ABSTRACT
As we have seen, optimization problems deal with finding the maximum or minimum of a function, called the objective function, subject to certain prescribed constraints. As opposed to concave programming, we minimize a given objective function with or without constraints in convex programming. Thus, given functions f , g 1 , … , g m $ f, g_1, \ldots , g_m $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_1.tif"/> and h 1 , … , h k $ h_1, \ldots , h_k $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_2.tif"/> defined on some domain D ⊂ R n $ D\subset { \mathbb R }^n $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_3.tif"/> , the minimization problem is stated as follows: Determine min x ∈ D f ( x ) $ \min \limits _{\mathbf{x}\in D} f(\mathbf{x}) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_4.tif"/> subject to the constraints g i ( x ) ≤ 0 $ g_i(\mathbf{x})\le 0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_5.tif"/> for all i = 1 , … , m $ i=1, \ldots , m $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_6.tif"/> and all h j ( x ) = 0 $ h_j(\mathbf{x})=0 $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_7.tif"/> for all j = 1 , … , k $ j=1, \ldots , k $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315202259/344878fe-3325-45a4-b7d8-5c91138061f2/content/inline-math5_8.tif"/> .
