# union of constraints

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?

• 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.
– D.W.
May 30 at 18:48

$$\begin{split} \min\ Z = &\ f(x) \\ \text{s.t.} \\ &\ \bigvee_{i\in D_k} \left[ \begin{gathered} Y_{ik} \\ g(x) \leq 0 \\ h(x) \leq 0 \\ \end{gathered} \right] \quad k \in K\\ &\ \Omega(Y) = True \\ &\ Y \in \{True, False\}^{p}\\ \end{split}$$
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.
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 $$<0$$.