# Chapter 4 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 4:...

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MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 4: Duality Theory 1 / 44

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Starting with a linear programming problem, called the primal LP, we introduce another linear programming problem, called the dual problem. Duality theory deals with the relation between these two LP problems. It is also a powerful theoretical tool that has numerous applications, provides insights and leads to another algorithm for linear programming. 2 / 44
Mathematical Motivation Consider the standard form problem which we call the primal problem ( P ): ( P ) min c 0 x s.t. Ax = b x 0 . Suppose we want to ﬁnd a good lower bound on the optimal value. Finding an upper bound is easy since (use any feasible solution). Let x * be an optimal solution, assumed to exist. Consider the relaxed problem in which the constraint Ax = b is replaced by a penalty in the objective p 0 ( b - Ax ), where p is a vector of the same dimension as b : min c 0 x + p 0 ( b - Ax ) s.t. x 0 . 3 / 44

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Let g ( p ) be the optimal cost for the relaxed problem, as a function of p . Then, g ( p ) c 0 x * + p 0 ( b - Ax * ) = c 0 x * . This implies that each p leads to a lower bound g ( p ) for the optimal cost c 0 x * . The problem ( D ) max g ( p ) s.t. no constraints which searches for the best (greatest) lower bound, is known as the dual problem ( D ). 4 / 44
Remark The relaxed problem is easy to solve: g ( p ) = p 0 b + min x 0 ( c 0 - p 0 A ) x = p 0 b + ± 0 , if c 0 - p 0 A 0 , -∞ , otherwise. Thus, the dual problem is a linear programming problem ( D ) max p 0 b s.t. p 0 A c 0 . One of the key results in linear programming duality is that the optimal cost for the dual linear program is equal to the optimal cost for the primal linear program. In other words, when the dual variables (also known as prices or Lagrange multipliers) are chosen according to an optimal solution to the dual problem, the option of violating the constraints Ax = b is of no value. 5 / 44

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Example 4.1 Formulate the dual to the linear program problem below and solve both the primal and dual problems graphically: min x 1 + x 2 s.t. x 1 + 2 x 2 - s 1 = 2 x 1 - s 2 = 1 x 1 , x 2 , s 1 , s 2 0 6 / 44
A local furniture company in Singapore manufactures desks, tables and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: ﬁnishing and carpentry. The amount of resource needed to make each type of furniture along with the availability of the resources and the revenue of the furniture is given in the table below: Resource Desk Table Chair Availability Lumber (board ft) 8 6 1 48 Finishing hours 4 2 1 . 5 20 Carpentry hours 2 1 . 5 0 . 5 8 Revenue (dollars per unit) 60 30 20 The company wants to maximize the total revenue revenue from sales of furniture. (a) Formulate the primal and dual linear program.

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## This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Chapter 4 - MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 4:...

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