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Solving Mixed Integer Nonlinear Programming Problem

There is a problem I want to solve. The function is:   

\begin{align} &\underset{a,b,\textbf{vec}}{\text{minimize} \text{ }\text{ }\text{ }\text{ } } f=\sum_{i=1}^{b}(\textbf{vec}_i)^{a}\\ &\text{subject to } \text{ }\text{ } f \geq F, \text{ } \sum_{i=1}^b \textbf{vec}_i = D, \text{ } a\leq N, b \leq D, \{\textbf{vec}_1,\cdots, \textbf{vec}_b\}\leq D \end{align} Basically

Basically, all the variables are non-negative values. I have two questions: (1) because the size of $\textbf{vec}$ is dependent on another parameters $b$, then the number of variables in the objective function are not fixed beforehand. Does it still a MINLP problem? (2) How to solve it? is there some open-source software can deal with it easily?

  1. Because the size of $\textbf{vec}$ is dependent on another parameters $b$, then the number of variables in the objective function are not fixed beforehand. Does it still a MINLP problem?

  2. How to solve it? is there some open-source software can deal with it easily?

Solving Mixed Integer Nonlinear Programming Problem

There is a problem I want to solve. The function is:  \begin{align} &\underset{a,b,\textbf{vec}}{\text{minimize} \text{ }\text{ }\text{ }\text{ } } f=\sum_{i=1}^{b}(\textbf{vec}_i)^{a}\\ &\text{subject to } \text{ }\text{ } f \geq F, \text{ } \sum_{i=1}^b \textbf{vec}_i = D, \text{ } a\leq N, b \leq D, \{\textbf{vec}_1,\cdots, \textbf{vec}_b\}\leq D \end{align} Basically, all the variables are non-negative values. I have two questions: (1) because the size of $\textbf{vec}$ is dependent on another parameters $b$, then the number of variables in the objective function are not fixed beforehand. Does it still a MINLP problem? (2) How to solve it? is there some open-source software can deal with it easily?

Mixed Integer Nonlinear Programming Problem

There is a problem I want to solve. The function is: 

\begin{align} &\underset{a,b,\textbf{vec}}{\text{minimize} \text{ }\text{ }\text{ }\text{ } } f=\sum_{i=1}^{b}(\textbf{vec}_i)^{a}\\ &\text{subject to } \text{ }\text{ } f \geq F, \text{ } \sum_{i=1}^b \textbf{vec}_i = D, \text{ } a\leq N, b \leq D, \{\textbf{vec}_1,\cdots, \textbf{vec}_b\}\leq D \end{align}

Basically, all the variables are non-negative values. I have two questions:

  1. Because the size of $\textbf{vec}$ is dependent on another parameters $b$, then the number of variables in the objective function are not fixed beforehand. Does it still a MINLP problem?

  2. How to solve it? is there some open-source software can deal with it easily?

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Solving Mixed Integer Nonlinear Programming Problem

There is a problem I want to solve. The function is: \begin{align} &\underset{a,b,\textbf{vec}}{\text{minimize} \text{ }\text{ }\text{ }\text{ } } f=\sum_{i=1}^{b}(\textbf{vec}_i)^{a}\\ &\text{subject to } \text{ }\text{ } f \geq F, \text{ } \sum_{i=1}^b \textbf{vec}_i = D, \text{ } a\leq N, b \leq D, \{\textbf{vec}_1,\cdots, \textbf{vec}_b\}\leq D \end{align} Basically, all the variables are non-negative values. I have two questions: (1) because the size of $\textbf{vec}$ is dependent on another parameters $b$, then the number of variables in the objective function are not fixed beforehand. Does it still a MINLP problem? (2) How to solve it? is there some open-source software can deal with it easily?