Handbook of Semidefinite Programming

Volume 27 of the series International Series in Operations Research & Management Science pp 361-419

Semidefinite Programming Relaxations of Nonconvex Quadratic Optimization

  • Yuri Nesterov
  • , Henry Wolkowicz
  • , Yinyu Ye

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Quadratically constrained quadratic programs, denoted Q2P, are an important modelling tool, e.g.: for hard combinatorial optimization problems, Chapter 12; and SQP methods in nonlinear programming, Chapter 20. These problems are too hard to solve in general. Therefore, relaxations such as the Lagrangian relaxation are used. The dual of the Lagrangian relaxation is the SDP relaxation. Thus SDP has enabled us to efficiently solve the Lagrangian relaxation and find good approximate solutions for these hard, possibly nonconvex, Q2P. This area has generated a lot of research recently. This has resulted in many strong and elegant theorems that describe the strength/performance of the bounds obtained from solving relaxations of these Q2P.