[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86276-en":3,"doc-seo-86276-105":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},86276,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Copositive Characterizations of Convex Hull Pricing","Wholesale electricity market unit commitment cannot generally be supported by any uniform linear energy pricing scheme due to nonconvex binary commitment constraints. Convex hull pricing (CHP) and copositive duality pricing (CDP) address this by deriving prices from relaxations of the unit commitment problem. The work defines a centralized convex hull price over the joint feasible set and proves, under non-degeneracy, its coincidence with the marginal CDP price. Numerical experiments on the Scarf example confirm the equivalence and quantify the semidefinite restriction’s induced gap.","Copositive Characterizations of Convex Hull Pricing  \nMadhusudan Ghosh 1 , Antoine Lesage-Landry2 , and Joshua Adam Taylor 1  \n1Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark, NJ, USA  \n2Department of Electrical Engineering, Polytechnique Montreal, GERAD, and Mila, Montreal, QC, Canada  \n[Email: mg2262@njit.edu](Email: mg2262@njit.edu), [antoine.lesage-landry@polymtl.ca](antoine.lesage-landry@polymtl.ca), [jat94@njit.edu](jat94@njit.edu)  \narXiv :2607 . 11590v1 [math .OC] 13 Jul 2026  \nAbstract—Due to the nonconvex binary constraints of unit commitment (UC), no uniform linear pricing scheme supports the optimal dispatch. Convex hull pricing (CHP) and copositive duality pricing (CDP) both address this problem. CHP derives the price from the subgradient of the value function of the convex hull relaxation of UC. CDP refers to several different pricing mechanisms that can be constructed from the dual multipliers of the completely positive programming reformulation. In this work, we define a centralized convex hull price over the joint feasible set of UC and prove that, under non-degeneracy, it coincides with the marginal copositive duality price. Numerical experiments on the Scarf example validate this equivalence and quantify the pricing gap introduced by the semidefinite restriction.  \nIndex Terms—Unit commitment, convex hull pricing, copositive programming, electricity markets, semidefinite relaxation  \nI. INTRODUCTION  \nWholesale electricity markets schedule generation by solving a unit commitment (UC) problem. The commitment variables are binary, which makes UC a mixed-integer linear program (MILP) with a non-convex value function [1] . A uniform set of energy prices therefore cannot, in general, both efficiently clear the market and leave every generator revenue adequate [2] . Operators close this gap with side payments, called uplift. Uplift covers a generator’s cost not recovered by its market revenue.  \nConvex hull pricing (CHP), introduced by Hogan and Ring [3] and formalized by Gribik, Hogan, and Pope [4], sets a uniform price that minimizes the total uplift payment. The price is a subgradient of the UC value function, evaluated over its convex hull relaxation. The convex hull relaxation is typically taken over the individual generator feasible sets. This admits a decentralized pricing interpretation and leads to somewhat lighter computations [5] .  \nIn a separate line of work, Guo, Bodur, and Taylor [6] introduced copositive duality pricing (CDP) . The UC problem is lifted into a completely positive program (CPP) using thereformulation of Burer [7] . The pricing mechanisms are constructed from the dual multipliers of the associated copositive program (COP) .  \nThis paper connects CHP and CDP. We define a centralized convex hull price over the joint feasible set of UC, rather than the per-generator convexification used in the literature.  \nThis work was supported by the National Science Foundation under Grant No. 2422849, and the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Alliance grant ALLRP 590233-23 .  \nWe prove that this price and the copositive duality price are subgradients of a common value function, and that they agree wherever that function is differentiable. At commitment transitions, where the value function is nondifferentiable the two prices may differ, though they share a common subgradient. We further show that the copositive duality price decomposes into two components, one linear in demand and one from the lift. Then we give a copositive formulation of the standard decentralized price, together with a more tractable semidefinite programming (SDP) restriction. Numerical experiments on Scarf’s example confirm the equivalence and measure the gap left by the restriction.  \nThe remainder of the paper is organized as follows. Section II defines the UC problem and its value function. Section III develops the convex hull price. ","cbCaicJTvyOBmd8P","https://ap.wps.com/l/cbCaicJTvyOBmd8P","pdf",1915120,3,1,5,"English","en",105,"# Introduction\n# Unit Commitment\n# Convex Hull Pricing\n# Copositive Duality Pricing\n# Equivalence of CHP and CDP\n# Semidefinite Restriction\n# Copositive Formulation of Decentralized CHP\n# Numerical Results","[{\"question\":\"Why can’t a uniform linear pricing scheme guarantee optimal dispatch in unit commitment?\",\"answer\":\"Unit commitment uses binary commitment decisions, making the optimization nonconvex. This prevents any single uniform linear pricing rule from both clearing the market efficiently and ensuring adequate generator revenues.\"},{\"question\":\"How does convex hull pricing (CHP) determine energy prices?\",\"answer\":\"CHP sets a uniform price as a subgradient of the unit commitment value function, evaluated using the convex hull relaxation. It aims to minimize total uplift payments.\"},{\"question\":\"What relationship is proven between the centralized convex hull price and copositive duality pricing (CDP)?\",\"answer\":\"The paper proves that, under non-degeneracy, the centralized convex hull price coincides with the marginal copositive duality price. At nondifferentiable commitment transitions, the prices may differ but share a common subgradient.\"}]",1784209975,13,{"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},"copositive-characterizations-of-convex-hull-pricing","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/copositive-characterizations-of-convex-hull-pricing/86276/",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","2026-07-27","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},"Why can’t a uniform linear pricing scheme guarantee optimal dispatch in unit commitment?","Question",{"text":75,"@type":76},"Unit commitment uses binary commitment decisions, making the optimization nonconvex. This prevents any single uniform linear pricing rule from both clearing the market efficiently and ensuring adequate generator revenues.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does convex hull pricing (CHP) determine energy prices?",{"text":80,"@type":76},"CHP sets a uniform price as a subgradient of the unit commitment value function, evaluated using the convex hull relaxation. It aims to minimize total uplift payments.",{"name":82,"@type":73,"acceptedAnswer":83},"What relationship is proven between the centralized convex hull price and copositive duality pricing (CDP)?",{"text":84,"@type":76},"The paper proves that, under non-degeneracy, the centralized convex hull price coincides with the marginal copositive duality price. 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