One way is to rewrite your constraints such that you get a penalty for being infeasible.

For a constraint of the form g_i(x)

This is a very good question.   In addition to the excellent answer by @Michael Patriksson, there is a bit more to add.

An overview of the convex feasibility problem is given in

Quasi-nonexpansivity and the convex feasibility problem

https://www.researchgate.net/publication/265900279_Quasi-nonexpansivity_and_the_convex_feasibility_problem

I requested the full-text for paper.   As soon as it arrive, I will post the link for the paper.

Another helpful source on the convex feasibility problem is

Algorithms for the quasiconvex feasibility problem

https://www.researchgate.net/publication/222406850_Algorithms_for_the_quasiconvex_feasibility_problem

I have also requested the full-text for this paper.

More to the point, see

From implicit convex feasibility to convex minimization

https://www.researchgate.net/publication/311396135_From_implicit_convex_feasibility_to_convex_minimization

See page 61 for an explanation.

Article Quasi-nonexpansivity and the Convex Feasibility Problem.

Article Algorithms for the quasiconvex feasibility problem

Article From implicit convex feasibility to convex minimization