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<article xsi:noNamespaceSchemaLocation="http://jats.nlm.nih.gov/publishing/1.1/xsd/JATS-journalpublishing1-mathml3.xsd" dtd-version="1.1" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><front><journal-meta><journal-id journal-id-type="publisher-id">LNE</journal-id><journal-title-group><journal-title>Lecture Notes in Education, Arts, Management and Social Science</journal-title></journal-title-group><issn>TBA</issn><eissn>2705-053X</eissn><publisher><publisher-name>WHIOCE PUBLISHING PTE. LTD.</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.18063/LNE.v3i8.858</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title>A New Projected Gradient Method for Unconstrained Problems</title><url>https://artdesignp.com/journal/LNE/3/8/10.18063/LNE.v3i8.858</url><author>LiuGuoyan</author><pub-date pub-type="publication-year"><year>2025</year></pub-date><volume>3</volume><issue>8</issue><history><date date-type="pub"><published-time>2025-09-26</published-time></date></history><abstract>In&amp;nbsp;this paper, we propose a new gradient projection method for problems without constraints. Based on the steepest descent method, there is a projection matrix P&amp;nbsp;constructed from the descending direction and the iterative direction [1&amp;ndash;4]. We gave a general proof of the convergence for the new projected gradient method in this paper. In numerical experiments using the CUTEr test problem library,&amp;nbsp;the new projected gradient method performs better than the classic fastest descent method.</abstract><keywords>Projection matrix,Global convergence,Unconstrained optimization</keywords></article-meta></front><body/><back><ref-list><ref id="B1" content-type="article"><label>1</label><element-citation publication-type="journal"><p>[1] Barzilai J, Borwein M, 1988, Two Point Step Gradient Methods. IMA,J.Numer.Anal, 8: 141&amp;ndash;148.
[2] Fridlander A, Martinez MJ, Molina B, et al., 1999, Gradient Method with Retards and Generalizations. SIAM J.Numer.Anal., 36: 275&amp;ndash;289.
[3] Raydan M, Svaiter BF, 2002, Relaxed Steepest Descent and Cauchy-Barzilai-Borwein Method. Compute Optimal Appl., 5: 167.
[4] Dai YH, Yuan Y, 2003, Alternate Minimization Gradient Method. IMA J.Anal. 23: 377&amp;ndash;393.
[5] Dai YH, 2006, The cyclic Barzilai-Borwein Method for Unconstrained Optimization. IMA Journal of Numerical Analysis, 26: 504&amp;ndash;627.
[6] Raydan M, 1993, On the Barzilai and Borwein Choice of Steplength for the Gradient Method. IMA J.Numer.Anal., 13: 321&amp;ndash;326.
[7] Raydan M, 1997, The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem, SIAM J. Optim., 7(1): 26&amp;ndash;33.
[8] Birgin EG, Chambouleyron I, Martinez JM, 1999, Estimation of the Optical Constants and the Thickness of Thin Films Using Unconstrained Optimization, J. Comput.Phys, 151: 862&amp;ndash;880.
[9] Birgin EG, Evtushenko YG, 1998, Automatic Differentiation and Spectral Projected Gradient Methods for Optimal Control Problems. Optim Methods Shoftw, 10, 125&amp;ndash;146.
[10] Birgin EG, Martinez JM, Mirada, 2000, Nonmonotone Spectral Projected Gradient Methods for Convex Sets. SIAM Journal on Optimization, 10(4): 1196&amp;ndash;1211.
[11] Dai YH, Liao LZ, n.d., R-Linear Convergence of the Barzilai and Borwein Gradient Method. Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences Research.
[12] Fletcher R, 1999, Low Storage Methods for Unconstrained Optimization. Lectures in Applied Mathematics (AMS), 26: 165&amp;ndash;179.
[13] Friedlander A, Martinez JM, Molina B, et al., 1999, Gradient Method with Retards and Generalization. J. Numer. Anal., 36, 275&amp;ndash;289.
[14] Raydan M, n.d., On the Barzilai and Borwein Choice of Step Length for the Gradient Method. J Anal., 13199: 321&amp;ndash;326.
[15] Yuan Y, 1991, A Modified BFGS Algorithm for Unconstrained Optimization. IMA J.Numer Anal, 11: 325&amp;ndash;332.
[16] Yuan Y, 1993, Numerical Methods for Nonlinear Programming, Shanghai Scientific and Technical Publishers.
[17] Wang W, 2002, Conjugate Hierarchy Algorithm and Its Convergence for Linear Constrained Optimization Problems. Journal of Natural Science of Northeast Normal University, 34(2): 12&amp;ndash;15.
[18] Ye L, 2005, A Memory Gradient Rosen Projection Algorithm Combining Conjugate Gradient Parameters for Linear or Nonlinear Constrained Optimization Problems. Journal of Sichuan University, 42(2): 652&amp;ndash;660.
[19] Liang Y, Jian J, 2003, A Conjugate Gradient Method for Linear Constrained Optimization. Operations Research and Management, 12(2): 31&amp;ndash;35.
[20] Calamai PH, More JJ, 1987, Projected Gradient Methods for Linearly Constrained Problems. Mathematical Programming, 39: 93&amp;ndash;116.
[21] Zhang XS, 1985, On the Convergence of Rosen&amp;rsquo;s Gradient Projection Method: Three-Dimensional, Case, Acta Mathematicae Applicate Sinica, 8: 125&amp;ndash;128.
[22] Du DZ, Remarks on the Convergence of Rosen&amp;rsquo;s Gradient Projection Method. MSRI Technique Report, 01718&amp;ndash;86.
[23] He G, 1987, Proof of Convergence of the Rosen&amp;rsquo;s Gradient Projection Method. Journal of Chengdu University of Science and Technology, 1(1987): 55&amp;ndash;68.
[24] Polak E, 1969, On the Convergence of Optimization Algorithms. Revue Francaised matique Infor et de Recherche Operrationelle, 3(16): 17&amp;ndash;34.
[25] Wang C, 1982, A Feasible Direction Method for Nonlinear Programming. Acta Mathematica Sinica, 25: 15&amp;ndash;19.
[26]&amp;nbsp;Bazaraa MS, Sherry CM, 1979, Nonlinear Programming: Theory and Algorithms, John Wiley &amp;amp; Sons, Inc.
[27] Grippo L, Ampariello FL, 1986, Lucidi A Nonmonotone Line Search Technique For Newton&amp;rsquo;s Method. Journal on Numerical SIAM Analysis, 23: 707&amp;ndash;716.
[28] Shi ZJ, 1996, A Class of Globally Convergent Conjugate Projection Gradient Method and Its Superlinear Convergence. Computational Mathematics, 11(4): 411&amp;ndash;421.
[29] Zhu Z, 2004, A New Conjugate Projection Gradient Algorithm and Its Superlinear Convergence, Chinese Journal of Applied Mathematics, 27(1): 149&amp;ndash;161.
[30] Liu WB, Dai YH, 1999, Minimization Algorithms Based on Supervisor and Searcher Co-Operation.I:--Faster and Robust Gradient Algorithms for Minimization Problems with Stronger Noises. Academy of Mathematic and Systems Sciences, Chinese Academy of Sciences.
[31] Hestenes MR, Stiefel EL, 1952, Methods of Conjugate Gradients for Solving Linear Systems. JRes Nat Bur Standards Sect. 5(49): 409&amp;ndash;436.
[32] Fletcher R, Reeves C, 1964, Function Minimization by Conjugate Gradients. Comput, 7: 149&amp;ndash;154.
[33] Toint PL, 1981, Towards an Efficient Sparsity Exploiting Newton Method for Minimization. Sparse Matrices and Their Uses, Academic Press London, England, 57&amp;ndash;88.
[34] Steihaug T, 1983, The Conjugate Gradient Method and Trust Region in Large Scale Optimization. SIAM Journal on Numerical Analysis, 20(1983): 626&amp;ndash;637.
[35] Dimitri P, Bertsekas DP, n.d., Projected Newton Methods For Optimization Problems With Simple Constraints SIAM J.Control and Optimization, 20(2): 2198.
[36] Hestenes MR, Stiefel EL, 1952, Methods of Conjugate Gradients for Solving Linear. JRes Nat Standards Sect, 49(5): 409&amp;ndash;436.
[37] Fletcher R, Reeves C, 1964, Function Minimization by Conjugate Gradients. Compute, 7: 149&amp;ndash;154.
[38] Bertsekas DP, 1976, On the Goldstein-Levitin-Polyak Gradient Projection Method. IEEE Transactions on Automatic Control, 21: 174&amp;ndash;184.
[39] Dai YH, Zhang HC, 2001, Adaptive Two-Point Stepsize Gradient Algorithm. Numerical Algorithms, 27: 377&amp;ndash;385.
[40] Zhang HC, Hager WW, 2004, A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization. SIAM Journal on Optimization, 14: 63&amp;ndash;85.</p><pub-id pub-id-type="doi"/></element-citation></ref></ref-list></back></article>
