Why equality constraints are not a problem for Constrained optimization problems (some methods just ignore them and focus on inequalities)
Thank you!
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If you have a constraint $h(x) = 0$, it could be equivalently rewritten as two inequality constraints $h(x) \le 0$ and $h(x) \ge 0$ – Andrei Kulunchakov Feb 24 '17 at 21:24
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thanks! do you mean equality so to say reduces number of constraints? I'd like to ask why working with h(x)=0-like constraints much easier? – kangarooo Feb 24 '17 at 22:10
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I think the only reason why authors omit equality constraints is to ease notation for their theory – Andrei Kulunchakov Feb 24 '17 at 22:14
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1Actually using $h(x)\geq0$ and $h(x)\leq0$ instead of $h(x)=0$ leads to constraint qualification failure and inapplicability of the Kuhn-Tucker approach. In my view a proper treatment of constrained optimization will deal with both types of constraints. It will come at the cost of notation, but the two types of constrains are not equivalent. – Jan Feb 24 '17 at 22:46
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Basically every well-known theorem or algorithm handles equality different than inequality. Just in the introduction some authors drop the distinction in favor of notational convenience. But both types of constraints are fundamentally different. – user251257 Feb 25 '17 at 02:29