Subtracting vectors to make a vector zero or negative while using as least vectors as possible

Question

I have the following linear algebra math problem:

Make vector a(a1,a2..a100) zero (or negative) by subtracting vectors like this: b(b1,b2...b100) c(c1,c2...c100) d(d1,d2..d100) and 400 vectors like this.

Each vector can only used once and the objective is to use as minimum vectors as possible.

How can I make vector a zero (or negative) by subracking at least as possible other vectors? The subtracting vector can only be used once.

Greetings, Allard

Answer 1

First beginning with $N_0 = 400, M = 100, v = (v_1,v_s, \cdots, v_M)$ and defining

$$ v^0 = (v_1^0,v_2^0,\cdots,v_M^0) \ \ \ \hbox{the reference vector} $$

$$ \sigma(v^0, v^k) = \max_{j \in M}\{v_j^0-v_j^k, 0\} $$

second choosing

$$ k^{*} = \arg\min_{k\in N_{k-1}}\sigma(v^0,v_k) $$

excluding $k^{*}$ from $N_{k-1}$ hence $N_k = N_{k-1}-\{k^{*}\}$

This loop follows until $N_{k} = \{\}$ or $\min_{k\in N_{k-1}}\sigma(v^0,v_k) = 0$