--- title: Optimizing over multinomial distributions | daily.dev description: When maximizing a concave differentiable function over a probability simplex (weights summing to 1, all non-negative), standard gradient ascent must be... canonical: https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Optimizing over multinomial distributions | daily.dev og:description: When maximizing a concave differentiable function over a probability simplex (weights summing to 1, all non-negative), standard gradient ascent must be... og:url: https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u og:image: https://api.daily.dev/og/posts/wv2VSxX2U.png og:image:alt: Optimizing over multinomial distributions og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Optimizing over multinomial distributions **[Erik Bernhardsson](https://daily.dev/sources/erikbernhardsson)** · 1 min read · 0 upvotes · 0 comments ## Summary When maximizing a concave differentiable function over a probability simplex (weights summing to 1, all non-negative), standard gradient ascent must be combined with a projection step back onto the constraint surface. Simple normalization by dividing by the sum is insufficient — an orthogonal projection is required to avoid negative values. A Python implementation of this projection algorithm is provided, iteratively distributing excess weight while clamping values at zero. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Similar posts on daily.dev - [A Gentle Introduction to Nonlinear Constrained Optimization with Piecewise Linear Approximations](https://daily.dev/posts/a-gentle-introduction-to-nonlinear-constrained-optimization-with-piecewise-linear-approximations-egnsu4dyn) · Towards Data Science · 1 upvotes · 0 comments - [Softmax, can you really derive the Jacobian? And should you care? — idlemachines](https://daily.dev/posts/softmax-can-you-really-derive-the-jacobian-and-should-you-care-idlemachines-nflk67kpf) · Hacker News · 0 upvotes · 0 comments --- Tags: [#machine-learning](https://daily.dev/tags/machine-learning), [#python](https://daily.dev/tags/python), [#gradient-descent](https://daily.dev/tags/gradient-descent) [View this post on daily.dev](https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Optimizing over multinomial distributions","url":"https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u"},"datePublished":"2026-03-26T08:35:21.934Z","dateModified":"2026-03-26T08:37:31.382Z","description":"When maximizing a concave differentiable function over a probability simplex (weights summing to 1, all non-negative), standard gradient ascent must be...","image":"https://media.daily.dev/image/upload/s--OHB84bZF--/f_auto/v1722860399/public/Placeholder%2010","thumbnailUrl":"https://media.daily.dev/image/upload/s--OHB84bZF--/f_auto/v1722860399/public/Placeholder%2010","isAccessibleForFree":true,"articleSection":"Erik Bernhardsson","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Erik Bernhardsson","logo":"https://media.daily.dev/image/upload/s--SlM88DTW--/f_auto,q_auto/v1774514112/logos/erikbernhardsson?_a=BAMAMiWQ0","url":"https://daily.dev/sources/erikbernhardsson"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/optimizing-over-multinomial-distributions-wv2vsxx2u","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"machine-learning,python,gradient-descent","timeRequired":"PT1M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Erik Bernhardsson","item":"https://daily.dev/sources/erikbernhardsson"},{"@type":"ListItem","position":3,"name":"Optimizing over multinomial distributions"}]} ```