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Unformatted text preview: Chapter 3: Standard Forms 1 Chek Beng Chua Introduction LP problems takes many form — three types of constraints and two types of objectives. Instead of developing theories and methods for all kinds of LP, we focus on TWO simple forms of LP problems • Standard Inequality Form (SIF) and • Standard Equality Form (SEF). We shall show that all LP problems is “equivalent” to one in SIF and one in SEF. Chapter 3: Standard Forms 2 Chek Beng Chua Definitions Standard Inequality Form (SIF) maximize c T x subject to Ax ≤ b x ≥ ⇐ = nonnegativity constraints 1. It is a maximization problem. 2. All variables are required to be nonnegative . These constraints are called nonnegativity constraints . 3. All other constraints are linear inequalities of the type “ ≤ ”. Example The factory production problem is in SIF. maximize n j =1 c j x j subject to n j =1 a ij x j ≤ b i ( i = 1 , 2 , . . . , m ) x j ≥ ( j = 1 , 2 , . . . , n ) Chapter 3: Standard Forms 3 Chek Beng Chua Standard Equality Form (SEF) maximize c T x subject to Ax = b x ≥ 1. It is a maximization problem. 2. All variables are required to be nonnegative . 3. All other constraints are linear equations . Example A reformulation of the transportation problem is in SEF. (Compare with LP on page 12.) maximize p i =1 q j =1 c ij x ij subject to q j =1 x ij = s i ( i = 1 , 2 , . . . , p ) p i =1 x ij = t i ( j = 1 , 2 , . . . , q ) x ij ≥ i = 1 , 2 , . . . , p, j = 1 , 2 , . . . , q Chapter 3: Standard Forms 4 Chek Beng Chua Example The problem of an overdetermined system of linear equa tions is not in SIF nor is it in SEF. minimize m i =1 y i subject to n j =1 a ij x j b i y i ≤ ( i = 1 , 2 , . . . , m ) n j =1 a ij x j b i + y i ≥ ( i = 1 , 2 , . . . , m ) Fortunately, this LP problem — and all LP problems for that matter — can be transformed into an “equivalent” LP in standard form (i.e., SEF or SIF). Chapter 3: Standard Forms 5 Chek Beng Chua Transformations We shall show that Every LP problem is “equivalent” to some LP problem in SIF (or SEF).problem in SIF (or SEF)....
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 Fall '97
 Wolkowicz
 Optimization, standard forms, Chek Beng Chua

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