2c:null 2d:["$","title",null,{"children":"Properties of Convex functions | Aditya Mahajan | Observable"}] 32:[] 30:{"id":"52997853f351d3c4","slug":"properties-of-convex-functions","title":"Properties of Convex functions","thumbnail":"1e0b49d6ed72b8401d5fe0a1b3ab0d6cb3c3d7385c88f6bb428f9146b79615af","access_level":"public","create_time":"$D2022-01-20T10:48:37.365Z","update_time":"$D2022-01-20T11:36:14.325Z","view_time":null,"first_public_time":"$D2022-01-20T11:36:47.129Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"ff25aa15160a9464","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":"starter_2024"},"version":89,"publish_version":89,"role":null,"authored":false,"authors":[{"type":"individual","id":"bb2fa893d524052c","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":null}],"members":[{"type":"individual","id":"bb2fa893d524052c","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Properties of Convex functions","cells":[{"id":0,"value":"# Properties of Convex functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"f = function(s) { return (s-5)**2 }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"We first show that average of convex functions is convex","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":14,"value":"average_plot = Plot.plot({\n grid: true,\n y: { domain: [0, 25] },\n marginRight: 40,\n marginTop: 40,\n\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(points, {filter: d => d.value <= 25 && d.type == \"noise\", \n x:\"state\", y:\"value\", z:\"noise\", stroke:\"gray\"}),\n Plot.text(points, Plot.selectLast({\n filter: d => d.value <= 25 && d.type == \"noise\", \n x: \"state\",\n y: \"value\",\n z: \"noise\",\n text: d => \"w = \" + d.noise,\n textAnchor: \"start\",\n dx: 3\n })),\n Plot.line(points, {filter: d => d.value <= 25 && d.type == \"average\", \n x:\"state\", y:\"value\", stroke:\"red\", strokeWidth: 4}),\n Plot.text(points, Plot.selectLast({\n filter: d => d.value <= 25 && d.type == \"average\", \n x: \"state\",\n y: \"value\",\n z: \"noise\",\n text: \"type\",\n textAnchor: \"start\",\n fill:\"red\",\n dy: -10\n }))\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":80,"value":"minimum_plot = Plot.plot({\n grid: true,\n y: { domain: [0, 25] },\n marginRight: 60,\n marginTop: 40,\n\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(points, {filter: d => d.value <= 25 && d.type == \"noise\", \n x:\"state\", y:\"value\", z:\"noise\", stroke:\"gray\"}),\n Plot.text(points, Plot.selectLast({\n filter: d => d.value <= 25 && d.type == \"noise\", \n x: \"state\",\n y: \"value\",\n z: \"noise\",\n text: d => \"f(s +\" + d.noise + \")\",\n textAnchor: \"start\",\n dx: 3\n })),\n Plot.line(points, {filter: d => d.value <= 25 && d.type == \"minimum\", \n x:\"state\", y:\"value\", stroke:\"red\", strokeWidth: 4}),\n Plot.text(points, Plot.selectLast({\n filter: d => d.value <= 25 && d.type == \"minimum\", \n x: \"state\",\n y: \"value\",\n text: \"type\",\n textAnchor: \"start\",\n fill:\"red\",\n dy: -10\n }))\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"points = {\n const n = 100\n const Smax = 10\n var points = new Array()\n var W = [0, 1, 2 ]\n var Pw = [1/3, 1/3, 1/3]\n\n var idx = 0\n for( var i = 0; i < n; i++) {\n var s = Smax*i/n\n var sum = 0\n var min = 10000 // CHECK: Large positive number\n for(var w = 0; w < W.length; w++) {\n sum += f(s + W[w])\n min = Math.min(min, f(s+W[w]))\n points[idx++] = { state: s, value: f(s + W[w]), noise:W[w], type: \"noise\" }\n }\n points[idx++] = { state: s, value: sum/W.length, type: \"average\" }\n points[idx++] = { state: s, value: min, type: \"minimum\" }\n }\n return points\n }","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1"}} 31:"$30:body"