Formulate $\min \ |a-x| + |b-y | + |c-z|$ $s.t. \ x+y+z = 3d$ Replace $x = 3d - y - z$, you have $\min |a-3d + y+ z| + |b-y| + |c-z|$. Linearize it by replacing: $\min t_1 + t_2 + t_3 \\ s.t. \ -... View answer Accepted answer 2 votes There are great theories behind most of the metaheuristics. Otherwise, it is not reliable to use them. One example is Simulated Annealing. It is approaching the true optimum if you increase the ... View answer Accepted answer 1 votes The optimal solution is at$x = [1.7,1.9]^T$, so your answer is correct. Also, the second coordinate (I believe this is$x_2$) is greater than the first one. Maybe "they shall differ at least 1" ... View answer 1 votes Firstly, please notice that the link you shared is not saying a point$x_i$satisfying the first set satisfies the second. Instead, it says that if you write such a classification in the$1^{st}$set, ... View answer Accepted answer 0 votes Ok, I solved. here they show that$(\sum\lambda_j g_j)^* =(\lambda_1g_1)^* \Delta (\lambda_2g_2)^*\Delta \ldots $where$\Delta$is the infimal convolution operator. Next, apply the definition of inf ... View answer 0 votes I just found a very helpful Wikipedia article, easy to understand: https://en.wikipedia.org/wiki/Integration_by_substitution#Application_in_probability View answer Accepted answer 0 votes First of all, it is hard to understand your main question. 'Simple but not simple' is not reflecting your wish to solve a nonconvex problem. I assume your variables are continuous, e.g.$\tau \in \...

Let $x_i,$ denote the type of alloy for all $i=1,2,\ldots,8$. For example, $x_1=iron, x_2=carbon$ etc. $UB_i$ denotes the upper bound possible for production of material $i$ (in percentage), $LB_i$ ...

Note that this is not a convex optimization model (check the hessian of the objective function). So in order to find the optimal point you may check all the KKT solutions. I am not writing them one by ...

They are not equivalent. There is no assumption about $b$. So $⟨a,x_i⟩+b$ can be i.e. $0.5$ when $y_i = 1$ and not fulfill the second condition. I think you should give more details about $a$ and $b$ ...

Solved it. Grötschel and Padberg (1979) proved that $0 \leq x \leq 1$ defines facets of the TSP polytope when n=5. This fact and redundant subtour elimination constraint gives the perfect formulation....