linearization algorithm for a class of linear ratios optimization problems. J Comput Appl Math 350:324–342

    Article  MathSciNet  MATH  Google Scholar 

  • Shen PP, Lu T (2018) Regional division and reduction algorithm for minimizing the sum of linear fractional functions. J Inequal Appl 63:1

    MathSciNet  MATH  Google Scholar 

  • Shen P, Zhu Z, Chen X (2019) A practicable contraction approach for sum of the generalized polynomial ratios problem. Eur J Oper Res 278:36–48

    Article  MathSciNet  MATH  Google Scholar 

  • Shi Z, Zhou H, Xia Y, Liu W, Wang Y (2020) Random noise attenuation of common offset gathers using iteratively reweighted ℓ2,1 norm minimization. IEEE Geosci Remote Sens Lett 1:1–5

    Google Scholar 

  • Sun Y, Chen H, Tao J (2018) Sparse signal recovery via minimax-concave penalty and ℓ1-norm loss function. IET Signal Proc 12(9):1091–1098

    Article  Google Scholar 

  • Sun Y, Tan X, Li X, Lei L, kaung G (2019) Sparse optimization problem with s-difference regularization. Signal Process

  • Tibshirani R (1996) Regression Shrinkage and Selection via the Lasso. J R Stat Soc Ser B (methodological) 58(1):267–288

    MathSciNet  MATH  Google Scholar 

  • Voronin S, Woerdeman HJ (2013) A new iterative firm-thresholding algorithms for inverse problems with sparsity constraints. Appl Comput Harmonic Anal 35:151–164

    Article  MathSciNet  MATH  Google Scholar 

  • Yin P, Lou Y, He Q, Xin J (2015) Minimization of L1–2 for compressed sensing. SIAM J Sci Comput 37(1):A536–A563

    Article  MATH  Google Scholar 

  • Zeng X, Figueiredo MAT (2014) Decreasing weighted sorted l1 regularization. IEEE Signal Process Lett 21(10):1240–1244

    Article  Google Scholar 

  • Zhang T (2013) Multi-stage convex relaxation for feature selection. Bernoulli Soc Math Stat Probab 19(58):2277–2293

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Polytechnic Tehran), Tehran, Iran

    S. M. Mirhadi & S. A. MirHassani

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  1. S. M. Mirhadi
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  2. S. A. MirHassani
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Correspondence to S. A. MirHassani.

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Mirhadi, S.M., MirHassani, S.A. A solution approach for cardinality minimization problem based on fractional programming. J Comb Optim 44, 583–602 (2022). https://doi.org/10.1007/s10878-022-00847-0

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