dual linear programming problem

For a given linear programming problem (LP1)

• minimize
  • $c_1^T{x}_1+c_2^T{x}_2$
• subject to
  • $A_{11}{x}_1+A_{12}{x}_2\ge b_1$
  • $A_{21}{x}_1+A_{22}{x}_2=b_2$
  • $x_1\ge 0$ The following linear programming problem (LP2) is called a dual problem
• maximize
  • $b_1^T{y}_1+b_2^T{y}_2$
• subject to
  • $A_{11}^T{y}_1+A_{21}^T{y}_2\le c_1$
  • $A_{12}^T{y}_1+A_{22}^T{y}_2=c_2$
  • $y_1\ge 0$ Since the maximize is changed and the direction of inequality is changed, we have to invert the sign and align it with LP1, like this
• minimize
  • $-b_1^T{y}_1-b_2^T{y}_2$
• subject to
  • $-A_{11}^T{y}_1-A_{21}^T{y}_2\ge -c_1$
  • $-A_{12}^T{y}_1-A_{22}^T{y}_2=-c_2$
  • $y_1\ge 0$

The correspondence between the variables LP1 and LP2 is as follows, so we can see that this transformation can be done twice to get back to the original Variable support

LP1	x1	x2		c1	c2	b1	b2	A11	A12	A21	A22
LP2	y1	y2		-b1	-b2	-c1	-c2	-A11^T	-A21^T	-A12^T	-A22^T

For simpler problems

• minimize

  • $c^{T}x$

• subject to

  • $Ax=b$
  • $x\ge 0$ The dual problem of the following

• minimize

  • $-b^{T}y$

• subject to

  • $-A^{T}y\ge -c$ Variable support | | x | | | c | | | b | | | A | | | — | — | — | — | — | — | — | — | — | — | — | — | | LP1 | x1 | x2 | | c1 | c2 | b1 | b2 | A11 | A12 | A21 | A22 | way of thinking

• Since x is positively constrained, this corresponds to x1.

• The coefficient of x1 in the objective function is c1

• The constraints are equations, and if we focus on x1, A21 and b2

• All other values are zero.

• Substituting for a dual problem

• A21 hangs on y2, which we will call y

• Note that y2 is not positively constrained.

• minimize

  • $c_1{x}_1+c_2{x}_2$

• subject to

  • $a_{11}{x}_1+a_{12}{x}_2\ge b_1\cdots (eq1)$
  • $a_{21}{x}_1+a_{22}{x}_2\ge b_2\cdots (eq2)$
  • $x_1\ge 0,x_2\ge 0$

• Consider y_1 \times eq1 + y_2 \times eq2$$(y_1 \ge 0, y_2 \ge 0)

  • $(a_{11}{y}_1+a_{21}{y}_2)x_1+(a_{12}{y}_1+a_{22}{y}_2)x_2\ge b_1{y}_1+b_2{y}_2$

• If the left-hand side is less than the objective function, then the objective function is greater than the right-hand side

  • $c_1{x}_1+c_2{x}_2\ge (a_{11}{y}_1+a_{21}{y}_2)x_1+(a_{12}{y}_1+a_{22}{y}_2)x_2\ge b_1{y}_1+b_2{y}_2$

• So, if we choose y so that the right-hand side is as large as possible, we get the following linear programming problem

• maximize

  • $b_1{y}_1+b_2{y}_2$

• subject to

  • $c_1\ge a_{11}{y}_1+a_{21}{y}_2$
  • $c_2\ge a_{12}{y}_1+a_{22}{y}_2$
  • $y_1\ge 0,y_2\ge 0$

• LP twins dual LP

• linear programming linear programming problem ref:

• study text Linear Programming (2) Dual Problem and Dual Theorem

• Frame Covers integer programming relaxed linear programming Application of Primal Dual Method http://www.akita-pu.ac.jp/system/elect/ins/kusakari/japanese/teaching/InfoMath/2008/note/14.pdf

---

This page is auto-translated from /nishio/双対線形計画問題 using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.