I have an equation of the form x/y+z=c, where x,y,z are decision variables (>0) and c is a constant. Is there a possible way of converting this to a linear form?

asked 28 May '12, 03:10

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abhik
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edited 28 May '12, 08:19

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fbahr ♦
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What types are (x) and (y) (continuous, integer, or binary)?

(28 May '12, 05:19) Ehsan ♦

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answered 28 May '12, 08:14

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fbahr ♦
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edited 04 Jul '12, 05:40

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If z occurs nowhere else, replace zy with a new variable u and it is linear. If z does appear elsewhere, substitution will not help, and the equation is quadratic (better than rational, but not linear).

(28 May '12, 15:13) Paul Rubin ♦♦

@Paul: Following AIMMS Modeling Guide, the product of two continuous variables, \(x_1 \cdot x_2\), can be converted into a separable form:

  1. Introduce two new (continuous) variables \(y_1\) and \(y_2\); defined as:
    \(y_1 = \frac{1}{2} \cdot (x_1 + x_2);\) \(y_2 = \frac{1}{2} \cdot (x_1 − x_2)\)

  2. Now, replace \(x_1 \cdot x_2\) by \(y_1^2 − y_2^2\) – which can be approximated by using piecewise linearization (AIMMS Modeling Guide, section 7.6) [...at the cost of having an approximate solution, of course].

(28 May '12, 16:05) fbahr ♦
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@Florian: Okay. I've seen the difference-of-squares trick before, but (as you note) while it can be approximated with piecewise linear functions, it's not truly linear. TANSTAAFL.

(28 May '12, 16:39) Paul Rubin ♦♦
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Asked: 28 May '12, 03:10

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