Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints

2011 ◽  
Vol 19 (1) ◽  
pp. 113-131 ◽  
Author(s):  
Chen Ling ◽  
Xinzhen Zhang ◽  
Liqun Qi
Keyword(s):  
Quadratic Optimization ◽  
Semidefinite Relaxation ◽  
Quadratic Constraints

Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization

2008 ◽  
Vol 19 (2) ◽  
pp. 503-523 ◽  
Author(s):  
Simai He ◽  
Zhi-Quan Luo ◽  
Jiawang Nie ◽  
Shuzhong Zhang
Keyword(s):  
Quadratic Optimization ◽  
Semidefinite Relaxation

scholarly journals Extending the Scope of Robust Quadratic Optimization

2021 ◽  
Author(s):  
Ahmadreza Marandi ◽  
Aharon Ben-Tal ◽  
Dick den Hertog ◽  
Bertrand Melenberg
Keyword(s):  
Optimization Problem ◽  
Quadratic Optimization ◽  
Uncertain Parameters ◽  
Quadratic Constraints ◽  
Sample Mean ◽  
Quadratic Optimization Problem ◽  
Statistical Confidence ◽  
Robust Counterpart ◽  
Uncertainty Set ◽  
Portfolio Optimization Problem

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations. As an application, we show how to construct a natural uncertainty set based on a statistical confidence set around a sample mean vector and covariance matrix and use this to provide a tractable reformulation of the robust counterpart of an uncertain portfolio optimization problem. We also apply the results of this paper to norm approximation problems. Summary of Contribution: This paper develops new theoretical results and algorithms that extend the scope of a robust quadratic optimization problem. More specifically, we derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We also consider hard quadratic constraints: those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints, we derive inner and outer tractable approximations.

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