I am trying to find global minima of a multivariable vector function. The function is as follows,
$$V(\vec{d}_{1},\vec{d}_{2},\vec{d}_{3},\vec{d}_{4})=\bigg[\bigg(\frac{1}{|\vec{d}_{1} + \vec{d}_{2}|}\bigg)^{6}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{|\vec{d}_{2} + \vec{d}_{3}|}\bigg)^{6}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{|\vec{d}_{3} + \vec{d}_{4}|}\bigg)^{6}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{|\vec{d}_{1} + \vec{d}_{2}+\vec{d}_{3}|}\bigg)^{6}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{|\vec{d}_{2} + \vec{d}_{3}+\vec{d}_{4}|}\bigg)^{6}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{|\vec{d}_{1} + \vec{d}_{2}+\vec{d}_{3}+\vec{d}_{4}|}\bigg)^{6}-1 \bigg]^{2}$$ Here the vectors ,$\vec{d}_{1},\vec{d}_{2},\vec{d}_{3},\vec{d}_{4}$ are unit vectors and they are lying on a 2 dimensional plane . Now my question is in the above function, each term is positive and hence can I claim that minimum value for each term is just $0$? Can we minimize each term separately? will that give correct answer? Suggest me some softwares where I can find minimum of such many variable functions
Edit: I am re expressing this in terms of $\theta$, angular variables. so $$V(\theta_{1},\theta_{2},\theta_{3})=\bigg[\bigg(\frac{1}{2+2\cos(\theta_{1})}\bigg)^{3}-1 \bigg]^{2} +\bigg[\bigg(\frac{1}{2+2\cos(\theta_{2})}\bigg)^{3}-1 \bigg]^{2} +\bigg[\bigg(\frac{1}{2+2\cos(\theta_{3})}\bigg)^{3}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{3+2\cos(\theta_{1})+2\cos(\theta_{2})+2\cos(\theta_{1}+\theta_{2})}\bigg)^{3}-1 \bigg]^{2} + \bigg[\bigg(\frac{1}{3+2\cos(\theta_{2})+2\cos(\theta_{3})+2\cos(\theta_{2}+\theta_{3})}\bigg)^{3}-1 \bigg]^{2}+\bigg[\bigg(\frac{1}{4+2\cos(\theta_{1})+2\cos(\theta_{2})+2\cos(\theta_{3})+2\cos(\theta_{1}+\theta_{2})+2\cos(\theta_{2}+\theta_{3})+2\cos(\theta_{1}+\theta_{2}+\theta_{3})}\bigg)^{3}-1 \bigg]^{2}$$