{"id":4814,"date":"2022-12-19T17:25:46","date_gmt":"2022-12-19T09:25:46","guid":{"rendered":"https:\/\/fanyuzhao.com\/?p=4814"},"modified":"2022-12-19T19:29:31","modified_gmt":"2022-12-19T11:29:31","slug":"kkt-for-optimisation","status":"publish","type":"post","link":"https:\/\/fanyuzhao.com\/?p=4814","title":{"rendered":"KKT for Optimisation"},"content":{"rendered":"\n

Optimisation<\/h4>\n\n\n\n

Suppose f, g_1, g_2, …, g_m<\/span> are differentiable.<\/p>\n\n\n\n

$$ \\max_{x_1, x_2,…, x_n} f(x_1, x_2, …, x_n) $$<\/p>\n\n\n\n

$$s.t. g_1(x_1, x_2, …, x_n)\\geq 0 $$<\/p>\n\n\n\n

$$\\quad g_2(x_1, x_2, …, x_n)\\geq 0 $$<\/p>\n\n\n\n

$$\\quad ……$$<\/p>\n\n\n\n

$$\\quad g_m(x_1, x_2, …, x_n)\\geq 0 $$<\/p>\n\n\n\n

There are m<\/span> constraints g_i, i \\in (1,m)<\/span>, and one object function f<\/span> to maximum. n<\/span> variables. <\/p>\n\n\n\n

KKT Conditions<\/h4>\n\n\n\n

Suppose the constraint qualification is satisfied. <\/strong>A local maximiser of the above problem \\bar{x} = (\\bar{x_1}, \\bar{x_2},… \\bar{x_n})<\/span> together with some \\bar{lambda_1}, \\bar{lambda_2}, …, \\bar{lambda_m} \\geq 0<\/span> satisfies the following:<\/p>\n\n\n\n

$$ \\frac{\\partial f}{\\partial x_i}(\\bar{x}) + \\lambda_1 \\frac{\\partial g_1}{\\partial x_i}(\\bar{x})+ \\lambda_2 \\frac{\\partial g_2}{\\partial x_i}(\\bar{x}) +…+ \\lambda_{m} \\frac{\\partial g_m}{\\partial x_i}(\\bar{x}) = 0$$<\/p>\n\n\n\n

for i=1,2,…,n<\/span>, and<\/strong> <\/p>\n\n\n\n

$$\\bar{\\lambda_k} g_k(\\bar{x})=0$$<\/p>\n\n\n\n

for k=1,2,…,m<\/span><\/p>\n\n\n\n