How to solve problems like $\min_x f(x) \\ g(x) < 0 \ or \ h(x)<0$?
I think we can solve $\min_x f(x) \\ g(x) < 0 $ and $\min_x f(x) \\ h(x) < 0 $, then we compare the solutions from these problems. Is there any other way to solve this problem?
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$\begingroup$ It depends. What form do $f,g,h$ take? Are they linear? convex functions? boolean formula? What is the domain for $x$? integers? real numbers? 0 or 1? I encourage you to edit your question to provide more context. $\endgroup$– D.W.Commented May 30, 2023 at 18:48
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2 Answers
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The problem seems can be formulated as a Generalized Disjunctive Programming as follows where each function you mentioned can be assumed as a disjunct of the whole set.
\begin{split} \min\ Z = &\ f(x) \\ \text{s.t.} \\ &\ \bigvee_{i\in D_k} \left[ \begin{gathered} Y_{ik} \\ g(x) \leq 0 \\ h(x) \leq 0 \\ \end{gathered} \right] \quad k \in K\\ &\ \Omega(Y) = True \\ &\ Y \in \{True, False\}^{p}\\ \end{split}
further you can solve the problem by introducing the axillary binary indicator variables, $Y_{ik}$ to transform each disjunct to whose linear or non-linear (in)equality constraints.
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You could introduce binary variable $y$ and a large enough constant $M$. You can then modify your constraints in the following linear way:
\begin{align} \min_x f(x) \\ g(x) < M \cdot y \\ h(x) < M \cdot (1-y)\\ y \in \{0,1\} \end{align}
If $y=0$, $g(x)$ must be $<0$ and if $y=1$, $h(x)$ must be $<0$.
