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I'm trying to solve an optimization problem with a linear objective function and mostly linear constraints. However, I do have several constraints of the form

$$\sum_{i=1}^m x_i\phi_i - \left(\sum_{i=1}^m x_i\right) \left(\sum_{j=m+1}^n x_j\right) \leq 0$$

In other words, part of this inequality has the form

$$(x_1 + \cdots + x_m)(x_{m+1} + \cdots + x_{n})$$

Since this does not lead to a positive semidefinite matrix and is therefore not convex, I'm wondering what techniques (if any) are available to efficiently solve this optimization problem.

Rodrigo de Azevedo
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user402078
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1 Answers1

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$$(x_1 + \cdots + x_m)(x_{m+1} + \cdots + x_{n}) = \mathrm x^{\top} \begin{bmatrix} 1_m\,\\ 0_{n-m}\end{bmatrix} \begin{bmatrix} 0_m\,\\ 1_{n-m}\end{bmatrix}^{\top} \mathrm x = \frac 12 \mathrm x^{\top} \begin{bmatrix} \mathrm O_m & 1_m 1_{n-m}^{\top}\\ 1_{n-m} 1_m^{\top} & \mathrm O_{n-m}\end{bmatrix} \mathrm x$$

which is an indefinite quadratic form. If the objective function and the other constraints are linear, then you have a non-convex quadratically constrained linear program (QCLP), which is almost certainly NP-hard.

Rodrigo de Azevedo
  • 23,223