# New bounds for nonconvex quadratically constrained quadratic programming

## TL;DR

This paper introduces new quadratic and cubic bounds for nonconvex quadratically constrained quadratic programming, connecting these bounds to semi-definite relaxations and comparing their effectiveness.

## Contribution

It proposes novel quadratic and cubic bounds using affine and quadratic multipliers, linking them to existing relaxations and providing comparative analysis.

## Key findings

- Most semi-definite relaxations are duals of quadratic bounds.
- Cubic bounds outperform some traditional bounds in certain cases.
- Comparison results show the effectiveness of proposed bounds.

## Abstract

In this paper, we study some bounds for nonconvex quadratically constrained quadratic programs. We propose two types of bounds for quadratically constrained quadratic programs, quadratic and cubic bounds. For quadratic bounds, we use affine functions as Lagrange multipliers. We demonstrate that most semi-definite relaxations can be obtained as the dual of a quadratic bound. In addition, we study bounds obtained by changing the ground set. For cubic bounds, in addition to affine multipliers we employ quadratic functions. We provide a comparison between the proposed cubic bound and typical bounds for standard quadratic programs. Moreover, we report comparison results of a quadratic and a cubic bound for some non-convex quadratically constrained quadratic programs.