[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86268-en":3,"doc-seo-86268-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86268,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","A Twin Gradient Method for Unconstrained Optimization","A new strategy for gradient-based unconstrained optimization is introduced through two parallel iterate sequences that cooperate via a Twin-Step principle. Instead of minimizing the objective separately, steplengths are chosen to minimize the Euclidean distance between the two simultaneously generated gradient-based processes at each iteration. Convergence of mutual distance is analyzed and shown to depend on the angle between search directions, with performance degrading near parallelism. A robustness hybrid, Twin-ABB min, switches to Adaptive Barzilai–Borwein when cooperation becomes ineffective, supported by strong numerical results.","arXiv :2607 . 11575v1 [math .NA] 13 Jul 2026  \nA Twin gradient method for unconstrained  \noptimization  \nAnna De Magistris 1,2 , Michiel E. Hochstenbach3 , and Gerardo Toraldo 1,2  \n1 Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Viale Lincoln 5, 81100 Caserta, Italy  \n2 Member of the INdAM research group GNCS  \n3 Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands  \nAbstract  \nWe propose a new strategy for gradient-based unconstrained optimization, involving two parallel sequences of iterates that cooperate to determine their stepsizes via a Twin-Step principle. Rather than minimizing the objective function individually, the algorithm selects steplengths that minimize the Euclidean distance between the two gradient based processes occurring simultaneously at each iteration. The theoretical analysis shows that the convergence of the mutual distance is governed by the angle between the search directions. In particular the effectiveness of the overall process degrades as the directions approach parallelism. To ensure robustness against collinearity, we introduce a hybrid framework, Twin-ABB min , which switches to the Adaptive Barzilai–Borwein method when the geometric cooperation becomes ineffective. Extensive and very promising numerical results evidence that the Twin phase creates favorable initial conditions for subsequent BB-type iterations.  \nKeywords: Unconstrained optimization, gradient method, Twin method, mutual step method.  \n1 Introduction  \nWe develop a new gradient method for unconstrained optimization problems  \n(1) min f (x),  \nx∈Rn  \nwhere f ∈ C 1 . Some theoretical results of the paper will need the assumption that ∇f is Lipschitz continuous with constant L. We will also show some additional theoretical results for the strictly convex quadratic optimization problem  \n(2) inRn f (x) = ~~1~~2 x⊤Ax − b⊤x,  \nwhere A ∈ Rn×n is symmetric positive definite with eigenvalues 0 \u003C λ 1 ≤ · · · ≤ λn, and b ∈ Rn. This problem is of paramount importance in a wide variety of applications, ranging from signal and image processing to machine learning and compressed sensing (see, e.g., [6, 17 , 24 , 27 , 29]) . Furthermore, the quadratic framework is an essential testing ground for algorithmic strategies that can be generalized to nonlinear, nonconvex, or large-scale optimization scenarios.  \nGradient-based methods of the form x k+1 = xk − αk ∇f(xk) are popular due to their simplicity, low computational cost per iteration, and minimal storage requirements. Although the Steepest Descent method guarantees global convergence and monotonic decrease of the objective function for a quadratic function, it is known to have slow convergence rates for ill-conditioned problems, as demonstrated by its well-known “zig–zag” behavior. To overcome the limitations of  \nSD while ensuring the simplicity of gradient iterations, significant research has focused on the selection of the stepsize. A breakthrough in this area has been achieved by Barzilai and Borwein (BB) [2], who propose a spectral stepsize derived from a two-point approximation of the secant equation:  \nαBkB1 = ss11sykk−11 , αBkB2 = sy11yykk−11 ,  \nwhere sk−1 = xk − xk−1 , yk−1 = gk − gk−1 . The BB methods generally offer far better performance compared to the SD method, despite not guaranteeing a monotonic decrease in the objective function. This success led to extensive research on spectral gradient methods, resulting in variants such as the Cyclic BB, Monotone Gradient methods, and Adaptive Barzilai–Borwein (ABB) strategies (see [8 , 22 , 31]) . In ABB we select, for a chosen τ ∈ (0 , 1) ,  \n􀀸  \nαAkBB = 􀀼􀀺  \nαBkB2  \nαBkB1  \nif ααB~~k~~BkB2B~~1~~ \u003C τ,  \notherwise.  \nThis is modified by Frassoldati et al. [13], introducing the ABB min strategy:  \n􀀸  \nαAkBB min = 􀀼􀀺  \nmin{αBB2j | j = max(1, k − Mα),..., k}αBkB1  \nif ααB~~k~~BkB2B~~1~~ \u003C τ, τ ∈ (0 , 1) otherwise  \nwhere Mα ","cbCaiozql3dM14UI","https://ap.wps.com/l/cbCaiozql3dM14UI","pdf",1729935,2,1,22,"English","en",105,"# Abstract\n# Introduction\n## Problem setup\n## Gradient methods and stepsize selection\n## Barzilai–Borwein and ABB strategies\n## Related acceleration and alternating stepsizes\n## Link to dualprocess methods and mutual step","[{\"question\":\"What problem does the Twin gradient method address?\",\"answer\":\"It targets unconstrained optimization problems of the form min f(x) over x in R^n using gradient-based iterations.\"},{\"question\":\"How are the two processes coupled to determine the stepsize?\",\"answer\":\"Two parallel gradient processes run simultaneously, and the steplength is selected to minimize the Euclidean distance between their iterates at each iteration (Twin-Step principle).\"},{\"question\":\"Why is a hybrid Twin-ABB min strategy introduced?\",\"answer\":\"When geometric cooperation becomes ineffective due to collinearity/near parallel search directions, the method switches to the Adaptive Barzilai–Borwein approach to maintain robustness and performance.\"}]",1784209932,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-twin-gradient-method-for-unconstrained-optimization","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-twin-gradient-method-for-unconstrained-optimization/86268/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the Twin gradient method address?","Question",{"text":75,"@type":76},"It targets unconstrained optimization problems of the form min f(x) over x in R^n using gradient-based iterations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the two processes coupled to determine the stepsize?",{"text":80,"@type":76},"Two parallel gradient processes run simultaneously, and the steplength is selected to minimize the Euclidean distance between their iterates at each iteration (Twin-Step principle).",{"name":82,"@type":73,"acceptedAnswer":83},"Why is a hybrid Twin-ABB min strategy introduced?",{"text":84,"@type":76},"When geometric cooperation becomes ineffective due to collinearity/near parallel search directions, the method switches to the Adaptive Barzilai–Borwein approach to maintain robustness and 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