Properties of optimization problem (convex, linear?)

Question

I have a set of optimization problems whose core is described as follows:

$A_{n\times q}.W_{q\times 1}=R$

$\sum_{1}^q w_i=1$

$w_i\geq 0, i=1...q$

where $A$ is given. We need to determine the $W$. The optimization problems could be minimizing the sum of R (i.e. $\min(\sum_1^n r_i)$) or maximizing the minimum element, i.e. $\max(minimum(r_i)_{i=1...n})$.

I'm looking for the properties of these problems, including linearity, convexity, etc. And if a Conjugate Gradient method or ILP solvers could be helpful.

Answer 1

Your problem seems to be a linear optimization (LO) problem. For the objective of minimizing the elements that's obvious, for maximizing the minimum, the formulation is given by: $$ \begin{align} \max_{z,w,r} \quad & z \\ \text{s.t.} \quad & Aw=r \\ & w^T e = 1 \\ & z \leq r_i \\ & w \geq 0 \end{align} $$ Free LO solvers like Clp or GLPK can solve this problem.