Consider the following linear program,

\( \min y \\ xc_1 \leq c_2 + yz,\\ x = x_1 + \dots + x_n,\\ z \leq x_1 + x_2, \\ z \leq x_2 + x_3, \\ \vdots\\ z \leq x_{n-1} + x_n, \\ x,x_1, \dots, x_n,y,z \geq 0 \)

where \(c_1, c_2\) are constants. This is an example of quadratically constrained linear program where I have 1 quadratic constraint. I wish to find out if this problem is NP-Hard or not. The quadratic constraint can be expressed in the form \( \vec{y}M\vec{y}^T \) where \( M \) for my problem is not positive semidefinite (and thus, non-convex).

Listing the questions:

  1. Can this problem be transformed into a linear program by taking logs?
  2. Is there any literature reference or reduction showing that linear programs with non-convex quadratic constraints is an NP-Hard problem?

asked 17 Sep '17, 12:02

karmanaut's gravatar image

karmanaut
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accept rate: 0%

edited 17 Sep '17, 15:30

Paul%20Rubin's gravatar image

Paul Rubin ♦♦
14.6k412

1

If you had (c_1*x-c_2)^2 <= yz then it would be a conic quadratic optimization problem (aka. SOCP) and it would be polynomial solvable.

(20 Sep '17, 05:27) Erling_MOSEK
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Asked: 17 Sep '17, 12:02

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Last updated: 20 Sep '17, 05:27

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