Would you please explain to me what is the technical definition or explanation behind the point that, in a convex optimization program or space, the equality constraint must be affine.
A simple explanation:
Let h(x) == constant be the equality constraint in the optimization problem and x is the optimization variable. The above equality constraint can be rewritten in terms of inequality constraints as follows:
h(x) = constant -- (2). If both (1) and (2) are satisfied then it is equivalent to the h(x)== constant.
If h(x) is a convex function then (1) is a convex constraint. However, (2) is not a convex constraint anymore. Similar argument holds when h(x) is a concave function. Thus, in order for both (1) and (2) to be a convex constraint h(x) has to be an affine function.
Dear Alireza,
I am totaly agree with Nathan, i add that the study of an optimization problem is called a program, the programming denomination being variously qualified to signify a specific framework, with its particular conditions and methods: linear programming when the functions at work are affine, convex programming for functions, or their opposite, convex. (equality or inequality defined be affine functions)
For more details about this subject i suggest you to see links in topic.
-Convex Optimization - SIAM
www.siam.org/books/mo19/MO19_ch8.pdf
-Alternative ways to formulate optimization problems - AASS
www.aass.oru.se/Research/Learning/drdv_dir/.../formulations.pdf
-Convex Optimization - University at Albany
https://www.albany.edu/~mark/classes/591/Boyd Convex_Optimization.pdf
Best regards
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