{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-06T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":524,"title":"Sequential Unconstrained Minimization (SUMT) using Interior Penalty","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_ ( _x_ ), given a function handle to _f_, and a starting guess, _x0_, subject to inequality constraints _g_ ( _x_ )\u003c=0 with function handle _g_. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function, _P_, is\r\n\r\n P(x,r) = -sum(log(-g(x)))/r\r\n\r\nwhere _r_ is the penalty parameter.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e ), given a function handle to \u003ci\u003ef\u003c/i\u003e, and a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality constraints \u003ci\u003eg\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e )\u0026lt;=0 with function handle \u003ci\u003eg\u003c/i\u003e. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function, \u003ci\u003eP\u003c/i\u003e, is\u003c/p\u003e\u003cpre\u003e P(x,r) = -sum(log(-g(x)))/r\u003c/pre\u003e\u003cp\u003ewhere \u003ci\u003er\u003c/i\u003e is the penalty parameter.\u003c/p\u003e","function_template":"function [x,fmin]=sumt_interior(f,g,x0,penalty_parameter)\r\nif nargin\u003c4 || isempty(penalty_parameter)\r\n   penalty_parameter=? % initialize penalty parameter values for SUMT loop\r\nend\r\n% You may find that fminsearch is not accurate enough for the unconstrained minimization","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 5;\r\n[xmin,fmin]=sumt_interior(f,g,x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nassert(norm(xmin-xcorrect)\u003c2e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\n\r\n%%\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 5;\r\n[xmin,fmin]=sumt_interior(f,g,x0,1) % 1 iteration for unit penalty value\r\nxr1=4;\r\nassert(norm(xmin-xr1)\u003c1e-4)\r\nassert(abs(fmin-f(xr1))\u003c1e-4)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [3; 3];\r\n[xmin,fmin]=sumt_interior(f,g,x0,1) % 1 iteration\r\nxr2=[1.8686\r\n     2.1221];\r\nassert(norm(xmin-xr2)\u003c1e-2)\r\nassert(abs(fmin-f(xr2))\u003c1e-2)\r\n\r\n%%\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [2.1; 0.1];\r\nr  = 2^12;\r\n[xmin,fmin]=sumt_interior(f,g,x0,r) % Final iteration\r\nxcorrect=[2; 0];\r\nassert(norm(xmin-xcorrect)\u003c5e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c2e-3)","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-24T03:23:26.000Z","updated_at":"2025-12-10T16:18:46.000Z","published_at":"2012-03-24T18:41:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality constraints\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e )\u0026lt;=0 with function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ P(x,r) = -sum(log(-g(x)))/r]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the penalty parameter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":523,"title":"Sequential Unconstrained Minimization (SUMT) using Exterior Penalty","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_, given a function handle to _f_, a starting guess, _x0_, subject to inequality and equality constraints with function handles _g_\u003c=0 and _h_=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e, given a function handle to \u003ci\u003ef\u003c/i\u003e, a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality and equality constraints with function handles \u003ci\u003eg\u003c/i\u003e\u0026lt;=0 and \u003ci\u003eh\u003c/i\u003e=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.\u003c/p\u003e","function_template":"function [x,fmin]=sumt_exterior(f,g,h,x0,penalty_parameter)\r\nif isempty(g), g=@(x)[]; end\r\nif isempty(h), h=@(x)[]; end\r\nif nargin\u003c5 || isempty(penalty_parameter)\r\n   penalty_parameter=? % initialize penalty parameter values for SUMT loop\r\nend\r\n% You may use fminsearch for the unconstrained minimization","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\n\r\n%%\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0,1) % 1 iteration for unit penalty value\r\nxr1=1.75;\r\nassert(norm(xmin-xr1)\u003c1e-4)\r\nassert(abs(fmin-f(xr1))\u003c1e-4)\r\n\r\n%% Haftka \u0026 Gurdal, Example 5.7.1\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\n[xmin,fmin]=sumt_exterior(f,[],h,x0)\r\nxcorrect=[40; 4]/11;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\n\r\n%%\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\nr  = [1, 5];\r\n[xmin,fmin]=sumt_exterior(f,[],h,x0,r) % 2 iterations\r\nxr2=[3.0769\r\n     0.3077];\r\nassert(norm(xmin-xr2,inf)\u003c1e-4)\r\nassert(abs(fmin-f(xr2))\u003c1e-4)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [0; 0];\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0)\r\nxcorrect=[2; 0];\r\nassert(norm(xmin-xcorrect)\u003c1e-4)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\n\r\n%%\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [5; 5];\r\nr  = [1, 2];\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0,r) % 2 iterations\r\nxr2=[1.9536\r\n    -0.0496];\r\nassert(norm(xmin-xr2)\u003c1e-4)\r\nassert(abs(fmin-f(xr2))\u003c1e-4)","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-24T03:18:31.000Z","updated_at":"2025-12-10T14:27:22.000Z","published_at":"2012-03-24T03:18:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality and equality constraints with function handles\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;=0 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":527,"title":"Augmented Lagrange Multiplier (ALM) Method","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_ ( _x_ ), given a function handle to _f_, and a starting guess, _x0_, subject to inequality and equality constraints with function handles _g_\u003c=0 and _h_=0, respectively. Use the Augmented Lagrangian Multiplier Method (a.k.a., the Method of Multipliers) and return the estimate of Lagrange multipliers _u_ for inequalities, and _v_ for equality constraints.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e ), given a function handle to \u003ci\u003ef\u003c/i\u003e, and a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality and equality constraints with function handles \u003ci\u003eg\u003c/i\u003e\u0026lt;=0 and \u003ci\u003eh\u003c/i\u003e=0, respectively. Use the Augmented Lagrangian Multiplier Method (a.k.a., the Method of Multipliers) and return the estimate of Lagrange multipliers \u003ci\u003eu\u003c/i\u003e for inequalities, and \u003ci\u003ev\u003c/i\u003e for equality constraints.\u003c/p\u003e","function_template":"function [x,fmin,u,v]=alm(f,g,h,x0,MaxIter)\r\nif isempty(g), g=@(x)0; end\r\nif isempty(h), h=@(x)0; end\r\nif nargin\u003c5, MaxIter=20; end","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin,u]=alm(f,g,[],x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nucorrect=0.5;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\nassert(abs(u-ucorrect)\u003c1e-3)\r\n\r\n%% Haftka \u0026 Gurdal, Example 5.7.1\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\n[xmin,fmin,~,v]=alm(f,[],h,x0)\r\nxcorrect=[40; 4]/11;\r\nvcorrect=-7.2727;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\nassert(abs(v-vcorrect)\u003c1e-2)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [0; 0];\r\n[xmin,fmin,u]=alm(f,g,[],x0)\r\nxcorrect=[2; 0];\r\nucorrect=[0.2; 0.6];\r\nassert(norm(xmin-xcorrect)\u003c1e-4)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\nassert(norm(u-ucorrect)\u003c1e-2)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2012-03-25T02:49:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-25T02:46:55.000Z","updated_at":"2025-09-01T07:55:40.000Z","published_at":"2012-03-25T02:49:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality and equality constraints with function handles\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;=0 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=0, respectively. Use the Augmented Lagrangian Multiplier Method (a.k.a., the Method of Multipliers) and return the estimate of Lagrange multipliers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eu\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for inequalities, and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ev\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e for equality constraints.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":524,"title":"Sequential Unconstrained Minimization (SUMT) using Interior Penalty","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_ ( _x_ ), given a function handle to _f_, and a starting guess, _x0_, subject to inequality constraints _g_ ( _x_ )\u003c=0 with function handle _g_. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function, _P_, is\r\n\r\n P(x,r) = -sum(log(-g(x)))/r\r\n\r\nwhere _r_ is the penalty parameter.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e ), given a function handle to \u003ci\u003ef\u003c/i\u003e, and a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality constraints \u003ci\u003eg\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e )\u0026lt;=0 with function handle \u003ci\u003eg\u003c/i\u003e. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function, \u003ci\u003eP\u003c/i\u003e, is\u003c/p\u003e\u003cpre\u003e P(x,r) = -sum(log(-g(x)))/r\u003c/pre\u003e\u003cp\u003ewhere \u003ci\u003er\u003c/i\u003e is the penalty parameter.\u003c/p\u003e","function_template":"function [x,fmin]=sumt_interior(f,g,x0,penalty_parameter)\r\nif nargin\u003c4 || isempty(penalty_parameter)\r\n   penalty_parameter=? % initialize penalty parameter values for SUMT loop\r\nend\r\n% You may find that fminsearch is not accurate enough for the unconstrained minimization","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 5;\r\n[xmin,fmin]=sumt_interior(f,g,x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nassert(norm(xmin-xcorrect)\u003c2e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\n\r\n%%\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 5;\r\n[xmin,fmin]=sumt_interior(f,g,x0,1) % 1 iteration for unit penalty value\r\nxr1=4;\r\nassert(norm(xmin-xr1)\u003c1e-4)\r\nassert(abs(fmin-f(xr1))\u003c1e-4)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [3; 3];\r\n[xmin,fmin]=sumt_interior(f,g,x0,1) % 1 iteration\r\nxr2=[1.8686\r\n     2.1221];\r\nassert(norm(xmin-xr2)\u003c1e-2)\r\nassert(abs(fmin-f(xr2))\u003c1e-2)\r\n\r\n%%\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [2.1; 0.1];\r\nr  = 2^12;\r\n[xmin,fmin]=sumt_interior(f,g,x0,r) % Final iteration\r\nxcorrect=[2; 0];\r\nassert(norm(xmin-xcorrect)\u003c5e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c2e-3)","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-24T03:23:26.000Z","updated_at":"2025-12-10T16:18:46.000Z","published_at":"2012-03-24T18:41:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality constraints\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e )\u0026lt;=0 with function handle\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. Use a logarithmic interior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of increasing penalty parameter values. That is, the penalty (barrier) function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ P(x,r) = -sum(log(-g(x)))/r]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the penalty parameter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":523,"title":"Sequential Unconstrained Minimization (SUMT) using Exterior Penalty","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_, given a function handle to _f_, a starting guess, _x0_, subject to inequality and equality constraints with function handles _g_\u003c=0 and _h_=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e, given a function handle to \u003ci\u003ef\u003c/i\u003e, a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality and equality constraints with function handles \u003ci\u003eg\u003c/i\u003e\u0026lt;=0 and \u003ci\u003eh\u003c/i\u003e=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.\u003c/p\u003e","function_template":"function [x,fmin]=sumt_exterior(f,g,h,x0,penalty_parameter)\r\nif isempty(g), g=@(x)[]; end\r\nif isempty(h), h=@(x)[]; end\r\nif nargin\u003c5 || isempty(penalty_parameter)\r\n   penalty_parameter=? % initialize penalty parameter values for SUMT loop\r\nend\r\n% You may use fminsearch for the unconstrained minimization","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\n\r\n%%\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0,1) % 1 iteration for unit penalty value\r\nxr1=1.75;\r\nassert(norm(xmin-xr1)\u003c1e-4)\r\nassert(abs(fmin-f(xr1))\u003c1e-4)\r\n\r\n%% Haftka \u0026 Gurdal, Example 5.7.1\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\n[xmin,fmin]=sumt_exterior(f,[],h,x0)\r\nxcorrect=[40; 4]/11;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\n\r\n%%\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\nr  = [1, 5];\r\n[xmin,fmin]=sumt_exterior(f,[],h,x0,r) % 2 iterations\r\nxr2=[3.0769\r\n     0.3077];\r\nassert(norm(xmin-xr2,inf)\u003c1e-4)\r\nassert(abs(fmin-f(xr2))\u003c1e-4)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [0; 0];\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0)\r\nxcorrect=[2; 0];\r\nassert(norm(xmin-xcorrect)\u003c1e-4)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\n\r\n%%\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [5; 5];\r\nr  = [1, 2];\r\n[xmin,fmin]=sumt_exterior(f,g,[],x0,r) % 2 iterations\r\nxr2=[1.9536\r\n    -0.0496];\r\nassert(norm(xmin-xr2)\u003c1e-4)\r\nassert(abs(fmin-f(xr2))\u003c1e-4)","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":12,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-24T03:18:31.000Z","updated_at":"2025-12-10T14:27:22.000Z","published_at":"2012-03-24T03:18:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality and equality constraints with function handles\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;=0 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=0, respectively. Use a quadratic exterior penalty for the sequential unconstrained minimization technique (SUMT) with an optional input vector of penalty parameter values that become increasingly larger.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":527,"title":"Augmented Lagrange Multiplier (ALM) Method","description":"Write a function to find the values of a design variable vector, _x_, that minimizes a scalar objective function, _f_ ( _x_ ), given a function handle to _f_, and a starting guess, _x0_, subject to inequality and equality constraints with function handles _g_\u003c=0 and _h_=0, respectively. Use the Augmented Lagrangian Multiplier Method (a.k.a., the Method of Multipliers) and return the estimate of Lagrange multipliers _u_ for inequalities, and _v_ for equality constraints.","description_html":"\u003cp\u003eWrite a function to find the values of a design variable vector, \u003ci\u003ex\u003c/i\u003e, that minimizes a scalar objective function, \u003ci\u003ef\u003c/i\u003e ( \u003ci\u003ex\u003c/i\u003e ), given a function handle to \u003ci\u003ef\u003c/i\u003e, and a starting guess, \u003ci\u003ex0\u003c/i\u003e, subject to inequality and equality constraints with function handles \u003ci\u003eg\u003c/i\u003e\u0026lt;=0 and \u003ci\u003eh\u003c/i\u003e=0, respectively. Use the Augmented Lagrangian Multiplier Method (a.k.a., the Method of Multipliers) and return the estimate of Lagrange multipliers \u003ci\u003eu\u003c/i\u003e for inequalities, and \u003ci\u003ev\u003c/i\u003e for equality constraints.\u003c/p\u003e","function_template":"function [x,fmin,u,v]=alm(f,g,h,x0,MaxIter)\r\nif isempty(g), g=@(x)0; end\r\nif isempty(h), h=@(x)0; end\r\nif nargin\u003c5, MaxIter=20; end","test_suite":"%% Haftka \u0026 Gurdal, Figure 5.7.1 example\r\nf = @(x) 0.5*x;\r\ng = @(x) 2-x;\r\nx0 = 0;\r\n[xmin,fmin,u]=alm(f,g,[],x0) %#ok\u003c*NOPTS\u003e\r\nxcorrect=2;\r\nucorrect=0.5;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-3)\r\nassert(abs(u-ucorrect)\u003c1e-3)\r\n\r\n%% Haftka \u0026 Gurdal, Example 5.7.1\r\nf = @(x) x(1).^2 + 10*x(2).^2;\r\nh = @(x) sum(x)-4;\r\nx0 = [0; 0];\r\n[xmin,fmin,~,v]=alm(f,[],h,x0)\r\nxcorrect=[40; 4]/11;\r\nvcorrect=-7.2727;\r\nassert(norm(xmin-xcorrect)\u003c1e-3)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\nassert(abs(v-vcorrect)\u003c1e-2)\r\n\r\n%% Vanderplaats, Figure 5-4 example\r\nf = @(x) sum(x);\r\ng = @(x) [x(1) - 2*x(2) - 2\r\n          8 - 6*x(1) + x(1).^2 - x(2)];\r\nx0 = [0; 0];\r\n[xmin,fmin,u]=alm(f,g,[],x0)\r\nxcorrect=[2; 0];\r\nucorrect=[0.2; 0.6];\r\nassert(norm(xmin-xcorrect)\u003c1e-4)\r\nassert(abs(fmin-f(xcorrect))\u003c1e-4)\r\nassert(norm(u-ucorrect)\u003c1e-2)","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":279,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2012-03-25T02:49:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-25T02:46:55.000Z","updated_at":"2025-09-01T07:55:40.000Z","published_at":"2012-03-25T02:49:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to find the values of a design variable vector,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, that minimizes a scalar objective function,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), given a function handle to\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a starting guess,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, subject to inequality and equality constraints with function handles\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e\u0026lt;=0 and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eh\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e=0, respectively. 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