Stochastic linear-quadratic control via primal-dual semidefinite programming

David D. Yao, Shuzhong Zhang, Xun Yu Zhou

Research output: Contribution to journalArticlepeer-review


We study stochastic linear-quadratic (LQ) optimal control problems over an infinite time horizon, allowing the cost matrices to be indefinite. We develop a systematic approach based on semidefinite programming (SDP). A central issue is the stability of the feedback control. We show that this can be effectively examined through the complementary duality of the SDP. Furthermore, we establish several implication relations among the SDP complementary duality, the (generalized) Riccati equation, and the optimality of the LQ control problem. Based on these relations, we propose a numerical procedure that provides a thorough treatment of the LQ control problem via primal-dual SDP: it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution. For the latter case, we develop an ε-approximation scheme that is asymptotically optimal.

Original languageEnglish (US)
Pages (from-to)87-111
Number of pages25
JournalSIAM Review
Issue number1
StatePublished - Mar 2004
Externally publishedYes


  • Complementary duality
  • Generalized Riccati equation
  • Mean-square stability
  • Semidefinite programming
  • Stochastic linear-quadratic control

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