# How to match my toy problem to primal dual explanatory formulas?

I am trying to understand weak and strong duality given a toy example LP program, which is to:

maximize $$m_D$$D + $$m_L$$L

s.t.
0.3 L + $$\ \ \ \ \ \ \ \$$D $$\leq$$ 1
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$ D $$\leq$$ 0.9
$$\ \ \ \ \$$ L + 0.25 D $$\leq$$ 1
0.7 L + 0.70 D $$\leq$$ 1

introducing slack variables:

maximize $$m_D$$D + $$m_L$$L + $$b_3y_3$$ + $$b_4y_4$$ + $$b_5y_5$$ + $$b_6y_6$$ (with $$b_3, b_4, b_5, b_6$$ = 0)

s.t.
0.3 L + $$\ \ \ \ \ \ \ \$$D + $$y_3$$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$= 1
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$ D $$\ \ \ \ \$$+$$y_4$$ $$\ \ \ \ \ \ \ \ \ \$$= 0.9
$$\ \ \ \ \$$ L + 0.25 D $$\ \ \ \ \ \ \ \ \ \$$+ $$y_5$$ $$\ \ \ \ \$$= 1
0.7 L + 0.70 D $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$+$$y_6$$ $$\$$= 1

I take the above as the dual because the primal (standard) is to $$minimize$$. However then I get too few equations in the primal because there are more coefficients ($$b$$) than constraints (in A) (?) and I don't know how or why the order between the dual and primal should correspond(?):

minimize $$x_1$$ + 0.9$$x_2$$ + $$x_3$$ + $$x_4$$

s.t.
0.3 $$x_1$$ + $$\ \ \ \ \ \ \ \ x_2$$ + $$x_3$$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$= $$m_D$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_2$$ $$\ \ \ \ \ \$$+ $$x_4$$ $$\ \ \ \ \ \ \ \ \ \ \$$= $$m_L$$
$$\ \ \ \ \ \ x_1$$ + 0.25 $$x_2$$ $$\ \ \ \ \ \ \ \ \ \ \ \$$+ $$x_5$$ $$\ \ \ \ \ \$$= $$b_3$$
0.7 $$x_1$$ + 0.70 $$x_2$$ $$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$$+ $$x_6$$ = $$b_4$$

(what about $$b_5$$ and $$b_6$$ and why would these first two equations pair with $$m_D$$ and $$m_L$$?)

What I did was to match my toy problem to a textbook primal dual formulaic description to derive the above.

• You should probably look at the indicator variables in the augmented form. – Julius Baer Oct 22 '19 at 19:25