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I was trying to find a name for the following model; is it mathematical programming, constraint programming, convex optimization, but as I can see, none of them has a continuous parameter $t$ like in the following example.

Let $x_i\ge 0$ ($i=1,...,n$) be real variables that we need to determine, $a_{ij},b_{ij},c_{ij}, d_i$ be given real constants. What is the type of the following model: $$min\sum_{i=1}^{n} x_i$$ s.t. $$a_{ij}+b_{ij}(t+x_i)^2+c_{ij}(t+x_j)^2\ge \epsilon, \forall t\in[min\{x_i,x_j\},max\{x_i+d_i,x_j+d_j\}], \forall i,j: i\ne j ?$$

To me, it looks like constraint programming, but there is this continuous parameter $t$. Also, the variables $x_i$ are in the segments for $t$.

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  • $\begingroup$ Are $b$ and $c$ nonnegative? That would allow a major simplification. $\endgroup$ – prubin Nov 27 '20 at 18:30
  • $\begingroup$ The constants are real numbers, i.e. they can be positive or negative. $\endgroup$ – HD2000 Nov 30 '20 at 8:33
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I would say that "mathematical program" and "optimization" would apply. Constraint programming (CP) is designed primarily for use with discrete variables. It can cope to some extent with continuous variables, but I doubt this problem would fit the CP paradigm very well. Whether the qualifier "convex" applies is hard to say, particularly without knowing the signs of $b$ and $c$.

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  • $\begingroup$ The way mathematical programming is defined (encyclopediaofmath.org/wiki/Mathematical_programming) leaves no room for parameter "t". $\endgroup$ – HD2000 Nov 26 '20 at 10:53
  • $\begingroup$ That definition is 10 years old. In recent years, there has been a merging of logic-based constraints into math programming. A good example is the indicator constraint, for instance $x = 0 \implies f(y) \ge 0$, which commercial MIP solvers now allow (when $f()$ is linear or perhaps quadratic) but which the 2010 definition excludes (barring a reformulation using binary variables). $\endgroup$ – prubin Nov 27 '20 at 18:25

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