You have to check that whether your objective function f(x) is convex or not. The function is convex if  f(ax+(1-a)y)

Dear Dr Seyed,

Thank you for your reply. The objective function is convex.

The problem of checking whether a function is convex is NP-hard. In fact, it is co-NP-complete even when the function is a quartic polynomial. See:

http://link.springer.com/article/10.1007%2Fs10107-011-0499-2

Fortunately, there are some simple sufficient conditions for convexity that work well for many functions of interest. See the book "Convex Optimization" by Boyd and Vandenberghe.  You can get it for free online:

http://stanford.edu/~boyd/cvxbook/

 Dear Professor Letchford,

Thank you very much. I appreciate your answer.

One sub-question might be whether the problem is always convex in the continuous variables, i.e., when all the binary/integer variables are fixed to arbitrary feasible values. If, for example, you know something about the sign of the expressions associated with the integer/binary variables then you might be able to say that the continuous part of the problem is convex for any fixed feasible integer/binary variable values. Is that possible? 

 Dear Prof Patriksson,

Thank you for your answer. In my case, setting the binary variables at fixed values will be an assumption to the model.

 Dear Dr Hassan,

Thank you very much for your reply. 

The objective function is convex. I tried to derive the constraints, but it is not easy to derive different sets of constraints with four different vectors. Some of the constraints are of the type '=' and the others are of the type '='. What I have understood is I have to check the definiteness of the whole system.

I am attaching a snapshot of the model to have a look. 

Variables:

y, v, u >=0

H is binary

The first set of constraints looks non-convex to me, since they are quadratic equations.  You might be able to linearise them, though, by introducing additional variables.  See, e.g., these papers:

http://www.tandfonline.com/doi/abs/10.1080/02522667.1984.10698780

http://www.sciencedirect.com/science/article/pii/S1572528604000210

Dear Prof Letchford,

Dear Dr Hijazi,

Thank you very much, I really appreciate your answers.

 Dear Dr Hassan,

Now I am implementing your solution, but I couldn't understand this construction. For example, why do you use v_{max} with the new variable z_n?

Thank you in advance.

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