I am considering minpack software package to solve my optimization problem ( this is the kind of question that I am facing), but I don't quite know what is the memory requirement and the speed of this package.
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Don't use MINPACK. It's over 30 years old, and better, more modern optimization software exists out there. More importantly, I've searched the MINPACK source code and perused the documentation (the PDFs are scanned images, and can't be searched), I don't see any options to accommodate the constraints in your problem.
It's not clear to me that your problem is convex. However, before considering global optimization algorithms, I'd try more modern optimization packages. The standard open-source constrained nonlinear optimization package is IPOPT, which has an excellent reputation in the optimization community. Its authors (primarily Andreas Wächter and Carl Laird) have won an award from INFORMS based on the quality of the software. It's probably overkill for your particular problem, but if you have any interest in solving similar case studies that are larger, it's well worth your while to use IPOPT.
There are also several other solvers out there. Arnold Neumaier has a web page listing many solvers; you should also check the solvers used by GAMS and AMPL, since Professor Neumaier's list does not contain some of those solvers. Since your problem is a constrained nonlinear least squares problem, you should also consider solvers that exploit that structure as well.
GAMS is an excellent modeling language for optimization problems; if you just need to solve the optimization problem, or you want to do some prototyping, it is a very good tool. The trial license should suffice for your small problem.
Given the size of your example, it's unlikely that any optimization package will have trouble with your problem with respect to memory and processing power requirements. Memory will definitely not be an issue, and processing power could only be an issue in cases where the problem is nonconvex and has nonzero local minima, or if your objective function is expensive to evaluate. Given that you're considering solving an optimization problem with this objective function using MINPACK, I'm going to assume that the objective function is not expensive to evaluate.
However, for large examples, more modern packages like IPOPT take advantage of sparsity, whereas MINPACK does not. Taking advantage of sparsity in the optimization algorithm reduces memory requirements and generally decreases the time it takes to solve a given problem.